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001f060c-6b7e-4705-a37d-f740a61334a2 | 4 | 0 | 0 | 2 | Hydrostatics 2 | Fluid Flow and Pressure | 3 | 4 | Water (density $\rho _{H_{2}O}=1000\,$kg/m$^3$) flows up the slanted pipe, which is at an angle of $\theta=30^\circ\,$to the horizontal, as shown below.
  $.
What is the difference between the values in parts 5b and 5a? Why is thi... | 4 | 3 | The pressure difference between the two points would be given by the hydrostatic formula: $p_1-p_2=\rho gL\tan\theta=1,000\times10\times0.05\tan30^\circ=290$Pa.
We cannot simply calculate the hydrostatic pressure difference between the two points. This is because the water is flowing, meaning that the pressure is not ... | The pressure difference would be given by $p_1-p_2=\rho gz$. Find $z$ (length of the opposite part of the triangle) by creating a triangle using $L$ and the angle $\theta$ given in the question. Substitute this and all values given back in to get a final numerical answer.
We cannot simply calculate the hydrostatic pre... | Fluid Flow and Pressure | 2 | 1 | 2 | Water (density $\rho _{H_{2}O}=1000\,$kg/m$^3$) flows up the slanted pipe, which is at an angle of $\theta=30^\circ\,$to the horizontal, as shown below.
  **: Use Gaussian elimination to solve the system of equations below (**Note**: these are the same equations as Q1, so you know the type of solution to expect).
***
You are asked to input the nature of the intersection of the planes. If the planes intersect at a point, input the point of intersection. If not, lo... | 0.666667 | 30 | $$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
x+y&=4\, .
\end{aligned}
$$
$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
4x+3y+z&=1\, .
\end{aligned}
$$
$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
2x+2y+2z&=12\, .
\end{aligned}
$$
$$
\begin{aligned}
-x+2y-2z&=1, \\
4x-y+6z&=2\, ,\\
2x+3y+2z&=4\, ,\\
\en... | 4 | 4 | Refer to **section 2.8** for Gaussian elimination rules. In question 1 (a), we found that this system of equations intersects at a point. Therefore, we will attempt to re-arrange the augmented matrix into the form:
$$
\left( \begin{array}{ccc|r} 1 & 0 & 0 & \ a\\ 0 & 1 & 0 & b\\ 0 & 0 & 1 & c \end{array} \right)
$$
... | Before starting Gaussian Elimination, ensure you have answered *Question 1* to identify the type of solution that you will expect.
***
Set-up the augmented matrix (**section 2.8**)...
***
... Attempt to manipulate the matrix into triangular form, e.g. :
$$
\left( \begin{array}{ccc|r} a & b & c & \ d\\ 0 & e & f ... | Gaussian Elimination | 2 | 6 | 1 | **(L8)**: Use Gaussian elimination to solve the system of equations below (**Note**: these are the same equations as Q1, so you know the type of solution to expect).
***
You are asked to input the nature of the intersection of the planes. If the planes intersect at a point, input the point of intersection. If not, lo... | L8: Use Gaussian elimination to solve the system of equations below Note: these are the same equations as Q1, so you know the type of solution to expect. If the planes intersect at a point, input the point of intersection. If not, look at the 'Final Answer' to check if your equation of intersection is correct. | 3 | 1 | 1 | 74 | 291 | 366 | 13 | 106 | 70 | 4 | 327 | 56 | 0 | 0 | 0 | 3,552 | 34 | 2.666667 | 2,368 | 22.666667 | 99 | 60 | 8 | 620 |
009e7a1e-33de-497a-8e79-b444d292db5a | 4 | 0 | 0 | 20 | Complex Analysis - Complex Numbers | Polar to Rectangular Form | 0 | 5 | Write the following complex numbers in Cartesian coordinates (rectangular form, $x+iy$):
| 0.333333 | 10 | $e^{-3\pi i/4}$
$e^{5\pi i/4}$
$3e^i$
$1/(\sqrt{3}e^{(\frac{\pi i}{3})})$
| 4 | 4 | Plotting the complex number in the Argand diagram,
***
Then, using trigonometry, we can break it into its $x$ and $y$ components:
***
?
***
Plot the complex number in the Argand Diagram...
***
... use trigonometry to find the $x$ (real) and $y$ (imaginary) components...
***
... the x component is $r\cos{\theta}$, ca... | Polar to Rectangular Form | 0 | 6 | 1 | Write the following complex numbers in Cartesian coordinates (rectangular form, $x+iy$):
$e^{-3\pi i/4}$
$e^{5\pi i/4}$
$3e^i$
$1/(\sqrt{3}e^{(\frac{\pi i}{3})})$
| Write the following complex numbers in Cartesian coordinates rectangular form, $x+iy$ : $e^{-3\pi i/4}$ $e^{5\pi i/4}$ $3e^i$ $1/(\sqrt{3}e^{(\frac{\pi i}{3})})$ | 1 | 1 | 2 | 16 | 73 | 575 | 72 | 105 | 217 | 4 | 70 | 16 | 73 | 0 | 0 | 1,488 | 11 | 1.333333 | 496 | 3.666667 | 14 | 95 | 4 | 145 |
012cc4b4-bb47-4ac4-a702-370084cef4cd | 1 | 2 | 0 | 13 | Gyroscopic Motion | Gyroscope in Vehicle | 5 | 2 | An experimental vehicle is fitted with a gyroscope to counteract completely the tendency of the vehicle to tip when rounding a bend.
The gyroscope rotor ... | 0.666667 | 35 | Find the angular velocity $\Omega$ that the gyroscope rotor should spin at if the car is travelling with velocity $v = 15~\mathrm{m/s}$.
In which direction should the gyroscope spin?
Observe the two diagrams below:
### **Diagram A**
***
Centripetal moment applied to the car:
$$
M=\cfrac{{mv}^2h}{R}~~\mathrm{(Equation~1)}
$$
***
Where:
$$
v=\omega_3R~~\mathrm{(Equation~2)}
$$
**... | Draw the car rounding a bend and introduce a suitable system for your vector conventions. The question may seem complicated but the solution is fairly straight forward.
***
Relate the car speed $v$ and the radius of the bend to its angular momentum around the centre of the bend.
***
You’ll then need to think of a... | Gyroscope in Vehicle | 2 | 4 | 2 | An experimental vehicle is fitted with a gyroscope to counteract completely the tendency of the vehicle to tip when rounding a bend.
The gyroscope rotor ... | Find the angular velocity $\Omega$ that the gyroscope rotor should spin at if the car is travelling with velocity $v = 15~\mathrm{m/s}$ . In which direction should the gyroscope spin? Observe the two diagrams below: ### Diagram A ### Diagram B Which diagram shows the correct change in angular momentum caused by the ... | 3 | 3 | 0 | 190 | 88 | 312 | 6 | 85 | 180 | 53 | 29 | 52 | 27 | 3 | 0 | 1,010 | 8 | 2 | 673.333333 | 5.333333 | 27 | 65 | 3 | 1,459 |
01e1b7d0-d39c-4cdf-8200-05e95de00232 | 4 | 0 | 0 | 6 | Additional Analysis - Thermodynamics and Mechanics | Verifying Engine Performance | 2 | 1 | As part of another design project, you are tasked with verifying whether a certain engine is suitable for driving the wheels of a lawn mower. You need to check that the engine will have a suitable cylinder capacity to generate the required 3kW of power at its minimum operating speed.
| Stage | Fi... | 3 | 1 |
The first step to this question is to use the picture of the engine to determine what type of engine it is. This engine is clearly a petrol engine as you can see a spark plug in the picture.
 (Challenge) Reciprocal Rule | 0 | 14 | In ordinary derivatives a function $y(x)$ implicitly defines $x=x(y)$ and their derivatives obey:
$$
\frac{dx}{dy} = \frac{1}{\left(\dfrac{dy}{dx}\right)}
$$
(check e.g. for $y=x^2$). In higher dimensions a similar relation holds, but it is important to keep track of which variable is being kept constant. The correct... | 0.666667 | 20 |  Confirm the above for $z=x^2-y^2$, $r=\sqrt{x^2+y^2}$ by calculating $\left(\frac{\partial y}{\partial x}\right)_{z}$, $\left(\frac{\partial x}{\partial y}\right)_{z}$, $\left(\frac{\partial y}{\partial x}\right)_{r}$, $\left(\frac{\partial x}{\partial y}\right)_{r}$. 
For $z=x^2-y^2$ confirm the cyclic rul... | 2 | 2 | Starting with the two partial derivatives where we hold $z$ constant:
***
$$
y(x,z) = (x^2-z)^{1/2}
$$
***
Using the chain rule and holding $z$ constant:
$$
\left(\dfrac{\partial y}{\partial x}\right)_z = \frac{x}{(x^2-z)^{1/2}} = \frac{x}{y}
$$
***
$$
x(y,z)=(y^2+z)^{1/2}
$$
***
$$
\therefore\left(\dfrac{\par... | Start by finding the partial derivatives holding $z$ constant...
***
Re-arrange $z=z(x,y)$ into the form $y=y(x,z)$. Then, find $\partial y/\partial x$ holding $z$ constant. Repeat for $x=x(y,z)$. Does the reciprocal rule hold?
***
Repeat the above process but instead holding $r$ constant. Again, does the recipro... | (A) (Challenge) Reciprocal Rule | 1 | 6 | 1 | In ordinary derivatives a function $y(x)$ implicitly defines $x=x(y)$ and their derivatives obey:
$$
\frac{dx}{dy} = \frac{1}{\left(\dfrac{dy}{dx}\right)}
$$
(check e.g. for $y=x^2$). In higher dimensions a similar relation holds, but it is important to keep track of which variable is being kept constant. The correct... | Confirm the above for $z=x^2-y^2$ , $r=\sqrt{x^2+y^2}$ by calculating $\left(\frac{\partial y}{\partial x}\right)_{z}$ , $\left(\frac{\partial x}{\partial y}\right)_{z}$ , $x=x(y)$0 , $x=x(y)$1 . For $z=x^2-y^2$ confirm the cyclic rule for partial derivatives $x=x(y)$3 | 2 | 1 | 0 | 94 | 650 | 1,223 | 146 | 111 | 113 | 27 | 386 | 30 | 156 | 0 | 0 | 1,685 | 13 | 1.333333 | 1,123.333333 | 8.666667 | 45 | 150 | 2 | 1,044 |
0315f302-37e4-420c-89aa-1674824e3cf5 | 4 | 0 | 0 | 15 | Maths for Life Scientists 04 - Logarithms | Henderson-Hasselbalch derivation | 3 | 2 | The acid dissociation constant, $K_{a}$ , is a measure of the strength of an acid ($\mathrm{HA}$), and is defined as the equilibrium constant for the reaction:
$$
\text{HA} \rightleftharpoons \text{H}^{+}+\text{A}^{-}
$$
It has a value of:
$$
K_{a}=\frac{[H^+][A^-]}{[HA]}
$$
To create a buffer solution, we can mix ... | 1 | 15 | Rearrange the first equation to give an expression for $[H^+]$ in terms of $K_a$, $[A^-]$ and $[HA]$
Take the base-10 logarithm of this equation, and simplify it, so the right hand side contains one term in $K_{a}$ and a second in $[HA]$ and $[A^-]$ . Enter $\log_{10}(x)$ as 'log10(x)'.
Using the definitions of $p... | 4 | 4 | $$
[H^+]=\frac{K_{a} \cdot [HA]}{[A^{-}]}
$$
$$
\log_{10}[H^+]=\log_{10}(\frac{K_a[HA]}{[A^-]})=\log_{10} K_a+\log_{10}(\frac{[HA]}{[A^{-}]})
$$
From the previous part:
$$
\log_{10} [H^+]=\log_{10}(\frac{K_a[HA]}{[A^-]})=\log_{10} K_a+\log_{10}(\frac{[HA]}{[A^{-}]})
$$
Multiply through by $-1$:
$$
-\log_{10} [H^+... | Henderson-Hasselbalch derivation | 1 | 5 | 0 | The acid dissociation constant, $K_{a}$ , is a measure of the strength of an acid ($\mathrm{HA}$), and is defined as the equilibrium constant for the reaction:
$$
\text{HA} \rightleftharpoons \text{H}^{+}+\text{A}^{-}
$$
It has a value of:
$$
K_{a}=\frac{[H^+][A^-]}{[HA]}
$$
To create a buffer solution, we can mix ... | The acid dissociation constant, $K_{a}$ , is a measure of the strength of an acid $\mathrm{HA}$ , and is defined as the equilibrium constant for the reaction: $ \text{HA} \rightleftharpoons \text{H}^{+}+\text{A}^{-} $ It has a value of: $ K_{a}=\frac{[H^+][A^-]}{[HA]} $ To create a buffer solution, we can mix to... | 6 | 8 | 1 | 202 | 292 | 505 | 0 | 33 | 0 | 111 | 183 | 204 | 311 | 0 | 0 | 604 | 6 | 4 | 604 | 6 | 25 | 79 | 4 | 1,078 | |
03568619-d5de-4cc4-bbca-ed5fca655d1a | 0 | 0 | 2 | 4 | Solving diffusion problems | Question 12 | 1 | 11 | An engine was left running in a large unventilated garage, resulting in a steady-state concentration of carbon monoxide, $C_{0}=24\ \mathrm{mg/m^{3}}$. At $t=0$ the engine is turned off and a large garage door is opened. Under the assumption that buoyancy effects are negligible and that the release can be regarded one-... | 1 | 25 | Assuming that there is only molecular diffusion: $D = 0.202\ \mathrm{cm^{2}/s}$.
Assuming the flow is turbulent: $D = 10^4\ \mathrm{cm^{2}/s}$.
| 2 | 2 | Given that the vertical and lateral extent of the garage door is large, the release may be treated as being one-dimensional. We also assume that the garage has infinite length and has an opening at $x=0$, which means that the initial condition of the concentration is given by
$$
C(x,t=0)=
\begin{cases}
C_{0}&\ \ \math... | Question 12 | 2 | 3 | 3 | An engine was left running in a large unventilated garage, resulting in a steady-state concentration of carbon monoxide, $C_{0}=24\ \mathrm{mg/m^{3}}$. At $t=0$ the engine is turned off and a large garage door is opened. Under the assumption that buoyancy effects are negligible and that the release can be regarded one-... | Under the assumption that buoyancy effects are negligible and that the release can be regarded one-dimensional, plot the concentration profile after one minute under the different conditions that follow. | 1 | 1 | 0 | 80 | 86 | 496 | 0 | 274 | 0 | 16 | 59 | 29 | 0 | 1 | 0 | 2,042 | 2 | 2 | 2,042 | 2 | 26 | 86 | 2 | 650 | |
037b6b33-24e8-4efa-a376-2e2d9e5261a7 | 0 | 0 | 1 | 9 | Op-Amps | Op-Amp Design | 4 | 0 | The most readily available resistor values are the ‘E12 series’: $1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2$ and $10~\Omega$ and factors of $10$ larger or smaller (e.g. $180~ \Omega$, $18 ~\mathrm{k}\Omega$, etc.).
| 0.333333 | 10 | Design a non-inverting amplifier with a gain of $263$, using fixed E12 resistors. A deviation of $\pm 1\%$ of the nominal gain is acceptable. Remember that resistors can be connected in parallel.
| 1 | 1 | A non-inverting op-amp stage appears as follows:
  
***
For a non-inverting op-amp, the gain can be calculated as follows:
  
$A_... | null | Op-Amp Design | 0 | 4 | 2 | The most readily available resistor values are the ‘E12 series’: $1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2$ and $10~\Omega$ and factors of $10$ larger or smaller (e.g. $180~ \Omega$, $18 ~\mathrm{k}\Omega$, etc.).
Design a non-inverting amplifier with a gain of $263$, using fixed E12 resistors. A deviat... | The most readily available resistor values are the ‘E12 series’: $1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2$ and $10~\Omega$ and factors of $10$ larger or smaller e.g $180~ \Omega$ , $18 ~\mathrm{k}\Omega$ , etc.. Design a non-inverting amplifier with a gain of $263$ , using fixed E12 resistors... | 1 | 0 | 2 | 58 | 108 | 48 | 0 | 112 | 1 | 32 | 14 | 40 | 100 | 0 | 0 | 716 | 5 | 0.333333 | 238.666667 | 1.666667 | 62 | 9 | 1 | 357 |
03baff5d-e329-4336-8cd5-a97e93fc94a9 | 2 | 0 | 0 | 11 | Turbomachinery | Relative Velocity and Flow Angle | 5 | 1 | At the inlet to a compressor stage, the absolute flow velocity is measured to be $250 ~\mathrm{m/s}$ at an angle of $17^{\circ}$ to the axis. 
| 0.333333 | 20 | If the blade speed is $350 ~\mathrm{m/s}$, calculate the velocity and flow angle relative to the rotor blades.
| 1 | 1 | Draw a velocity triangle, for which:
***
$C_2 = 250~\mathrm{m/s}$
  
$\alpha = 17^{\circ}$
  
$U = 350~\mathrm{m/s}$
***
where... | null | Relative Velocity and Flow Angle | 1 | 4 | 2 | At the inlet to a compressor stage, the absolute flow velocity is measured to be $250 ~\mathrm{m/s}$ at an angle of $17^{\circ}$ to the axis. 
If the blade speed is $350 ~\mathrm{m/s}$, calculate the velocity and flow angle relative to the rotor blades.
| If the blade speed is $350 ~\mathrm{m/s}$ , calculate the velocity and flow angle relative to the rotor blades. | 1 | 1 | 4 | 42 | 48 | 446 | 0 | 126 | 1 | 18 | 19 | 18 | 18 | 0 | 0 | 1,233 | 11 | 0.333333 | 411 | 3.666667 | 12 | 115 | 2 | 216 |
0592be77-e032-469d-bfca-4d9408f3632b | 1 | 1 | 0 | 9 | Introduction to Control Systems | D.C. Motor | 6 | 32 | The speed of a D.C. motor is governed by a proportional control system with unity feedback. The shaft of the motor is subject to an external torque causing a maximum speed reduction of $50~\mathrm{rad/s}$. What is the value of $K_\mathrm{P}$ needed to sustain the speed within $1\%$ of the desired value $100~\mathrm{rad... | 1 | 25 | The parameters of the D.C. motor are:
  
$K_\mathrm{e} = 5~\mathrm{V/krpm}$
$K_\mathrm{t} = 4~\mathrm{Ncm/A}$
$R_\mathrm{a} = 2~\Omega$
$J = 0.1~\mathrm{Ncm/krpm}$
$K_\mathrm{f} = 0$ (No frictional losses)
| 1 | 1 | From the lecture notes Section 3.5.3 (see inside for full derivation), the gain and time constant of a D.C. motor are as follows:
  
$K = \frac{K_\mathrm{t}}{K_\mathrm{f}R_\mathrm{a}+K_\mathrm{e}K_\mathrm{t}}$
  
$\tau = \frac{JR_\mathrm{a}}{K_\mathrm{f}R_\mathrm{a}+K_\mathrm{e}K_\mathrm{t}}$
*... | null | D.C. Motor | 2 | 4 | 2 | The speed of a D.C. motor is governed by a proportional control system with unity feedback. The shaft of the motor is subject to an external torque causing a maximum speed reduction of $50~\mathrm{rad/s}$. What is the value of $K_\mathrm{P}$ needed to sustain the speed within $1\%$ of the desired value $100~\mathrm{rad... | What is the value of $K_\mathrm{P}$ needed to sustain the speed within $1\%$ of the desired value $100~\mathrm{rad/s}$ ? | 1 | 1 | 8 | 72 | 185 | 2,116 | 0 | 298 | 1 | 16 | 137 | 19 | 39 | 1 | 0 | 3,806 | 25 | 1 | 3,806 | 25 | 34 | 333 | 2 | 640 |
05b6833a-94f9-46a4-8e93-018225fb5cd0 | 0 | 0 | 1 | 9 | Signal Conditioning | Passive High-Pass Filter Proof | 5 | 0 | Using complex impedances develop the gain and phase shift relationships between the input and output voltages of the passive high-pass filter below and draw a Bode diagram of the filter.
| 0.666667 | 20 | 
| 1 | 1 | Use the potential divider equation to describe the relationship between input and output voltages:
***
$V_\mathrm{o} = \frac{Z_\mathrm{R}}{Z_\mathrm{R}+Z_\mathrm{C}}V_\mathrm{i}$
  
where $Z_\mathrm{R}$ and $Z_\mathrm{C}$ are the impedances of the resistor and capacitor respectively.
***
From the notes:... | null | Passive High-Pass Filter Proof | 1 | 4 | 2 | Using complex impedances develop the gain and phase shift relationships between the input and output voltages of the passive high-pass filter below and draw a Bode diagram of the filter.
}$
***
The saturation pressure, $P_\mathrm{sat}$, at $20^{\circ}\mathrm{C}$, can be found from Data and Formula book (Table E19):
  
$P_\mathrm{sat}(20^{\circ}) = 0.02339$ bar
***
Substituting this valu... | Air-Vapour Mixtures Basics | 0 | 4 | 2 | A room contains air at $20^{\circ}\mathrm{C}$ and $0.98$ bar with a relative humidity of $85\%$. Determine:
The partial pressure of the dry-air component.
The specific humidity of the air.
| Determine: The partial pressure of the dry-air component. | 1 | 1 | 0 | 30 | 34 | 351 | 0 | 102 | 0 | 13 | 0 | 8 | 0 | 0 | 0 | 851 | 5 | 0.666667 | 283.666667 | 1.666667 | 8 | 106 | 2 | 160 | |
061a2ba6-c482-4891-be40-1a18f21b89bf | 6 | 0 | 0 | 15 | Maths for Life Scientists 02 - Graphs | Michaelis-Menten equation | 1 | 0 | The Michaelis-Menten equation describes enzyme kinetics.
$$
v=\frac{v_{max}[S]}{K_{M}+[S]}
$$
* $v$ Velocity of enzyme-catalysed reaction ($\mathrm{mmol \cdot s^{-1}}$)
* $v_{max}$ Maximum rate of the reaction ($\mathrm{mmol \cdot s^{-1}}$)
* $K_{M}$ Michaelis constant ($\mathrm{mM}$)
* $[S]$ ... | 1 | 12 | Sketch the graph of this equation: $v$ is the $y$-variable, $[S]$ is the $x$-variable. What is the horizontal limit in terms of its variables?
Next, we will prove an important property of the Michalis-Menten equation. Find the value of $v$ when $[S]$=$K_{M}$.
This graph is not very user-friendly as far as extracting ... | 6 | 6 | As $[S] \to \infty$ adding the finite value $K_M$ to $[S]$ becomes less and less significant compared to the size of $[S]$ alone. Therefore, $v =\frac{v_{max} [S]}{K_M + [S]} \to \frac{v_{max} [S]}{[S]}=v_{max}$. The horizontal asymptote is therefore at $v=v_{max}$.
* $v_{max}$ Maximum rate of the reaction ($\mathrm{mmol \cdot s^{-1}}$)
* $K_{M}$ Michaelis constant ($\mathrm{mM}$)
* $[S]$ ... | $ v=\frac{v_{max}[S]}{K_{M}+[S]} $ $v$ Velocity of enzyme-catalysed reaction $\mathrm{mmol \cdot s^{-1}}$ $v_{max}$ Maximum rate of the reaction $\mathrm{mmol \cdot s^{-1}}$ $K_{M}$ Michaelis constant $\mathrm{mM}$ $[S]$ Concentration of substrate $\mathrm{mM}$ Sketch the graph of this equation: $... | 11 | 10 | 0 | 253 | 266 | 635 | 0 | 106 | 0 | 209 | 130 | 181 | 345 | 0 | 0 | 1,182 | 8 | 6 | 1,182 | 8 | 17 | 61 | 6 | 1,416 | |
06696572-f255-45b9-bafa-cabbe184e559 | 2 | 1 | 0 | 4 | Incipient Motion and Bed Load Sediment Transport | Question 1 | 5 | 0 | A wide freshwater stream has a smooth granular bed with a bed slope of $S = 0.002$, a uniform flow depth of $h =2.0\ \mathrm{m}$ and a median grain size of $d_{50} =2\ \mathrm{mm}$.
You may wish to use [the Shields diagram](https://bb.imperial.ac.uk/webapps/blackboard/execute/content/file?cmd=view\&content_id=_2543151... | 0.333333 | 10 | Compute the bed shear stress, $\tau_0$ $\mathrm{[N/m^2]}$.
What is the critical bed shear stress, $\tau_{cr}\ \mathrm{[N/m^2]}$, for this channel?
Is the channel bed stable in terms of initiation of motion for this flow?
| 3 | 3 | The bed sear stress is computed as: $\tau_0 = \rho g h S = 39.240\ \mathrm{N/m^2}$.
The shear velocity $u_*$ is obtained from
$$
u_* = \sqrt{\dfrac{\tau_0}{\rho}} = \sqrt{\dfrac{39.2}{1000}} = 0.198\ \mathrm{m/s}.
$$
The shear Reynolds number is
$$
\mathrm{Re}_* = \dfrac{u_* d_{50}}{\nu} = \dfrac{0.198\cdot 2 \tim... | Question 1 | 0 | 3 | 3 | A wide freshwater stream has a smooth granular bed with a bed slope of $S = 0.002$, a uniform flow depth of $h =2.0\ \mathrm{m}$ and a median grain size of $d_{50} =2\ \mathrm{mm}$.
You may wish to use [the Shields diagram](https://bb.imperial.ac.uk/webapps/blackboard/execute/content/file?cmd=view\&content_id=_2543151... | Compute the bed shear stress, $\tau_0$ $\mathrm{[N/m^2]}$ . What is the critical bed shear stress, $\tau_{cr}\ \mathrm{[N/m^2]}$ , for this channel? Is the channel bed stable in terms of initiation of motion for this flow? | 3 | 2 | 0 | 70 | 103 | 391 | 0 | 104 | 0 | 34 | 55 | 34 | 54 | 0 | 0 | 827 | 4.5 | 1 | 275.666667 | 1.5 | 22 | 111 | 3 | 490 | |
071be152-c0bf-4be1-8a82-3514e08b18ea | 1 | 0 | 0 | 19 | Vector Algebra | Force Balance | 0 | 4 | null | 0.333333 | 5 | (L2) A weight of mass 10 kg is attached to a wall with a string, and is pulled horizontally with a force $\vec{F}$ so that it is in equilibrium, as in the diagram. Find the magnitude of the force $F$ required for the string to make an angle of $\theta=60^\circ$ to the *normal* to the wall.
 for $T$:
$$
T = mg/\sin{\theta}
$$
***
Inserting this result into eq. (1):
$$
F = \frac{mg}{\sin\theta}\cos\theta = \frac{mg}{\tan\theta} = \frac{10\times9.81}{\tan{60^\circ}}=56.6... | * One approach is to equate vertical and horizontal forces, then solve each equation separately, but actually there is no need to do this; there is a shorter way.
***
* A shorter approach is to draw a triangle, and use trigonometric to solve directly for $F$.
| Force Balance | 0 | 6 | 1 | (L2) A weight of mass 10 kg is attached to a wall with a string, and is pulled horizontally with a force $\vec{F}$ so that it is in equilibrium, as in the diagram. Find the magnitude of the force $F$ required for the string to make an angle of $\theta=60^\circ$ to the *normal* to the wall.
 Incomplete Descriptions | 5 | 5 | **(L8)**: Use Gaussian elimination to find the general solution of the following equations:
$$
\begin{aligned}
3x_1+x_2-2x_3+x_4&=-8\, ,\\
x_1-2x_2-2x_3+6x_4&=-3\, ,\\
2x_1-x_2-3x_3+4x_4&=-7\, .
\end{aligned}
$$
| 1 | 25 | **(L8)**: Use Gaussian elimination to find the general solution of the following equations:
$$
\begin{aligned}
3x_1+x_2-2x_3+x_4&=-8\, ,\\
x_1-2x_2-2x_3+6x_4&=-3\, ,\\
2x_1-x_2-3x_3+4x_4&=-7\, .
\end{aligned}
$$
| 1 | 1 | This system of 3 equations has 4 unknowns, so it is incomplete. However, we may still find an infinity of solutions using row reduction. Setting up the augmented matrix:
***
$$
\left( \begin{array}{ c c c c | c }
3 & 1 & -2 & 1 & -8\\
1 & -2 & -2 & 6 & -3\\
2 & -1 & -3 & 4 & -7
\end{array} \right)
$$
We begin by ma... | This system of 3 equations has 4 unknowns, so it is incomplete. However, we may still find an infinity of solutions using row reduction.
***
Write down the $3\times4$ augmented matrix for the system of equations.
***
Use Gaussian elimination (**section 2.8**) to introduce a triangular matrix of zeros in one of th... | (Challenge) Incomplete Descriptions | 2 | 6 | 1 | **(L8)**: Use Gaussian elimination to find the general solution of the following equations:
$$
\begin{aligned}
3x_1+x_2-2x_3+x_4&=-8\, ,\\
x_1-2x_2-2x_3+6x_4&=-3\, ,\\
2x_1-x_2-3x_3+4x_4&=-7\, .
\end{aligned}
$$
**(L8)**: Use Gaussian elimination to find the general solution of the following equations:
$$
\begin{alig... | L8: Use Gaussian elimination to find the general solution of the following equations: $ \begin{aligned} 3x_1+x_2-2x_3+x_4&=-8\, ,\\ x_1-2x_2-2x_3+6x_4&=-3\, ,\\ 2x_1-x_2-3x_3+4x_4&=-7\, . | 1 | 0 | 1 | 28 | 220 | 434 | 24 | 97 | 105 | 14 | 119 | 21 | 0 | 0 | 0 | 1,770 | 16 | 1 | 1,770 | 16 | 94 | 87 | 1 | 378 |
07558792-2199-462d-8f3e-bf701a870259 | 1 | 0 | 0 | 18 | Lorentz transformations, velocity addition, spacetime diagrams, invariant intervals, four-vectors | Perpendicular velocity addition formula | 1 | 1 | In Lecture 4 we derived the velocity addition formula for a particle moving with speed $u$ along the $x$ axis, in a frame moving at velocity $v$ in the $x$ direction. If the particle velocity is not just directed along the $x$ axis, then the velocity transformation formula needs to be written as $u_x^\prime = \frac{u_x... | 0.333333 | 15 | In Lecture 4 we derived the velocity addition formula for a particle moving with speed $u$ along the $x$ axis, in a frame moving at velocity $v$ in the $x$ direction. If the particle velocity is not just directed along the $x$ axis, then the velocity transformation formula needs to be written as $u_x^\prime = \frac{u_x... | 1 | 1 | We know that $y=u_y t$ and $y^\prime = u_y^\prime t^\prime$. We also know $y^\prime = y$ as the frame only moves in the $x$ direction. 
***
The Lorentz transformation for $t$ only involves the $x$ coordinate for which $x=u_xt$, so this gives
***
   
$$
ct^\prime = \g... | How can you write $u'_y$ in terms of $y'$ and $t'$?
***
   
What do you know about the relative direction of $y'$ and $v$? How des this knowledge let us easily relate $y$ and $y'$?
***
   
As $y'$ is perpendicular to $v$, there will be no length contraction in ... | Perpendicular velocity addition formula | 1 | 6 | 1 | In Lecture 4 we derived the velocity addition formula for a particle moving with speed $u$ along the $x$ axis, in a frame moving at velocity $v$ in the $x$ direction. If the particle velocity is not just directed along the $x$ axis, then the velocity transformation formula needs to be written as $u_x^\prime = \frac{u_x... | If the particle velocity is not just directed along the $x$ axis, then the velocity transformation formula needs to be written as $u_x^\prime = \frac{u_x - v}{1- vu_x/c^2}.$ Use the same method as in the lecture to work out the transformation rule for one of the perpendicular velocity components, e.g $u_y$ . | 1 | 1 | 0 | 159 | 116 | 316 | 72 | 60 | 112 | 80 | 63 | 47 | 46 | 0 | 0 | 535 | 3 | 0.333333 | 178.333333 | 1 | 16 | 37 | 1 | 814 |
09429e36-172e-4f9c-8e4d-206d50606944 | 7 | 0 | 0 | 19 | Eigenvectors | Invariants of Rotations | 8 | 1 | **(L14)**: The matrices below represent rotations in $\mathbb{R}^3$ about the $x$-axis ($\text{R}_1$) and about the $y$-axis ($\text{R}_2$), each by 90$^\circ$ in the counter-clockwise direction:
$$
\text{R}_1=
\left(\begin{array}{ccr}
1&\hskip12pt 0&\hskip3pt 0\\
0&\hskip12pt 0&\hskip3pt -1\\
0&\hskip12pt 1&\hskip3pt... | 0.666667 | 25 | Find the real eigenvalues of $\text{R}_1$ and $\text{R}_2$ (denoted $\lambda_1$, $\lambda_2$ respectively). Determine the normalised eigenvectors corresponding to the eigenvalues.
Find the products ${\mathbf{\text{R}}}_1{\mathbf{\text{R}}}_2$ and ${\mathbf{\text{R}}}_2{\mathbf{\text{R}}}_1$ and show they do not commut... | 3 | 3 | Solving the characteristic equation for $\text{R}_1$ (see section **3.19**). 
***
$$
\det(\text{R}_1-\lambda\mathbb{I}_3)=
\left|\begin{array}{crr}
1-\lambda&0&\hskip3pt 0\\
0&-\lambda&\hskip3pt -1\\
0&1&\hskip3pt -\lambda
\end{array}\right|=\lambda^2(1-\lambda)+(1-\lambda)=(1-\lambda)(\lambda^2+1)=0,
$$
which ... | Set up and solve the characteristic equations $p(\lambda)=\det{(\text{R} - \lambda\mathbb{I}_3)}$ for each rotation matrix (see **section 3.19**). 
***
After finding the eigenvalue, in each case solve $(\text{R}- \lambda \mathbb{I}_3)\mathbf{\underline{x}}=0$ for the eigenvector $\mathbf{\underline{x}}=(x,y,z)$... | Invariants of Rotations | 2 | 6 | 1 | **(L14)**: The matrices below represent rotations in $\mathbb{R}^3$ about the $x$-axis ($\text{R}_1$) and about the $y$-axis ($\text{R}_2$), each by 90$^\circ$ in the counter-clockwise direction:
$$
\text{R}_1=
\left(\begin{array}{ccr}
1&\hskip12pt 0&\hskip3pt 0\\
0&\hskip12pt 0&\hskip3pt -1\\
0&\hskip12pt 1&\hskip3pt... | $ Find the real eigenvalues of $\text{R}_1$ and $\text{R}_2$ denoted $\lambda_1$ , $x$0 respectively. Determine the normalised eigenvectors corresponding to the eigenvalues. Find the products $x$1 and $x$2 and show they do not commute. Determine the rotation axes of $x$1 and $x$2 represent these as norm... | 4 | 3 | 1 | 93 | 590 | 475 | 249 | 83 | 114 | 61 | 259 | 32 | 244 | 0 | 0 | 5,211 | 12 | 2 | 3,474 | 8 | 73 | 62 | 7 | 987 |
0b9fe9b9-efd5-40d5-a797-1efe86da2328 | 1 | 0 | 4 | 14 | The Cauchy equations | Components in Cartesian coordinates | 8 | 1 | Using three-dimensional Cartesian coordinates write all the components of: | 0.666667 | 20 | $$
\vec{\mathcal{T}}^{\left(\hat{n}\right)}=\underline{\underline{{\sigma}}}^T\hat{n}
$$
$$
\vec{\mathcal{T}}^{\left(\hat{\mathrm{e}}_y\right)}=\underline{\underline{\sigma}}^T\hat{\mathrm{e}}_y
$$
$$
\mathrm{div}\left(\vec{\mathcal{T}}^{\left(\hat{\mathrm{e}}_y\right)}\right)
$$
$$
\overrightarrow{\mathrm{div}}\lef... | 5 | 0 | Components in Cartesian coordinates | 1 | 4 | 2 | Using three-dimensional Cartesian coordinates write all the components of:$$
\vec{\mathcal{T}}^{\left(\hat{n}\right)}=\underline{\underline{{\sigma}}}^T\hat{n}
$$
$$
\vec{\mathcal{T}}^{\left(\hat{\mathrm{e}}_y\right)}=\underline{\underline{\sigma}}^T\hat{\mathrm{e}}_y
$$
$$
\mathrm{div}\left(\vec{\mathcal{T}}^{\left... | Using three-dimensional Cartesian coordinates write all the components of: $ \vec{\mathcal{T}}^{\left(\hat{n}\right)}=\underline{\underline{{\sigma}}}^T\hat{n} $ $ \vec{\mathcal{T}}^{\left(\hat{\mathrm{e}}_y\right)}=\underline{\underline{\sigma}}^T\hat{\mathrm{e}}_y $ $ \mathrm{div}\left(\vec{\mathcal{T}}^{\left(\h... | 1 | 0 | 0 | 14 | 496 | 0 | 0 | 0 | 0 | 5 | 506 | 14 | 486 | 0 | 0 | 0 | 0 | 2.5 | 0 | 0 | 15 | 0 | 5 | 562 | ||
0ba3880d-0abb-480d-a028-280c196df8ca | 1 | 0 | 0 | 17 | N3 and momentum | Walking on a plank | 2 | 2 | A woman of mass 60 kg stands at one end of a 10 m plank of mass 20 kg, which itself lies on a (frictionless) frozen lake. She walks to the other end of the plank. By using the principle of conservation of momentum, work out how far she has travelled relative to the lake.
\[Those of you who already know about the centr... | 0.666667 | 10 | Distance $x_\text{w}$ travelled by woman relative to lake:
| 1 | 1 | Call the mass of the woman $m_{\text{w}}$ and her velocity $v_{\text{w}}$. The mass and velocity of the plank are $m_{\text{p}}$ and $v_{\text{p}}$. Since no externally applied horizontal forces are acting on the system consisting of the woman and the plank, its total momentum, which starts off equal to zero, must rema... | Call the mass of the woman $m_{\text{w}}$ and her velocity $v_{\text{w}}$. The mass and velocity of the plank are $m_{\text{p}}$ and $v_{\text{p}}$.
***
What is the initial momentum of the system? 
***
Use conservation of momentum to express this in terms of $m_{\text{w}}$, $v_{\text{w}}$, $m_{\text{p}}$, a... | Walking on a plank | 0 | 6 | 1 | A woman of mass 60 kg stands at one end of a 10 m plank of mass 20 kg, which itself lies on a (frictionless) frozen lake. She walks to the other end of the plank. By using the principle of conservation of momentum, work out how far she has travelled relative to the lake.
\[Those of you who already know about the centr... | By using the principle of conservation of momentum, work out how far she has travelled relative to the lake. | 1 | 1 | 3 | 105 | 12 | 797 | 162 | 194 | 143 | 8 | 12 | 19 | 0 | 0 | 0 | 1,583 | 5 | 0.666667 | 1,055.333333 | 3.333333 | 0 | 71 | 1 | 464 |
0ba7ab1f-f9af-439a-b379-e4b2a1fb959f | 1 | 0 | 0 | 9 | Introduction to Control Systems | Capacitor Selection | 6 | 11 | Find the value of the capacitor $C$ needed in the circuit below to ensure a step response faster than $2~\mathrm{ms}$. The resistor values are $R_1 = 1~\mathrm{k\Omega}$ and $R_2 = 4~\mathrm{k\Omega}$.
| 0.666667 | 10 | 
| 1 | 1 | The circuit shown is an active low-pass filter, for which the transfer function can be written as follows:
  
$|H| = \frac{R_2}{R_1}\frac{1}{\sqrt{1+(\omega R_2C)^2}}$
***
This can be re-written in complex form:
  
$|H| = \frac{R_2}{R_1}\frac{1}{1+j\omega R_2C}$
***
Setting $s = j\omega$:
&... | null | Capacitor Selection | 0 | 4 | 2 | Find the value of the capacitor $C$ needed in the circuit below to ensure a step response faster than $2~\mathrm{ms}$. The resistor values are $R_1 = 1~\mathrm{k\Omega}$ and $R_2 = 4~\mathrm{k\Omega}$.
, 442 (1963)] established the equality of gravitational and inertial mass, $m_G$ and $m_I$, to an accuracy of 3 parts in $10^{11}$.
| 0.666667 | 25 | Suppose that $m^{\text{Cu}}_{G}/m^{\text{Cu}}_{I} = 1$ but $m^{\text{Pb}}_{G}/m^{\text{Pb}}_{I} = 1 + 3 \times 10^{-11}$. If copper and lead balls were dropped together from the top of the Tower of Pisa, which would hit the ground first?
Given that the tower is 60 m high, what is the separation of the two balls when ... | 3 | 3 | The acceleration $a$ of a falling object with inertial mass $m_I$ and gravitational mass $m_G$ is given by
***
$$
F = m_G g = m_I a \qquad \Rightarrow \qquad
a = \frac{m_G g}{m_I}.
$$
***
The value of $m_G/m_I$ is larger for lead, so $a_{\text{Pb}} > a_{\text{Cu}}$ and the lead ball hits the ground first.
Th... | The gravitational force uses the gravitational mass $m_G$, whereas N2 uses inertial mass $m_I$. 
***
The gravitational force is equal to the N2 force. Hence solve for the 'inertial' acceleration $a$ of N2.
***
This should tell you which ball accelerates faster, and therefore which ball hits the ground first.... | Gravitational vs. inertial mass | 2 | 6 | 1 | Using a torsion balance, Roll, Krotkov and Dicke [Ann. Phys. **26**(3), 442 (1963)] established the equality of gravitational and inertial mass, $m_G$ and $m_I$, to an accuracy of 3 parts in $10^{11}$.
Suppose that $m^{\text{Cu}}_{G}/m^{\text{Cu}}_{I} = 1$ but $m^{\text{Pb}}_{G}/m^{\text{Pb}}_{I} = 1 + 3 \times 10^{-1... | Phys. If copper and lead balls were dropped together from the top of the Tower of Pisa, which would hit the ground first? Given that the tower is 60 m high, what is the separation of the two balls when the first hits the ground? Would the separation change if you moved the Tower of Pisa to the Moon, where $g$ is appr... | 5 | 6 | 3 | 119 | 116 | 1,028 | 143 | 311 | 159 | 87 | 105 | 77 | 3 | 0 | 0 | 2,260 | 12 | 2 | 1,506.666667 | 8 | 20 | 251 | 3 | 632 |
0bf0e821-13b2-4de7-9c7c-2c2cf4b5c13b | 6 | 0 | 0 | 8 | Thick Walled Cylinders | Thin Vs. Thick Walled Cylinders | 5 | 0 | A steel pipe with $0.125 \text{ m}$ internal diameter, wall thickness $10 \text{ mm}$, is subjected to an internal pressure of $11.0\text{ MPa}$. Use the notation: $K=\frac{r_\text{o}}{r_\text{i}}$ ratio of outer radius to inner radius, $P=$ internal pressure, and external pressure $=0$.
| 0.333333 | 25 | What are the maximum hoop, radial and shear stresses, if the axial stress is zero **when treated as a thin cylinder**.
What are the maximum hoop, radial and shear stresses, if the axial stress is zero **when treated as a thick cylinder**. 
   
Show that the maximum hoop stress is given by: $\sigma_\th... | 2 | 2 | When using the thin-walled assumption and equations from ME1-SAN:
   
$$
\sigma_\theta=\frac{Pr}{t}=\frac{11\times0.0625}{0.01}=\boxed{68.75\text{ MPa}}
$$
   
***
For thin-walled assumption, because $r>>t$, the radial stress is negligible:
   
$$
\sigma_r=\boxed{0 \text{ MPa}}
$$
... | Thin Vs. Thick Walled Cylinders | 2 | 4 | 2 | A steel pipe with $0.125 \text{ m}$ internal diameter, wall thickness $10 \text{ mm}$, is subjected to an internal pressure of $11.0\text{ MPa}$. Use the notation: $K=\frac{r_\text{o}}{r_\text{i}}$ ratio of outer radius to inner radius, $P=$ internal pressure, and external pressure $=0$.
What are the maximum hoop, radi... | What are the maximum hoop, radial and shear stresses, if the axial stress is zero when treated as a thin cylinder. What are the maximum hoop, radial and shear stresses, if the axial stress is zero when treated as a thick cylinder. Show that the maximum hoop stress is given by: $\sigma_\theta=P\frac{K^2+1}{K^2-1}$ . | 3 | 5 | 0 | 94 | 121 | 1,400 | 0 | 215 | 0 | 55 | 36 | 53 | 36 | 0 | 0 | 2,438 | 12 | 0.666667 | 812.666667 | 4 | 26 | 280 | 6 | 538 | |
0c44ba33-2e71-4e4f-9ea3-d60b94200685 | 1 | 1 | 0 | 17 | Rotational motion, centre-of-mass frame and conservative forces | Forces and circular motion | 4 | 4 | Any object moving in a circle is continuously being accelerated towards the centre with centripetal acceleration is given by the formula
$$
a_{\text{centripetal}} = -\omega^2 r = -\frac{v_{\phi}^2}{r},
$$
where $\omega = v_{\phi}/r$ is the angular velocity and $v_{\phi}$ is the speed of the mass around the circumfere... | 0.666667 | 60 | Any object moving in a circle is continuously being accelerated towards the centre with centripetal acceleration is given by the formula
$$
a_{\text{centripetal}} = -\omega^2 r = -\frac{v_{\phi}^2}{r},
$$
where $\omega = v_{\phi}/r$ is the angular velocity and $v_{\phi}$ is the speed of the mass around the circumfere... | 1 | 1 | As it swings down to the bottom, the mass $m$ loses gravitational potential energy $mgl(1 - \cos\theta)$. By energy conservation, this must be equal to its kinetic energy $K$ at the bottom of the swing:
$$
K = mgl(1 - \cos\theta).
$$
Since $K = \frac{1}{2}mv^2$, the speed $v_{\phi}$ of the mass at the bottom of the s... | How can you use Conservation of Energy to get an expression for the velocity of the mass as a function of $\theta$ after it is released from rest at a starting angle of $\theta_0$? At what value of $\theta$ is the speed of the mass greatest?
***
What force is responsible for lifting the larger mass? Can you write an... | Forces and circular motion | 2 | 6 | 1 | Any object moving in a circle is continuously being accelerated towards the centre with centripetal acceleration is given by the formula
$$
a_{\text{centripetal}} = -\omega^2 r = -\frac{v_{\phi}^2}{r},
$$
where $\omega = v_{\phi}/r$ is the angular velocity and $v_{\phi}$ is the speed of the mass around the circumfere... | Analyse the experiment mathematically, assuming for simplicity that the string has been replaced by a light but stiff rod so that the starting angle can range from $0$ to $\pi$ radians. Using energy conservation and the concept of centripetal acceleration, find a formula for the minimum angle of release required fo... | 4 | 3 | 1 | 290 | 218 | 519 | 57 | 354 | 170 | 145 | 118 | 80 | 20 | 1 | 0 | 1,990 | 3 | 0.666667 | 1,326.666667 | 2 | 26 | 85 | 2 | 1,836 |
0c714910-48e8-4c59-bed0-de230105e923 | 3 | 0 | 0 | 15 | Maths for Life Scientists 01 - Algebra | Heart Function | 0 | 1 | The amount of oxygen and blood pumped by the mammalian heart can be modelled by the following:
$$
V_{O_{2}}=Q(C_{a}-C_{v})
$$
$$
Q=f\cdot S
$$
* $V_{O_{2}}$ Volume of oxygen consumed $(\mathrm{ml\cdot min^{-1}})$
* $Q$ Cardiac output ($\mathrm{mL \cdot min^{-1}}$)
* $C_{a}$ Oxygen content in arterial... | 1 | 6 | Rearrange the second equation, $Q = f S$ so that we could work out stroke volume when cardiac output and heartbeat frequency are known.
Rearrange the equations so we can calculate stroke volume if we know the difference in oxygen content between arterial and venous blood, as well as the volume of oxygen consumed and t... | 3 | 3 | $$
Q=S\cdot f
$$
Divide through by $f$
$$
\frac{Q}{f}=\frac{S f}{f}
$$
Cancel $f$ in fraction on RHS:
$$
\frac{Q}{f}=S
$$
Swap LHS and RHS
$$
S=\frac{Q}{f}
$$
$$
S=\frac{Q}{f}
$$
And
$$
V_{O_{2}}=Q(C_{a}-C_{v})
$$
Rearrange second for $Q$:
$$
Q=\frac{V_{O_{2}}}{(C_{a}-C_{v})}
$$
Substitute in this value fo... | Heart Function | 0 | 5 | 0 | The amount of oxygen and blood pumped by the mammalian heart can be modelled by the following:
$$
V_{O_{2}}=Q(C_{a}-C_{v})
$$
$$
Q=f\cdot S
$$
* $V_{O_{2}}$ Volume of oxygen consumed $(\mathrm{ml\cdot min^{-1}})$
* $Q$ Cardiac output ($\mathrm{mL \cdot min^{-1}}$)
* $C_{a}$ Oxygen content in arterial... | Rearrange the second equation, $ Q=f\cdot S $5 so that we could work out stroke volume when cardiac output and heartbeat frequency are known. Rearrange the equations so we can calculate stroke volume if we know the difference in oxygen content between arterial and venous blood, as well as the volume of oxygen consume... | 4 | 5 | 2 | 197 | 346 | 376 | 0 | 44 | 0 | 100 | 114 | 105 | 55 | 0 | 0 | 517 | 8.5 | 3 | 517 | 8.5 | 51 | 58 | 3 | 1,165 | |
0c82de4f-4a34-49dd-9d48-1605e94b1af5 | 1 | 1 | 0 | 2 | Hydrostatics 1 | Hydrostatic Pressures | 2 | 4 | Manometers are instruments used to measure pressure based on columns of liquid. The figure below shows a u-bend manometer, usually used to measure small pressures.
$ about the $x$-axis. His 'friend' Banxy has told him that he will surely need an infinite amount of paint - and he senses th... | 1 | 15 | In search of an Arts Council grant, the artist Damien Worst wishes to paint the **inside** surface of the musical funnel obtained by rotating the curve $y = {1\over x}$ for $(1\leq x\leq \infty)$ about the $x$-axis. His 'friend' Banxy has told him that he will surely need an infinite amount of paint - and he senses th... | 1 | 1 | You can show that the volume would be finite using the formula for volumes of revolution about the $x$ axis.
***
The volume,
$$
\begin{aligned}
V = \pi \int_1^\infty {1\over x^2}\mathrm{d}x = \pi
\end{aligned}
$$
is finite.
***
Now you can find an expression for the surface area of the funnel, and compare this ... | If you were painting a wall, how would you go about calculating the volume of paint that you would need to buy?
***
What conclusion can you draw from this? Consider when a finite/infinite amount of paint would be required based on the required thickness of the paint.
***
You can show that the volume would be fini... | Challenge problem: Last Question - A paradox! | 1 | 6 | 1 | In search of an Arts Council grant, the artist Damien Worst wishes to paint the **inside** surface of the musical funnel obtained by rotating the curve $y = {1\over x}$ for $(1\leq x\leq \infty)$ about the $x$-axis. His 'friend' Banxy has told him that he will surely need an infinite amount of paint - and he senses th... | In search of an Arts Council grant, the artist Damien Worst wishes to paint the inside surface of the musical funnel obtained by rotating the curve $y = {1\over x}$ for $(1\leq x\leq \infty)$ about the $x$ -axis. Can you help Damien to resolve this apparent paradox? ! | 3 | 1 | 4 | 210 | 72 | 252 | 14 | 179 | 104 | 105 | 41 | 43 | 36 | 0 | 0 | 1,053 | 5 | 1 | 1,053 | 5 | 14 | 33 | 1 | 992 |
0ce1cf22-a410-4a4f-8800-780ed360edeb | 0 | 0 | 2 | 14 | Introduction to turbulence | Time-averaged continuity equation | 16 | 4 | Consider a turbulent and incompressible flow. | 0.666667 | 20 | Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that:
$$
\frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}+\frac{\partial \bar{w}}{\partial z} = 0.
$$
Show that the velocity fluctuations also satisfy the continuity equation, that is:
$$
\frac{\... | 2 | 0 | Time-averaged continuity equation | 1 | 4 | 2 | Consider a turbulent and incompressible flow.Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that:
$$
\frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}+\frac{\partial \bar{w}}{\partial z} = 0.
$$
Show that the velocity fluctuations also satisfy ... | Consider a turbulent and incompressible flow.Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that: $ \frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}+\frac{\partial \bar{w}}{\partial z} = 0. $ Show that the velocity fluctuations also satisfy t... | 3 | 3 | 0 | 34 | 219 | 0 | 0 | 0 | 0 | 29 | 243 | 34 | 215 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 12 | 0 | 2 | 421 | ||
0d2632f2-1c09-48e8-b0ec-13b3bf89315c | 1 | 0 | 0 | 15 | Maths for Life Scientists 07 - Trigonometry | Clinometry | 6 | 4 | An ecologist is using a clinometer to estimate the height (and therefore the biomass) of a tree. A clinometer is a tube attached to a protractor with a weighted string (see the diagram below), which can be used to measure angles between the ground and a distant point. The ecologist stands exactly $10 \, \mathrm{m}$ fro... | 1 | 7 | What is the *overall* height of the tree to the closest metre?
| 1 | 1 | $$
1\,\mathrm{m} + 10\,\mathrm{m} \times \tan(90\degree -42\degree) = 12.1\,\mathrm{m}
$$
| null | Clinometry | 0 | 5 | 0 | An ecologist is using a clinometer to estimate the height (and therefore the biomass) of a tree. A clinometer is a tube attached to a protractor with a weighted string (see the diagram below), which can be used to measure angles between the ground and a distant point. The ecologist stands exactly $10 \, \mathrm{m}$ fro... | An ecologist is using a clinometer to estimate the height and therefore the biomass of a tree. What is the overall height of the tree to the closest metre? | 2 | 1 | 0 | 110 | 41 | 80 | 0 | 1 | 1 | 10 | 0 | 29 | 0 | 1 | 0 | 80 | 1 | 1 | 80 | 1 | 7 | 19 | 1 | 694 |
0d36ee3e-25fd-49a8-9452-894589501cb0 | 0 | 4 | 0 | 19 | Gaussian Elimination | Types of Solution | 5 | 0 | **(L7/8)**: For each of the following sets of simultaneous equations, determine if there is:
1. A unique solution
2. No solutions
3. An infinity of solutions
| 0.333333 | 30 | $$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
x+y&=4\, .
\end{aligned}
$$
$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
4x+3y+z&=1\, .
\end{aligned}
$$
$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
2x+2y+2z&=12\, .
\end{aligned}
$$
$$
\begin{aligned}
-x+2y-2z&=1, \\
4x-y+6z&=2\, ,\\
2x+3y+2z&=4\, ,\\
\en... | 4 | 4 | (**Section 2.8**): We start by evaluating $\Delta$. If $\Delta\ne0$, there are unique solutions. If $\Delta=0$, we check for the other two cases:
* For no solutions, *one of* $\Delta_{1,2~\text{or}~3}$ is non-zero. 
* For an infinity of solutions, *all of* $\Delta_{1,2~\text{and}~3}$ must be zero. 
***
... | Refer to **section 2.7**: what are the Cramer's Rule conditions for unique solutions, no solutions or an infinity of solutions? **Note:** In some cases, you *may* be able to deduce the type of solution by inspecting the simultaneous equations: are the planes independent?
***
Start by evaluating $\Delta$ (to speed th... | Types of Solution | 2 | 6 | 1 | **(L7/8)**: For each of the following sets of simultaneous equations, determine if there is:
1. A unique solution
2. No solutions
3. An infinity of solutions
$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
x+y&=4\, .
\end{aligned}
$$
$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
4x+3y+z&=1\, .
\end{aligned}
... | L7/8: For each of the following sets of simultaneous equations, determine if there is: 1. | 1 | 1 | 6 | 30 | 291 | 2,189 | 284 | 650 | 416 | 4 | 327 | 15 | 0 | 0 | 0 | 4,928 | 19 | 1.333333 | 1,642.666667 | 6.333333 | 99 | 471 | 4 | 423 |
0d4b9dc3-b242-493c-982a-e08fefe374f5 | 0 | 0 | 1 | 11 | Entropy Generation, Entropy Balance and Exergy | Counterflow Heat Exchanger | 6 | 4 | Consider a counterflow heat exchanger in which the fluids on the hot and cold sides have inlet temperatures $T_\mathrm{hi}$ and $T_\mathrm{ci}$ respectively, while the temperature of the environment is $T_0$. The fluids can be taken to be the same gas on each side, assumed to behave as a perfect gas with constant-press... | 1 | 15 | $\frac{\dot{B}_\mathrm{destroyed}}{\dot{m}c_\mathrm{p}T_0} = \frac{\dot{I}}{\dot{m}c_\mathrm{p}T_0} = \mathrm{ln}(1+K_1\Delta T + K_2(\Delta T)^2)$
  
where
  
$K_1 = \frac{T_\mathrm{hi}- T_\mathrm{ci}}{T_\mathrm{hi}T_\mathrm{ci}}$
  
and 
  
$K_2 = -\frac{1}{T_\math... | 1 | 1 | It can be helpful to draw a diagram of the heat exchanger:
***
***
$\dot{B}_\mathrm{destroyed} = \dot{I} = T_0\dot{S}_\mathrm{gen}$
***
where $\dot{S... | null | Counterflow Heat Exchanger | 1 | 4 | 2 | Consider a counterflow heat exchanger in which the fluids on the hot and cold sides have inlet temperatures $T_\mathrm{hi}$ and $T_\mathrm{ci}$ respectively, while the temperature of the environment is $T_0$. The fluids can be taken to be the same gas on each side, assumed to behave as a perfect gas with constant-press... | Show that the exergy destruction $\dot{B}_\mathrm{destroyed}$ or irreversibility $\dot{I}$ can be expressed in dimensionless form by: $\frac{\dot{B}_\mathrm{destroyed}}{\dot{m}c_\mathrm{p}T_0} = \frac{\dot{I}}{\dot{m}c_\mathrm{p}T_0} = \mathrm{ln}(1+K_1\Delta T + K_2(\Delta T)^2)$ where $K_1 = \frac{T_\mathrm{hi... | 1 | 2 | 0 | 125 | 354 | 1,274 | 0 | 81 | 1 | 9 | 263 | 22 | 259 | 0 | 0 | 1,789 | 14 | 1 | 1,789 | 14 | 22 | 99 | 1 | 949 |
0d81759e-b284-40f7-854b-fce603b34a62 | 1 | 0 | 0 | 13 | Momentum and Impulse 3 | Rolling Wheel Step | 4 | 0 | A wheel is rolling along a horizontal path towards a step which lies across its path.
The wheel, of radius $r = 0.5~\mathrm{m}$ and radius of gyration $... | 0.666667 | 30 | Find the angular velocity $\dot{\theta}_2$ of the wheel at the instant when it leaves the path and starts to mount the step. Assume no slip and no rebound from the step.
*(Take rotation in the clockwise direction to be positive)*
| 1 | 1 | Free body diagram and kinematic diagram:
***
Starting with the equation for angular momentum about a general point:
$$
\begin{aligned}
H_O=I_G\do... | Let’s take angular momentum about a fixed point to make our lives easier and apply the conservation of momentum principle. What would be a suitable point? 
***
The top tip of the step serves this purpose very well in this case for example. 
Remember that the total angular momentum about a point is the sum... | Rolling Wheel Step | 2 | 4 | 2 | A wheel is rolling along a horizontal path towards a step which lies across its path.
The wheel, of radius $r = 0.5~\mathrm{m}$ and radius of gyration $... | Find the angular velocity $\dot{\theta}_2$ of the wheel at the instant when it leaves the path and starts to mount the step. | 1 | 2 | 2 | 88 | 99 | 1,063 | 9 | 99 | 262 | 40 | 16 | 22 | 16 | 1 | 0 | 1,651 | 9 | 0.666667 | 1,100.666667 | 6 | 25 | 123 | 1 | 632 |
0da53862-6330-4f62-a563-840b35c837dd | 2 | 0 | 0 | 14 | Dimensional Analysis | Aircraft propeller | 4 | 3 | Tests on a model propeller in a wind tunnel at sea level (air density $\rho = 1.2\,\mathrm{kg/m}^3$) gave the following results for the thrust at a number of forward velocities.
  
$$
\begin{array} {c|ccccc} \mathrm{U (m/s)}&\mathrm{0}&\mathrm{10}&\mathrm{15}&\mathrm{20}&\mathrm{30}\\ \hline \mathrm{Thrust ... | 0.666667 | 20 | Using dimensional analysis find the non-dimensional parameters which govern this observed behavior.
Using experimental data given in the table, find the thrust generated by a geometrically similar propeller of diameter $3\,\mathrm{m}$, spinning at $1500\,\mathrm{rpm}$ at a forward velocity of $45\,\mathrm{m/s}$, while ... | 2 | 0 | Aircraft propeller | 1 | 4 | 2 | Tests on a model propeller in a wind tunnel at sea level (air density $\rho = 1.2\,\mathrm{kg/m}^3$) gave the following results for the thrust at a number of forward velocities.
  
$$
\begin{array} {c|ccccc} \mathrm{U (m/s)}&\mathrm{0}&\mathrm{10}&\mathrm{15}&\mathrm{20}&\mathrm{30}\\ \hline \mathrm{Thrust ... | Using dimensional analysis find the non-dimensional parameters which govern this observed behavior. Using experimental data given in the table, find the thrust generated by a geometrically similar propeller of diameter $3\,\mathrm{m}$ , spinning at $1500\,\mathrm{rpm}$ at a forward velocity of $45\,\mathrm{m/s}$ , ... | 2 | 3 | 0 | 106 | 317 | 0 | 0 | 0 | 0 | 62 | 53 | 56 | 53 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 53 | 0 | 2 | 820 | ||
0df0dd8e-3185-427a-9f46-f4e28cddca81 | 1 | 0 | 0 | 10 | Creep | Larson-Miller Parameter | 8 | 2 | The creep rupture properties of Nimonic 105 are shown in the figure below. Using this figure, estimate the maximum operating temperature of a gas turbine blade made out of this material which is to withstand a stress of $150\text{ MPa}$ for a duration of $10,000$ hours. 
   
$$
   
Where $T$ is the temperature, $t_r$ is the time to rupture, and $C=T(20+\log t)$.
   
***
Using values read from the graph at $150\text{ MPa}$:
   
 Collision | 9 | 4 | A particle starts at position $\vec{r}_{0}=(0, 1, 0)$ with a velocity $\vec{v} = (2, 1, 3)$. It collides with a wall whose position is given by the equation $x+y+z = 4$. (You may assume throughout this question that all physical quantities are given in dimensionless units, and there are no external forces acting on th... | 0.666667 | 20 | Give an expression for the trajectory of the particle before the collision. 
Find the position at which the particle strikes the wall. 
Find the velocity of the particle after the collision with the wall, assuming the collision is elastic.
| 3 | 3 | $$
\vec{r}(t) = \vec{r}_0 + \vec{v}t=\begin{pmatrix}0\\1\\0\end{pmatrix}+\begin{pmatrix}2\\1\\3\end{pmatrix}t=\begin{pmatrix}2t\\1+1t\\3t\end{pmatrix}
$$
From part (a): $x=2 t, \quad y=1+t, \quad z=3 t$. Inserting this into the plane equation:
***
$$
\begin{array}{l l }
\text{in} \qquad x+y+z=4, \quad
& \Rightarrow... | $\vec{r}(t) = \vec{r}_0 + \vec{v}t$
Your goal is to find the intersection of the linear trajectory of the particle and the plane.
***
Express your trajectory equation from part (a) as $x=x(t),y=y(t),z=z(t)$. 
***
Insert these results into the equation of the plane and solve for the time of intersection.
*... | 2020 Q1(i) Collision | 1 | 6 | 1 | A particle starts at position $\vec{r}_{0}=(0, 1, 0)$ with a velocity $\vec{v} = (2, 1, 3)$. It collides with a wall whose position is given by the equation $x+y+z = 4$. (You may assume throughout this question that all physical quantities are given in dimensionless units, and there are no external forces acting on th... | Give an expression for the trajectory of the particle before the collision. Find the position at which the particle strikes the wall. Find the velocity of the particle after the collision with the wall, assuming the collision is elastic. | 3 | 4 | 0 | 89 | 47 | 1,106 | 80 | 140 | 154 | 39 | 0 | 39 | 0 | 0 | 0 | 2,422 | 12 | 2 | 1,614.666667 | 8 | 20 | 139 | 3 | 487 |
0f54ab92-904b-4523-b04d-54193f227921 | 3 | 1 | 0 | 2 | The Bernoulli Equation | Bernoulli's with a Pump | 6 | 9 | (Based on P3.139, White) The horizontal pump shown below discharges water at $20^\circ$C at a volume flux of $Q = 57 m^3 /h$. The diameters of the inlet and outlet pipes are $D_1 =$ 9 cm and $D_2 =$ 3 cm, respectively, whilst the measured pressures are $p_1 =$ 120 kPa and $p_2 =$ 400 kPa. In this question you may negl... | 0.666667 | 30 | What are the heads $p_{T1}$ and $p_{T2}$ before and after the pump, respectively?
Why can’t the Bernoulli equation be used to equate the two heads?
Neglecting losses, what power in Kilowatts is delivered to the water by the pump?
| 3 | 2 | The heads are given by $p_{T1}=p_1+\rho u_1^2/2$, and $p_{T2}=p_2+\rho u_2^2/2$, where $u_1$ and $u_2$ are the respective velocities (assumed uniform). Mass conservation implies that the flux both sides of the pump is equal to $Q$, and therefore $Q=u_1A_1=u_2A_2$, where $A_1=\frac{\pi}4D_1^2$ and $A_2=\frac{\pi}4D_2^2$... | Use the equation for total head as shown in Question 7.1 to find expressions for the total head upstream and downstream. Keep in mind that we can neglect any contribution due to height. We can also refer to the upstream velocity as $u_1$ and the downstream as $u_2$.
***
The heads are given by $p_{T1}=p_1+\rho u_1^2/... | Bernoulli's with a Pump | 2 | 1 | 2 | (Based on P3.139, White) The horizontal pump shown below discharges water at $20^\circ$C at a volume flux of $Q = 57 m^3 /h$. The diameters of the inlet and outlet pipes are $D_1 =$ 9 cm and $D_2 =$ 3 cm, respectively, whilst the measured pressures are $p_1 =$ 120 kPa and $p_2 =$ 400 kPa. In this question you may negl... | What are the heads $p_{T1}$ and $p_{T2}$ before and after the pump, respectively? Why can’t the Bernoulli equation be used to equate the two heads? Neglecting losses, what power in Kilowatts is delivered to the water by the pump? | 3 | 3 | 1 | 101 | 61 | 491 | 203 | 101 | 178 | 39 | 16 | 39 | 16 | 1 | 0 | 836 | 7.5 | 2 | 557.333333 | 5 | 22 | 217 | 4 | 658 |
10703359-bbd8-4c73-b85e-57c27e140d16 | 3 | 0 | 0 | 15 | Maths for Life Scientists 04 - Logarithms | Radiolabelled mRNA | 3 | 1 | If a population of cells is exposed to radioactive UTP labelled with $^{32}\mathrm{P}$ (a radioactive isotope of phosphorus), the mRNA being formed at that moment will be labelled. As the mRNA is degraded, the amount of labelled mRNA decreases exponentially with time (we will assume no radioactive uracil is recycled):
... | 1 | 7 | Rearrange the left-hand equation for $\lambda$
Calculate the half-lives of histone H3 and fructose-*bis*-phosphatase mRNAs from the data below. How much longer lived is FBP mRNA than H3 mRNA?
| mRNA | Mass of labelled mRNA at $t=0 \,\mathrm{min}$ ($\mathrm{\mu g \cdot L^{-1}}$) | Mass of labelled mRNA at $t=180... | 3 | 3 | $\lambda=\frac{-\ln{\frac{L}{L_0}}}{t}=\frac{\ln{L_0}-\ln{L}}{t}$
There are lots of other ways of writing this!
Half-lives are $10$ and $0.5$
$10/0.5= 20$
| mRNA | Calculation | $H$ |
| :--------- | :------------------------------------------------------... | Radiolabelled mRNA | 0 | 5 | 0 | If a population of cells is exposed to radioactive UTP labelled with $^{32}\mathrm{P}$ (a radioactive isotope of phosphorus), the mRNA being formed at that moment will be labelled. As the mRNA is degraded, the amount of labelled mRNA decreases exponentially with time (we will assume no radioactive uracil is recycled):
... | As the mRNA is degraded, the amount of labelled mRNA decreases exponentially with time we will assume no radioactive uracil is recycled: $ L=L_{0}\cdot e^{-\lambda \cdot t} $ $ \lambda = \frac{\ln{2}}{H} $ $L$ Mass of labelled mRNA embedded $L_{0}$ Mass of labelled mRNA at beginning of experiment embedded $... | 3 | 3 | 0 | 181 | 183 | 303 | 0 | 18 | 0 | 87 | 87 | 131 | 189 | 3 | 1 | 621 | 2.5 | 3 | 621 | 2.5 | 27 | 60 | 3 | 1,356 | |
10cb0c12-9685-4f7f-9bc4-fe7f3022a872 | 3 | 0 | 0 | 8 | Complex Strains and Strain Gauges | Multiple Strain Gauges On Shaft | 3 | 2 | Two strain gauges are fitted at $\pm45^{\circ}$ to the axis of a $75\text{ mm}$ diameter steel shaft. The shaft is rotating, and in addition to transmitting power, it is subjected to an unknown bending moment and a direct thrust. The readings of the gauges are recorded, and it is found that the maxima or minima values ... | 1 | 40 | Calculate the transmitted torque.
Calculate the applied bending moment.
Calculate the end thrust.
| 3 | 3 | First draw a diagram of the problem:
  
The strain gauges readings pre rotation:
   
$$
e_{45}=-0.00060 \hspace{30pt} e_{-45}=+0.00030
... | Multiple Strain Gauges On Shaft | 2 | 4 | 2 | Two strain gauges are fitted at $\pm45^{\circ}$ to the axis of a $75\text{ mm}$ diameter steel shaft. The shaft is rotating, and in addition to transmitting power, it is subjected to an unknown bending moment and a direct thrust. The readings of the gauges are recorded, and it is found that the maxima or minima values ... | Calculate the transmitted torque. Calculate the applied bending moment. Calculate the end thrust. | 3 | 3 | 7 | 122 | 94 | 3,340 | 0 | 491 | 0 | 13 | 0 | 13 | 0 | 0 | 0 | 5,600 | 21 | 3 | 5,600 | 21 | 43 | 595 | 3 | 621 | |
117dcbfd-a250-41f3-8eeb-4be2993a67c3 | 0 | 1 | 0 | 15 | Maths for Life Scientists 02 - Graphs | Binding of calcium by calmodulin protein | 1 | 1 | The binding of calcium ions to the signalling protein calmodulin can be modelled by the equation:
$$
\theta=\frac{C^{n}}{K_{d}+C^n}
$$
Where:
* $\theta$ is the fraction of calmodulin molecules bound to calcium 
* $C$ is the concentration of calcium
* $n$ is a constant, greater than zero
* $K_{d}$ is... | 1 | 3 | Consider a plot of $\theta$ ($Y$-axis) against $C$ ($X$-axis).
Which one of the following is true?
| 1 | 1 | As $C$ becomes very large compared to $K_d$, $θ$ tends to $C^{n} / ( C^{n} ) = 1 $
| null | Binding of calcium by calmodulin protein | 0 | 5 | 0 | The binding of calcium ions to the signalling protein calmodulin can be modelled by the equation:
$$
\theta=\frac{C^{n}}{K_{d}+C^n}
$$
Where:
* $\theta$ is the fraction of calmodulin molecules bound to calcium 
* $C$ is the concentration of calcium
* $n$ is a constant, greater than zero
* $K_{d}$ is... | Which one of the following is true? | 1 | 1 | 0 | 81 | 72 | 28 | 0 | 13 | 1 | 20 | 17 | 7 | 0 | 0 | 0 | 74 | 1 | 1 | 74 | 1 | 3 | 4 | 1 | 430 |
11b0aaf1-ec0d-4b38-b8fb-084e6ce20788 | 5 | 0 | 0 | 19 | Dot Product | (Challenge) Flux | 1 | 7 | **(L3)**: A solar panel of unit area ($A = 1\text{m}^2$) is placed on the equator on the equinox (i.e. a day of 12 hours of sunlight with the sun passing directly overhead).
| 1 | 40 | Given that the normal to the solar panel is along the $x$-axis and the sun moves in the $xy$ plane, give an expression for the unit vector $\hat{\mathbf{k}}$ along the direction of the sunlight when it makes an angle $\theta$ to the normal.
Give an expression for the angle $\theta$ as a function of the hour $t$ assumi... | 4 | 4 | *North Pole is up*. $\mathbf{\hat{n}}$ is the normal to the solar panel, which is $(1,0,0)$.
***
From the diagram, we see $\hat{\mathbf{k}} = (-\cos\theta... | Draw the $xy$ plane including:
* The normal to the solar panel.
* The direction of the sunlight vector, at angle $\theta$ to the -axis.
***
Can you now deduce the unit vector $\hat{\mathbf{k}}$? ...
***
... The sunlight vector points *towards* the origin.
Think how the power incident on the solar panel woul... | (Challenge) Flux | 2 | 6 | 1 | **(L3)**: A solar panel of unit area ($A = 1\text{m}^2$) is placed on the equator on the equinox (i.e. a day of 12 hours of sunlight with the sun passing directly overhead).
Given that the normal to the solar panel is along the $x$-axis and the sun moves in the $xy$ plane, give an expression for the unit vector $\hat{\... | Given that the normal to the solar panel is along the $x$ -axis and the sun moves in the $xy$ plane, give an expression for the unit vector $\hat{\mathbf{k}}$ along the direction of the sunlight when it makes an angle $\theta$ to the normal. Give an expression for the angle $\theta$ as a function of the hour ... | 7 | 9 | 0 | 236 | 109 | 205 | 128 | 114 | 318 | 204 | 100 | 203 | 80 | 0 | 0 | 2,622 | 13 | 4 | 2,622 | 13 | 17 | 37 | 5 | 1,073 |
11dff774-e524-47dd-bfa9-b66c2bd3dcc9 | 0 | 4 | 0 | 11 | Basic Concepts | Entropy Change | 0 | 6 | Entropy change, in general, is defined by the two relationships
  
$\Delta S = \int(\frac{\delta Q}{T})_{\mathrm{rev}}$
  
and
  
$\Delta S_{\mathrm{irrev}} > \Delta S_{\mathrm{rev}}$
| 0.333333 | 5 | Does the entropy of a system remain constant during reversible and irreversible adiabatic processes?
What happens to the entropy when heat is transferred from the system to the environment?
| 2 | 2 | Entropy change can be calculated as
  
$\Delta S = \int(\frac{\delta Q}{T})_{\mathrm{rev}}$
***
For a reversible, adiabatic process, $\delta Q_{\mathrm{rev}}= 0$.
***
Therefore:
  
$\Delta S_{\mathrm{rev}} = 0$
***
For an irreversible, adiabatic process, $\delta Q_{\mathrm{irrev}}= 0$, but... | Entropy Change | 0 | 4 | 2 | Entropy change, in general, is defined by the two relationships
  
$\Delta S = \int(\frac{\delta Q}{T})_{\mathrm{rev}}$
  
and
  
$\Delta S_{\mathrm{irrev}} > \Delta S_{\mathrm{rev}}$
Does the entropy of a system remain constant during reversible and irreversible adiabatic processes... | Entropy change, in general, is defined by the two relationships $\Delta S = \int(\frac{\delta Q}{T})_{\mathrm{rev}}$ and $\Delta S_{\mathrm{irrev}} > \Delta S_{\mathrm{rev}}$ Does the entropy of a system remain constant during reversible and irreversible adiabatic processes? What happens to the entropy when heat is... | 2 | 2 | 1 | 45 | 97 | 353 | 0 | 99 | 0 | 29 | 0 | 42 | 97 | 0 | 0 | 829 | 8 | 0.666667 | 276.333333 | 2.666667 | 4 | 19 | 4 | 351 | |
1222244a-4ad8-4452-9ade-2959a563ef7e | 1 | 0 | 0 | 11 | Combustion | Water Gas | 7 | 11 | Gaseous fuel must be provided for combustion in gas turbines. In a gasifier, solid carbon and steam (as oxidant) form gaseous carbon monoxide and hydrogen. This mixture is known as "water gas" and is used as fuel for turbines. The steam supplied to the gasifier at $1$ atm is dry-saturated and the carbon is at $25^{\cir... | 1 | 20 | Determine the magnitude and the direction of the heat transfer per kilogram of carbon burnt.
| 1 | 1 | Write the balanced equation for the reaction:
***
$\mathrm{C}+\mathrm{H_2O} \rightarrow \mathrm{CO}+\mathrm{H_2}$
***
The equation is already balanced.
***
Apply the SFEE, bearing in mind that there is no work transfer and that kinetic and potential energy changes can be neglected:
  
$\frac{\dot{Q}}{... | null | Water Gas | 1 | 4 | 2 | Gaseous fuel must be provided for combustion in gas turbines. In a gasifier, solid carbon and steam (as oxidant) form gaseous carbon monoxide and hydrogen. This mixture is known as "water gas" and is used as fuel for turbines. The steam supplied to the gasifier at $1$ atm is dry-saturated and the carbon is at $25^{\cir... | Determine the magnitude and the direction of the heat transfer per kilogram of carbon burnt. | 1 | 1 | 2 | 117 | 48 | 687 | 0 | 116 | 1 | 15 | 0 | 15 | 0 | 0 | 0 | 1,236 | 10 | 1 | 1,236 | 10 | 6 | 113 | 1 | 619 |
12a5ea53-b0fa-4108-895a-62af8577bc67 | 0 | 0 | 1 | 19 | Cross Product | Angular Momentum | 2 | 7 | **(L5)**: A particle of mass $m$ with position vector $\vec{r}$ relative to origin $O$ experiences a force $\vec{F}$, producing a torque $\vec{\tau}=\vec{r}\times\vec{F}$. The angular momentum of the particle around $O$ is given by $\vec{L}=\vec{r}\times m\vec{v}$, where $\vec{v}$ is the velocity of the particle. ... | 0.666667 | 5 | **(L5)**: A particle of mass $m$ with position vector $\vec{r}$ relative to origin $O$ experiences a force $\vec{F}$, producing a torque $\vec{\tau}=\vec{r}\times\vec{F}$. The angular momentum of the particle around $O$ is given by $\vec{L}=\vec{r}\times m\vec{v}$, where $\vec{v}$ is the velocity of the particle. ... | 1 | 1 | We have $\vec{L} = \vec{r}\times(m\vec{v})$. To differentiate this with respect to time, we apply the rule for differentiating vector cross products (**section 1.18**): 
***
$$
\displaystyle \rightarrow \qquad \frac{d\vec{L}}{d t} = \frac{d}{dt} \left( \vec{r}\times(m\vec{v}) \right) = m\left( \frac{d \vec{r}}{d... | To begin, it may help you to look at **section 1.19** and **1.20** to see how cross products are applied to physics scenarios. 
***
To find $d\vec{L}/dt$, refer to **section 1.18**... 
***
... The method is very similar to the product rule for differentiation. 
***
Recall that $\vec{v} = d\vec{r... | Angular Momentum | 0 | 6 | 1 | **(L5)**: A particle of mass $m$ with position vector $\vec{r}$ relative to origin $O$ experiences a force $\vec{F}$, producing a torque $\vec{\tau}=\vec{r}\times\vec{F}$. The angular momentum of the particle around $O$ is given by $\vec{L}=\vec{r}\times m\vec{v}$, where $\vec{v}$ is the velocity of the particle. ... | Show that the rate of change of angular momentum $d\vec{L}/dt$ is equal to the applied torque. | 1 | 1 | 0 | 122 | 226 | 500 | 55 | 73 | 71 | 61 | 114 | 16 | 13 | 0 | 0 | 810 | 4 | 0.666667 | 540 | 2.666667 | 14 | 34 | 1 | 702 |
12ff524b-68b8-40aa-9d3f-2a68d3fc82aa | 5 | 0 | 1 | 8 | Complex Strains and Strain Gauges | Mohr's Strain Circle 2 | 3 | 5 | A mild steel beam of rectangular cross-section, $12.5 \text{ mm}$ wide, $100\text{ mm}$ deep, carries loads which cause a bending moment $M$ and a shear force $F$ at a section $\text{YY}$ along its length. An electric resistance strain gauge rosette, placed at the section as shown below indicates the following strains:... | 0.666667 | 40 | Use Mohr's strain circle to determine the magnitude and direction of the principal strains at the site of the rosette.
Use an analytical method to confirm the answers found in (a).
Calculate the bending moment acting at this cross-section.
Calculate the shear force acting across this section, assuming that it is un... | 4 | 4 | When drawing a Mohr's strain circle, first an $x$-axis is drawn and the three different strain gauges are drawn out:
   
  
***
Th... | Mohr's Strain Circle 2 | 2 | 4 | 2 | A mild steel beam of rectangular cross-section, $12.5 \text{ mm}$ wide, $100\text{ mm}$ deep, carries loads which cause a bending moment $M$ and a shear force $F$ at a section $\text{YY}$ along its length. An electric resistance strain gauge rosette, placed at the section as shown below indicates the following strains:... | An electric resistance strain gauge rosette, placed at the section as shown below indicates the following strains: $ \text{along xx}=3.5\times10^{-4} \\ \text{along aa}=-0.75\times10^{-4}\\ \text{along bb}=4.25\times10^{-4} $ Use Mohr's strain circle to determine the magnitude and direction of the principal strains a... | 4 | 2 | 2 | 109 | 150 | 1,767 | 0 | 427 | 0 | 56 | 0 | 74 | 102 | 1 | 0 | 4,821 | 20 | 2.666667 | 3,214 | 13.333333 | 40 | 398 | 6 | 1,088 | |
1306d7c8-8825-4ea0-9ba6-1c1ae1c1034b | 1 | 0 | 0 | 2 | The Continuity Equation and Navier Stokes Equations | Continuity Equation | 7 | 0 | (Based on P4.9, White) An incompressible flow has a velocity field of the form 
${\bf u}=4xy^2 {\bf i}+f(y){\bf j}-zy^2{\bf k}$
What is $f(y)$?
| 0.333333 | 5 | (Based on P4.9, White) An incompressible flow has a velocity field of the form 
${\bf u}=4xy^2 {\bf i}+f(y){\bf j}-zy^2{\bf k}$
What is $f(y)$?
| 1 | 1 | The continuity equation for incompressible flow is
$\nabla\cdot{\bf u}=0 \quad\Rightarrow\quad 4y^2+\frac{df}{dy}-y^2=0 \quad\Rightarrow\quad f=c-y^3$ where $c$ is constant.
| The continuity equation for incompressible flow is
$ \nabla\cdot{\bf u}=0 $
Expand this by getting the divergence of $\bf{u}$.
***
Rearrange the equation you have to isolate the $f$ term on one side. Integrate to get an expression for $f$.
| Continuity Equation | 0 | 1 | 2 | (Based on P4.9, White) An incompressible flow has a velocity field of the form 
${\bf u}=4xy^2 {\bf i}+f(y){\bf j}-zy^2{\bf k}$
What is $f(y)$?
(Based on P4.9, White) An incompressible flow has a velocity field of the form 
${\bf u}=4xy^2 {\bf i}+f(y){\bf j}-zy^2{\bf k}$
What is $f(y)$?
| Based on P4.9, White An incompressible flow has a velocity field of the form ${\bf u}=4xy^2 {\bf i}+f(y){\bf j}-zy^2{\bf k}$ What is $f(y)$ ? | 1 | 1 | 0 | 38 | 96 | 97 | 35 | 12 | 39 | 19 | 53 | 19 | 48 | 0 | 0 | 157 | 0.5 | 0.333333 | 52.333333 | 0.166667 | 14 | 16 | 1 | 252 |
13ef2460-a035-442a-82fa-5666c31e06fa | 4 | 0 | 0 | 11 | Combustion | Combustion With Air | 7 | 9 | Consider combustion of oil and wood in air.
| 1 | 20 | Calculate the air requirements and the atmospheric nitrogen in the products if air is used for stoichiometric combustion of $100~\mathrm{kg}$ of oil.
 Calculate the stoichiometric air requirements and excess air in the products if $100~\mathrm{kg}$ of wood is burnt with $30\%$ excess air.
| 2 | 2 | Since the nitrogen in the air mixture does not react, the mass of the $\mathrm{O_2}$ in the reactant air mixture and the mass of the $\mathrm{CO_2}$ and $\mathrm{H_2O}$ in the products will be the same as in Q9a:
  
Mass $\mathrm{O_2} = 337.1~\mathrm{kg}$
  
Mass $\mathrm{CO_2} = 318.3~\mathrm{k... | Combustion With Air | 1 | 4 | 2 | Consider combustion of oil and wood in air.
Calculate the air requirements and the atmospheric nitrogen in the products if air is used for stoichiometric combustion of $100~\mathrm{kg}$ of oil.
 Calculate the stoichiometric air requirements and excess air in the products if $100~\mathrm{kg}$ of wood is burnt with... | Calculate the air requirements and the atmospheric nitrogen in the products if air is used for stoichiometric combustion of $100~\mathrm{kg}$ of oil. Calculate the stoichiometric air requirements and excess air in the products if $100~\mathrm{kg}$ of wood is burnt with $30\%$ excess air. | 2 | 2 | 5 | 51 | 40 | 472 | 0 | 154 | 0 | 43 | 40 | 43 | 40 | 0 | 0 | 1,160 | 6 | 2 | 1,160 | 6 | 10 | 155 | 4 | 288 | |
13f4f722-af7c-4bfe-b4ee-b4018b2cf9b2 | 1 | 0 | 0 | 11 | Turbomachinery | Non-Isentropic Turbine | 5 | 4 | Steam enters a turbine at $600^{\circ}\mathrm{C}$ and $20$ bar, and leaves at $0.1$ bar (all total conditions). 
| 0.333333 | 20 | If the turbine total-to-total isentropic efficiency is $92\%$, what is the specific work output?
| 1 | 1 | Apply the SFEE, bearing in mind that there is no heat transfer and kinetic and potential energy changes can be neglected:
  
$-\frac{\dot{W}}{\dot{m}} = h_{03} - h_{01}$
***
The enthalpies can be found using the steam tables.
***
At state $1$, the vapour is superheated. From Table E26 at $600^{\circ}\ma... | null | Non-Isentropic Turbine | 1 | 4 | 2 | Steam enters a turbine at $600^{\circ}\mathrm{C}$ and $20$ bar, and leaves at $0.1$ bar (all total conditions). 
If the turbine total-to-total isentropic efficiency is $92\%$, what is the specific work output?
| If the turbine total-to-total isentropic efficiency is $92\%$ , what is the specific work output? | 1 | 1 | 0 | 32 | 38 | 636 | 0 | 152 | 1 | 15 | 6 | 15 | 6 | 0 | 0 | 1,306 | 9 | 0.333333 | 435.333333 | 3 | 10 | 200 | 1 | 184 |
148040a4-9421-4696-b8b9-7d976a131fde | 1 | 0 | 0 | 16 | Functions - Stationary points, Series, and Sums | Stationary Points | 7 | 5 |  Show that the function 
$$
u(x, y) = x^4 + 4x^2y^2 - 2x^2 + 2y^2 -1
$$
has three stationary points - two local minima and one saddle point.
| 0.666667 | 15 |  Show that the function 
$$
u(x, y) = x^4 + 4x^2y^2 - 2x^2 + 2y^2 -1
$$
has three stationary points - two local minima and one saddle point.
| 1 | 1 | From section 5.5 of the notes, stationary points of a function $u(x,y)$ are where
$$
\begin{aligned}
{\partial u\over\partial x} = 0 = {\partial u\over\partial y}
\end{aligned}
$$
By finding these partial derivatives and applying this condition, find the three stationary points of $u$.
***
$$
\begin{aligned}
{\p... | From section 5.5 of the notes, stationary points of a function $u(x,y)$ are where
$$
\begin{aligned}
{\partial u\over\partial x} = 0 = {\partial u\over\partial y}
\end{aligned}
$$
By finding these partial derivatives and applying this condition, find the three stationary points of $u$.
***
You can determine thei... | Stationary Points | 1 | 6 | 1 |  Show that the function 
$$
u(x, y) = x^4 + 4x^2y^2 - 2x^2 + 2y^2 -1
$$
has three stationary points - two local minima and one saddle point.
 Show that the function 
$$
u(x, y) = x^4 + 4x^2y^2 - 2x^2 + 2y^2 -1
$$
has three stationary points - two local minima and one saddle point.
| Show that the function $ u(x, y) = x^4 + 4x^2y^2 - 2x^2 + 2y^2 -1 $ has three stationary points - two local minima and one saddle point. | 1 | 0 | 2 | 34 | 68 | 1,033 | 352 | 194 | 87 | 17 | 46 | 17 | 32 | 0 | 0 | 2,408 | 5 | 0.666667 | 1,605.333333 | 3.333333 | 32 | 185 | 1 | 244 |
148e2545-9224-4659-8e6e-732dd07642ca | 9 | 0 | 0 | 9 | Linear Networks and Components | Resistor Conventions | 0 | 0 | Calculate the resistor nominal value and upper and lower limits for each of the following parameters:
| 0.333333 | 15 | The resistor parameters are given in the table below:
| *a* | *b* | *c* | *tol* |
| :------- | :------- | :---- | :-------- |
| *Yellow* | *Orange* | *Red* | *Nothing* |
 = 3400~\Omega$
$R_{\mathrm{max}} = 4300 + (20\% \times 4300) = 5160~\Omega$
$R = ab \times 10^c$
***
$R = 40 \times 10^3 = 40000~\Omega$
***
Go... | Resistor Conventions | 1 | 4 | 2 | Calculate the resistor nominal value and upper and lower limits for each of the following parameters:
The resistor parameters are given in the table below:
| *a* | *b* | *c* | *tol* |
| :------- | :------- | :---- | :-------- |
| *Yellow* | *Orange* | *Red* | *Nothing* |
:** Find whether $\vec{p}\times\vec{q}$ points into or out of the page for the following pairs of vectors:
| 0.333333 | 5 | 
:** Find whether $\vec{p}\times\vec{q}$ points into or out of the page for the following pairs of vectors:
.
| 1 | 1 | Consider first the complex cosine function:
$$
\begin{equation*}
\cos z=\cos x\cosh y-i\,\sin x\sinh y\, .
\end{equation*}
$$
***
Find the magnitude of this function and simplify as follows:
$$
\begin{align*}
|\cos z|
&=\bigl(\cos^2x\cosh^2y+\sin^2x\sinh^2y\bigr)^{1/2}\\
\end{align*}
$$
***
Simplify this by mak... | Use the hyperbolic/trigonometric formula for $\cos{z}$ in terms of $x$ and $y$ (refer to **section 2.6** of the notes).
***
Find the magnitude of cos z using this formula. 
***
Can you use trigonometric/hyperbolic identities to simplify your expression?
***
Does this simplified formula prove that $\cos{z}... | Unbounded Sine/Cosine | 0 | 6 | 1 | Explain why the magnitudes of $\cos{z}$ and $\sin{z}$ are unbounded (i.e., can go to infinity).
| null | 1 | 0 | 3 | 15 | 18 | 71 | 33 | 41 | 56 | 15 | 18 | 1 | 0 | 0 | 0 | 774 | 5 | 0.333333 | 258 | 1.666667 | 0 | 6 | 1 | 81 |
150d356e-54d3-4aa5-9b2b-94fa0a483d7b | 1 | 0 | 1 | 14 | Isentropic flow relations | Super Hornet low-pass speed | 6 | 0 | Suppose that you were lucky enough to take the following picture of a Super Hornet flying slightly above the speed of sound at sea level. The weather conditions are just perfect so that a Mach cone is made visible thanks to a condensation phenomenon.
$$
\footnotesize
\text{Figure 2: Triangular element along plane A.}
$$
   
***
Usi... | Complex Stresses | 1 | 4 | 2 | At a central point in a boiler rivet, the material of the rivet is undergoing a shear stress of $77 \text{ MPa}$ while resisting movement between the boiler plates and a tensile stress of $62 \text{ MPa}$ due to the extension of the rivet.
   
. The isentropic total-to-total efficiency is $0.75$. Calculate:
| 0.333333 | 20 | The temperature of the air leaving the compressor.
The specific work transfer to the air.
| 2 | 2 | Calculate the temperature for an isentropic compression:
  
$\frac{T_{04\mathrm{s}}}{T_{02}} = (\frac{P_{04}}{P_{02}})^{\frac{\gamma - 1}{\gamma}}$
***
Substituting in numbers and rearranging for $T_{04\mathrm{s}}$:
  
$T_{04\mathrm{s}} = (22+273.15)(\frac{10}{1})^{\frac{1.4-1}{1.4}} = 569.8~\... | Non-Isentropic Compressor | 1 | 4 | 2 | Air enters a compressor at $1$ bar, $22^{\circ}\mathrm{C}$ and is compressed to $10$ bar (all total conditions). The isentropic total-to-total efficiency is $0.75$. Calculate:
The temperature of the air leaving the compressor.
The specific work transfer to the air.
| Calculate: The temperature of the air leaving the compressor. | 1 | 0 | 1 | 40 | 35 | 538 | 0 | 86 | 0 | 15 | 0 | 9 | 0 | 0 | 0 | 1,005 | 5 | 0.666667 | 335 | 1.666667 | 9 | 161 | 2 | 227 | |
16eb7876-b1be-42cc-8deb-e26ead614877 | 1 | 0 | 0 | 15 | Maths for Life Scientists 03 - Arithmetic | Melting DNA | 2 | 1 | When double-stranded DNA is heated, the two strands separate (‘melt’). The temperature ($T_{m}$) in degrees Celsius at which this melting occurs depends on the composition of bases in the DNA, because C and G are held together by more hydrogen bonds than A and T.
If we call the percentage of bases that are guanine or ... | 1 | 10 | What is the melting temperature of a sample of DNA that is $21\%$ adenine?
| 1 | 1 | If the DNA is $21\%$ A, it is also $21\%$ T, and therefore $58\%$ G+C ($29\%$ each), so $T_m=\frac{P_{G+C}+175}{2.5}=\frac{58+175}{2.5}= 93\,\degree\mathrm{C}$
| null | Melting DNA | 0 | 5 | 0 | When double-stranded DNA is heated, the two strands separate (‘melt’). The temperature ($T_{m}$) in degrees Celsius at which this melting occurs depends on the composition of bases in the DNA, because C and G are held together by more hydrogen bonds than A and T.
If we call the percentage of bases that are guanine or ... | $ P_{G+C}=2.5T_{m}-175 $ What is the melting temperature of a sample of DNA that is $21\%$ adenine? | 1 | 2 | 0 | 86 | 53 | 94 | 0 | 20 | 1 | 14 | 6 | 15 | 28 | 0 | 0 | 140 | 1 | 1 | 140 | 1 | 13 | 34 | 1 | 429 |
1782eae8-f4c0-4a16-a3d7-5906736d742c | 1 | 2 | 1 | 17 | Collisions and centre-of-mass frame | Chemical reaction | 3 | 8 | In the chemical reaction $\text{NO} + \text{O}_3 \rightarrow
\text{NO}_2 + \text{O}_2$, there is an intermediate step where the $\text{NO}$ and $\text{O}_3$ molecules combine to form an activated complex $[\text{NO--O}_3]^{\ast}$. The diagram below shows an empirical potential energy curve representing the three diff... | 0.666667 | 30 | Find the velocity of the $[\text{NO--O}_3]^{\ast}$ complex after the collision.
Show that the collision is inelastic. What has happened to the kinetic energy lost in the collision?
Will this collision lead to the formation of $\text{NO}_2$? Explain your answer.
Two oxygen atoms collide in isolation. Could the col... | 4 | 4 | The total momentum before the collision that forms the complex is
***
$$
p_{\text{tot}} = m_{\text{NO}} u_{\text{NO}} + m_{\text{O}_3}
u_{\text{O}_3} = (14 + 16)m_u u_{\text{NO}} + (3\times 16)m_u u_{\text{O}_3},
$$
where $m_u \approx 1.66 \times 10^{-27}$kg is the atomic mass unit. Setting the value of $u... | Find the total mass of the $\text{NO}$ and $\text{O}_3$ molecules, and hence also the $[\text{NO--O}_3]^{\ast}$ complex. 
***
Use conservation of momentum to find the velocity of the complex.
The overall kinetic energy of a system decreases in an inelastic collision.
***
Therefore, show that the kinetic ene... | Chemical reaction | 2 | 6 | 1 | In the chemical reaction $\text{NO} + \text{O}_3 \rightarrow
\text{NO}_2 + \text{O}_2$, there is an intermediate step where the $\text{NO}$ and $\text{O}_3$ molecules combine to form an activated complex $[\text{NO--O}_3]^{\ast}$. The diagram below shows an empirical potential energy curve representing the three diff... | An $\text{NO}$ molecule travelling in the $x $\text{NO}$2 u_{text{NO}} = -550,text{m}cdottext{s}^{-1}$ and an $\text{O}_3$ molecule travelling at $u_{text{O}_3} = +550,text{m}cdottext{s}^{-1} $\text{NO}$5 text{NO--O}_3^{ast} $\text{NO}$6 text{NO--O}_3^{ast} $\text{NO}$7 text{NO}_2 $\text{NO}$8 text{O}_2$ molecule i... | 1 | 0 | 1 | 231 | 415 | 1,224 | 57 | 512 | 298 | 61 | 50 | 29 | 183 | 0 | 0 | 3,659 | 12 | 2.666667 | 2,439.333333 | 8 | 68 | 280 | 4 | 1,648 |
1791621e-fcec-45c6-8e95-497341743143 | 2 | 0 | 0 | 16 | Functions - Differentiation | Higher-order Derivatives | 0 | 5 | Find the third derivative of
$$
x^3\cos(5x+ 1)
$$
| 0.333333 | 15 | ...directly
...using the Leibniz formula (lectures).
| 2 | 2 | To find the derivative directly, find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$, from there $\displaystyle {\mathrm{d}^2y \over \mathrm{d}x^2}$, and from there $\displaystyle {\mathrm{d}^3y \over \mathrm{d}x^3}$.
Now you can have a go at finding these.
***
You can use the product rule to calculate $\displaysty... | To find the derivative directly, find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$, from there $\displaystyle {\mathrm{d}^2y \over \mathrm{d}x^2}$, and from there $\displaystyle {\mathrm{d}^3y \over \mathrm{d}x^3}$.
Now you can have a go at finding these.
***
You can use the product rule to calculate $\displayst... | Higher-order Derivatives | 1 | 6 | 1 | Find the third derivative of
$$
x^3\cos(5x+ 1)
$$
...directly
...using the Leibniz formula (lectures).
| Find the third derivative of $ x^3\cos(5x+ 1) $ ...directly ...using the Leibniz formula lectures. | 1 | 1 | 7 | 12 | 17 | 1,638 | 559 | 142 | 109 | 6 | 0 | 12 | 15 | 0 | 0 | 2,195 | 9 | 0.666667 | 731.666667 | 3 | 4 | 295 | 2 | 88 |
17c2d6dd-27f7-4911-a94d-2db22124026e | 1 | 0 | 0 | 24 | Green's Theorem and Curl | Stokes' Theorem | 4 | 3 | **\[Boas 6.11.3]** Use Stokes' Theorem to evaluate:
$$
\iint_S (\nabla\times\vec{V})\cdot{d\vec{S}}
$$
where $\vec{V} = x^2\mathbf{\hat{i}} + z^2\mathbf{\hat{j}} -y^2\mathbf{\hat{k}}$ and $S$ is the part of the surface $z=4-x^2-y^2$ above the $x-y$ plane.
| 0.333333 | 10 | **\[Boas 6.11.3]** Use Stokes' Theorem to evaluate:
$$
\iint_S (\nabla\times\vec{V})\cdot{d\vec{S}}
$$
where $\vec{V} = x^2\mathbf{\hat{i}} + z^2\mathbf{\hat{j}} -y^2\mathbf{\hat{k}}$ and $S$ is the part of the surface $z=4-x^2-y^2$ above the $x-y$ plane.
| 1 | 3 | Stokes' Theorem states:
$$
\iint_{S}(\nabla\times\vec{V})\cdot d\vec{S}=\oint_{C,\text{ccw}}{\vec{V}\cdot d\vec{r}}
$$
where the RHS is the line integral evaluated anti-clockwise around the loop $C$ that encloses the opening of the surface. This is a circle of radius $2$ in the $(x,y)$ plane. 
***
Setting $z... | Start by sketching the surface $z=4-x^2-y^2$. To do this, consider the equation of the surface at a fixed $z$ (e.g., $z=0,4$). What happens as $z$ increases?
***
There are two ways to apply Stokes' Theorem to this problem:
1. Reduce the surface integral to a line integral by direct application of Stokes' theorem.
... | Stokes' Theorem | 0 | 6 | 1 | **\[Boas 6.11.3]** Use Stokes' Theorem to evaluate:
$$
\iint_S (\nabla\times\vec{V})\cdot{d\vec{S}}
$$
where $\vec{V} = x^2\mathbf{\hat{i}} + z^2\mathbf{\hat{j}} -y^2\mathbf{\hat{k}}$ and $S$ is the part of the surface $z=4-x^2-y^2$ above the $x-y$ plane.
**\[Boas 6.11.3]** Use Stokes' Theorem to evaluate:
$$
\iin... | Boas 6.11.3 Use Stokes' Theorem to evaluate: $ \iint_S ( abla\times\vec{V})\cdot{d\vec{S}} $ where $\vec{V} = x^2\mathbf{\hat{i}} + z^2\mathbf{\hat{j}} -y^2\mathbf{\hat{k}}$ and $S$ is the part of the surface $z=4-x^2-y^2$ above the $x-y$ plane. | 1 | 1 | 1 | 46 | 274 | 802 | 122 | 169 | 183 | 23 | 145 | 23 | 133 | 0 | 0 | 1,644 | 11 | 0.333333 | 548 | 3.666667 | 34 | 120 | 1 | 450 |
184e6da2-45c4-4ea3-b27e-2354a37e2d22 | 4 | 0 | 2 | 20 | Complex Analysis - Analytic Functions and ODEs | Coupled Differential Equations | 2 | 6 | Consider two systems labelled $A$ and $B$ which are weakly coupled according to the ODEs
$$
\begin{aligned}
\frac{dx_A}{dt} &= \phantom{\varepsilon} x_A + \varepsilon x_B\\
\frac{dx_B}{dt} &= \varepsilon x_A + \phantom{\varepsilon} x_B \ ,
\end{aligned}
$$
where $\varepsilon \ll 1$.
| 0.666667 | 30 | Show that if we define two new variables,
$$
\begin{aligned}
x_1 &= x_A + x_B \nonumber \\
x_2 &= x_A - x_B \nonumber
\end{aligned}
$$
then the ODE expressed in terms of $x_1$ and $x_2$ are not coupled.
Solve the two separate ODEs in $x_1$ and $x_2$ and use this to find the general solutions for $x_A$ and $x_B$.
... | 4 | 4 | Starting with the definitions of $x_1$ and $x_2$ and differentiating with respect to $t$:
***
$$
\begin{aligned}
\frac{dx_1}{dt} &= \frac{dx_A}{dt} + \frac{dx_B}{dt} \\
\frac{dx_2}{dt} &= \frac{dx_A}{dt} - \frac{dx_B}{dt}
\end{aligned}
$$
***
Next, we insert $dx_A/dt$ and $dx_B/dt$ into equations (1) and (2):
*... | See sections **3.1** and **5.1.2** to understand the term *coupled differential equation*.
***
Start by differentiating $x_1$ and $x_2$ with respect to $t$.
***
Insert the expressions for $dx_A/dt$ and $dx_B/dt$ into your equations.
***
Now express the equations in terms of $x_1$ and $x_2$...
***
... Does t... | Coupled Differential Equations | 2 | 6 | 1 | Consider two systems labelled $A$ and $B$ which are weakly coupled according to the ODEs
$$
\begin{aligned}
\frac{dx_A}{dt} &= \phantom{\varepsilon} x_A + \varepsilon x_B\\
\frac{dx_B}{dt} &= \varepsilon x_A + \phantom{\varepsilon} x_B \ ,
\end{aligned}
$$
where $\varepsilon \ll 1$.
Show that if we define two new va... | Show that if we define two new variables, $ \begin{aligned} x_1 &= x_A + x_B onumber \\ x_2 &= x_A - x_B onumber \end{aligned} $ then the ODE expressed in terms of $x_1$ and $x_2$ are not coupled. Solve the two separate ODEs in $x_1$ and $x_2$ and use this to find the general solutions for $x_A$ and $B$0 .... | 4 | 4 | 3 | 149 | 400 | 1,568 | 163 | 325 | 198 | 130 | 254 | 118 | 267 | 0 | 0 | 2,818 | 23 | 2.666667 | 1,878.666667 | 15.333333 | 53 | 225 | 6 | 931 |
18cc9219-397a-486a-9c41-ca1e76ce2007 | 2 | 0 | 0 | 16 | Functions - Integrals, Centre of Mass, Volumes and Surface Areas | Area enclosed and Centre of Mass | 4 | 2 | Consider the elliptical lamina bounded by the curve $\displaystyle {{x^2}\over{ a^2}} + {{y^2}\over{b^2}} = 1$.
(lamina = thin sheet)
| 0.666667 | 20 | Find the area the ellipse encloses.
 Assuming the lamina to be of uniform density, find the position of the centre of mass of the part that lies in the first quadrant ($x\geq 0, y\geq 0)$.
| 2 | 4 | By considering symmetry, what do you notice about the ellipse?
You can integrate over just a section of the ellipse, and then consider the symmetry of the curve to find the total area, which will make the integral easier to evaluate. What limits could you use?
***
Try using just the region in the first quadrant. You... | To find the area under a curve specified in Cartesian coordinates, you can rearrange to the form $y=...$ and then use regular integration.
What limits should you use? You could try sketching the curve.
***
You should have an equation for $y$ containing a $\pm$. How should you deal with this?
Is there a way t... | Area enclosed and Centre of Mass | 1 | 6 | 1 | Consider the elliptical lamina bounded by the curve $\displaystyle {{x^2}\over{ a^2}} + {{y^2}\over{b^2}} = 1$.
(lamina = thin sheet)
Find the area the ellipse encloses.
 Assuming the lamina to be of uniform density, find the position of the centre of mass of the part that lies in the first quadrant ($x\geq 0, ... | lamina = thin sheet Find the area the ellipse encloses. Assuming the lamina to be of uniform density, find the position of the centre of mass of the part that lies in the first quadrant $x\geq 0, y\geq 0)$ . | 2 | 2 | 17 | 48 | 68 | 3,870 | 616 | 667 | 498 | 34 | 19 | 37 | 16 | 0 | 0 | 6,941 | 21 | 1.333333 | 4,627.333333 | 14 | 13 | 498 | 2 | 274 |
19205aba-f145-48b6-aa1e-c5c321c4bfa3 | 1 | 0 | 3 | 21 | Fourier Series Evaluation and Properties | (Challenge) Orthonormal Polynomials | 0 | 8 | Consider the interval $[-1,1]$ and develop a series of polynomials of successively higher order that form an orthonormal set. Let $p_n(x)$ denote a polynomial of order $n$. 
| 1 | 30 | Let $p_0(x)=\sqrt{\frac{1}{2}}$ . Show that $p_1(x)= \sqrt{\frac{3}{2}}x$ is normalised and is orthogonal to $p_0(x)$.
Find a second order polynomial $p_2(x)$ which is orthogonal to both $p_0$ and $p_1$, and is normalised.
Describe a method to find $p_n$ satisfying the criteria for an orthonormal set given that all $... | 4 | 4 | To show that $p_1(x)$ is normalised, we must show that the inner product $ \braket{p_1 | p_1} $ (see **section 2.3** of the notes) is 1 in the interval $[-1,1]$.
***
$$
\begin{aligned}
\braket{p_1 | p_1}
&= \int_{-1}^1{\sqrt{\frac{3}{2}}x\sqrt{\frac{3}{2}}x dx} \\
&= \int_{-1}^1{\frac{3}{2}x^2... | To show that $p_1(x)$ is normalised, show that the inner product $ \braket{p_1 | p_1} $ (see **section 2.3** of the notes) is 1.
***
To show that $p_0(x)$ and $p_1(x)$ are orthogonal, show that their inner product $ \braket{p_0 | p_1} $ is 0. 
Express $p_2(x)$ as a general second-order polynomial (you may us... | (Challenge) Orthonormal Polynomials | 2 | 6 | 1 | Consider the interval $[-1,1]$ and develop a series of polynomials of successively higher order that form an orthonormal set. Let $p_n(x)$ denote a polynomial of order $n$. 
Let $p_0(x)=\sqrt{\frac{1}{2}}$ . Show that $p_1(x)= \sqrt{\frac{3}{2}}x$ is normalised and is orthogonal to $p_0(x)$.
Find a second order p... | Consider the interval $[-1,1]$ and develop a series of polynomials of successively higher order that form an orthonormal set. Show that $p_1(x)= \sqrt{\frac{3}{2}}x$ is normalised and is orthogonal to $p_0(x)$ . Find a second order polynomial $p_2(x)$ which is orthogonal to both $p_0$ and $p_1$ , and is norma... | 5 | 8 | 2 | 108 | 115 | 8 | 46 | 248 | 96 | 80 | 97 | 98 | 83 | 0 | 0 | 1,839 | 7 | 4 | 1,839 | 7 | 21 | 1 | 4 | 579 |
1929083d-ee49-47da-8f1a-7a423af519bd | 0 | 0 | 8 | 14 | Introduction to turbulence | Time-average operator | 16 | 3 | Let $f$ and $g$ be two spatio-temporal functions (i.e. they both depend on $x$, $y$, $z$ and $t$). We assume that $f$ and $g$ are *bounded*, meaning that there exists a strictly positive real constant $M$ such that $|f|\leq M$ and $|g|\leq M$ for any point in space and time. Let $\lambda$ be a real constant. The time-a... | 0.666667 | 30 | $$
\bar{\bar{f}} = \bar{f}
$$
$$
\overline{f + \lambda g} = \bar{f} + \lambda \bar{g}
$$
$$
\overline{f^\prime} = 0
$$
$$
\overline{f^\prime \bar{g}} = \overline{g^\prime \bar{f}} = 0
$$
$$
\overline{fg} = \bar{f}\bar{g} + \overline{f^\prime g^\prime}
$$
$$
\overline{\partial f/\partial x} = \partial\bar{f}/\partial x
... | 8 | 0 | Time-average operator | 2 | 4 | 2 | Let $f$ and $g$ be two spatio-temporal functions (i.e. they both depend on $x$, $y$, $z$ and $t$). We assume that $f$ and $g$ are *bounded*, meaning that there exists a strictly positive real constant $M$ such that $|f|\leq M$ and $|g|\leq M$ for any point in space and time. Let $\lambda$ be a real constant. The time-a... | prime or dash i.e $g$6 represents the fluctuating component of the function, that is: $g$7 Show that: $g$8 $g$9 $x$0 $x$1 $x$2 $x$3 $x$4 $x$5 if $x$6 Hint: you need to use the assumption that $f$ is bounded. | 1 | 1 | 0 | 145 | 560 | 0 | 0 | 0 | 0 | 21 | 438 | 49 | 36 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 50 | 0 | 8 | 1,138 | ||
1a604759-45f4-43de-907c-3426ec48d8d8 | 1 | 0 | 0 | 24 | Gradient and Divergence | Divergence Theorem: Sphere | 3 | 7 | Evaluate the surface integral $\oiint_{S}{\vec{B}\cdot d\vec{S}}$ where $\vec{B}=r\mathbf{\hat{r}}$ in spherical polar coordinates and $S$ is a sphere of radius $2$ centred at the origin. Apply the divergence theorem to this integral and show that $\iiint_{V}{(\nabla\cdot\vec{B})dV}$gives the same answer.
| 0.333333 | 15 | Evaluate the surface integral $\oiint_{S}{\vec{B}\cdot d\vec{S}}$ where $\vec{B}=r\mathbf{\hat{r}}$ in spherical polar coordinates and $S$ is a sphere of radius $2$ centred at the origin. Apply the divergence theorem to this integral and show that $\iiint_{V}{(\nabla\cdot\vec{B})dV}$gives the same answer.
| 1 | 3 | On a sphere of radius $a=2$: 
$$
d\vec{S} = a^2\sin\theta\,d\theta\,d\phi \,\mathbf{\hat{r}} = 4\sin\theta\,d\theta\,d\phi\,\mathbf{\hat{r}}
$$
This result was derived in lectures (you may wish to prove it to yourself by the cross product). 
***
On the surface of the sphere, $\vec{B} = r\mathbf{\hat{r}} =... | (**Surface Integral**) You can use the result derived in lectures that $d\vec{S}$ for a sphere of radius $a$ is given by $a^2\sin\theta d\theta d\phi\mathbf{\hat{r}}$. 
***
(**Surface Integral**) Hence find $\vec{B}\cdot d\vec{S}$. Evaluate this over the limits of $\theta$ and $\phi$ for the sphere. 
***... | Divergence Theorem: Sphere | 1 | 6 | 1 | Evaluate the surface integral $\oiint_{S}{\vec{B}\cdot d\vec{S}}$ where $\vec{B}=r\mathbf{\hat{r}}$ in spherical polar coordinates and $S$ is a sphere of radius $2$ centred at the origin. Apply the divergence theorem to this integral and show that $\iiint_{V}{(\nabla\cdot\vec{B})dV}$gives the same answer.
Evaluate the ... | Evaluate the surface integral $\oiint_{S}{\vec{B}\cdot d\vec{S}}$ where $\vec{B}=r\mathbf{\hat{r}}$ in spherical polar coordinates and $S$ is a sphere of radius $2$ centred at the origin. Apply the divergence theorem to this integral and show that $\iiint_{V}{( abla\cdot\vec{B})dV}$ gives the same answer. | 2 | 3 | 0 | 76 | 206 | 813 | 250 | 149 | 127 | 38 | 104 | 38 | 101 | 0 | 0 | 1,422 | 8 | 0.333333 | 474 | 2.666667 | 10 | 104 | 1 | 538 |
1b6227b3-1883-4c7a-a50b-e03bb8de05db | 0 | 2 | 0 | 16 | Functions - Stationary points, Series, and Sums | Ratio Test | 7 | 0 | Use the ratio test to determine whether each of these two series is convergent
| 0.333333 | 10 | $$
\begin{aligned}
\sum_{n=0}^\infty {1\over (n + 1)!}
\end{aligned}
$$
$$
\begin{aligned}
\sum_{n=0}^\infty {(3 - 4i)^n \over n!}
\end{aligned}
$$
| 2 | 2 | For the ratio test of a sum $\displaystyle \sum _{n = 0}^\infty u_n$, find
$$
\begin{aligned} L = \lim_{n\to\infty} \left| {u_{n+1}\over u_n}\right|
\end{aligned}
$$
Do you remember how the convergence/divergence of the series depends on $L$?
***
If $L<1$, it is convergent. If $L>1$, it is divergent. If $L=1$ you n... | For the ratio test of a sum $\displaystyle \sum _{n = 0}^\infty u_n$, find
$$
\begin{aligned} L = \lim_{n\to\infty} \left| {u_{n+1}\over u_n}\right|
\end{aligned}
$$
Do you remember how the convergence/divergence of the series depends on $L$?
***
If $L<1$, it is convergent. If $L>1$, it is divergent. If $L=1$ need... | Ratio Test | 0 | 6 | 1 | Use the ratio test to determine whether each of these two series is convergent
$$
\begin{aligned}
\sum_{n=0}^\infty {1\over (n + 1)!}
\end{aligned}
$$
$$
\begin{aligned}
\sum_{n=0}^\infty {(3 - 4i)^n \over n!}
\end{aligned}
$$
| Use the ratio test to determine whether each of these two series is convergent $ \begin{aligned} \sum_{n=0}^\infty {1\over (n + 1)!} | 1 | 1 | 1 | 16 | 129 | 35 | 35 | 31 | 25 | 2 | 146 | 21 | 0 | 0 | 0 | 1,752 | 8 | 0.666667 | 584 | 2.666667 | 16 | 5 | 2 | 194 |
1c0e2dba-6dfb-4939-a870-6a263d24cb65 | 2 | 0 | 0 | 11 | Entropy Generation, Entropy Balance and Exergy | Gas Turbine Engine | 6 | 2 | Gas enters the turbine of a gas turbine engine at $1500 ~\mathrm{K}$, with a mass flow rate of $500 ~\mathrm{kg/s}$, and expands adiabatically through a pressure ratio of $20$. The turbine isentropic efficiency is $0.90$. The ambient temperature is $288 ~\mathrm{K}$. The gas can be treated as perfect, with $c_\mathrm{p... | 0.666667 | 15 | Calculate the change in the exergy flow rate of the gas stream.
 Compare the value obtained in part (a) with the turbine power in units of $\mathrm{MW}$.
| 2 | 2 | Assuming kinetic and potential energy changes can be neglected:
  
$\Delta \dot{B}_\mathrm{flowing\space fluid} = \dot{m}[(h_2-h_1) -T_0(s_2-s_1)]$
***
For a perfect gas:
  
$s_2 -s_1 = c_\mathrm{p} \mathrm{ln}(\frac{T_{02}}{T_{01}}) − R \mathrm{ln}( \frac{P_{02}}{P_{01}} )$
***
$\frac{P_{01... | Gas Turbine Engine | 1 | 4 | 2 | Gas enters the turbine of a gas turbine engine at $1500 ~\mathrm{K}$, with a mass flow rate of $500 ~\mathrm{kg/s}$, and expands adiabatically through a pressure ratio of $20$. The turbine isentropic efficiency is $0.90$. The ambient temperature is $288 ~\mathrm{K}$. The gas can be treated as perfect, with $c_\mathrm{p... | Regarding all properties as stagnation total values and the efficiency as the total-to-total value: Calculate the change in the exergy flow rate of the gas stream. Compare the value obtained in part a with the turbine power in units of $\mathrm{MW}$ . | 2 | 2 | 4 | 97 | 130 | 1,063 | 0 | 187 | 0 | 28 | 13 | 42 | 13 | 0 | 0 | 1,987 | 17 | 1.333333 | 1,324.666667 | 11.333333 | 32 | 296 | 2 | 544 | |
1c0fb18f-35b0-4e37-a35b-4ef651e69dc3 | 0 | 0 | 3 | 14 | Boundary layer equations | Wind tunnel data processing | 13 | 3 | Referring to data from the wind tunnel in Figure 14.6, measured velocity profiles, $u(y)$, are given at different locations, $x$.
To complete this question you will need to use that data. You can begin by reading from the graph but later will need the raw data and instructions are provided.
Figure 14.6 is replicate... | 1 | 45 | Based on reading the graph, estimate (roughly!) the boundary layer thickness, $\delta_{99}$, for the ten profiles. Comment on the ratio $\delta(x)/x$ where $x$ is the distance along the wall. Add you own data from the Wind Tunnel and check if it agrees.
Derive a method to estimate the average value, $V$, for a given ve... | 3 | 0 | Wind tunnel data processing | 2 | 4 | 2 | Referring to data from the wind tunnel in Figure 14.6, measured velocity profiles, $u(y)$, are given at different locations, $x$.
To complete this question you will need to use that data. You can begin by reading from the graph but later will need the raw data and instructions are provided.
Figure 14.6 is replicate... | Based on reading the graph, estimate roughly! Comment on the ratio $\delta(x)/x$ where $x$ is the distance along the wall. Add you own data from the Wind Tunnel and check if it agrees. Derive a method to estimate the average value, $V$ , for a given velocity profile. You need to use a control volume analysis to an... | 8 | 4 | 0 | 287 | 78 | 0 | 0 | 0 | 0 | 226 | 69 | 90 | 34 | 1 | 0 | 0 | 0 | 3 | 0 | 0 | 10 | 0 | 3 | 1,547 | ||
1c7a7f6c-55d0-445b-9aef-f251ced69b63 | 2 | 2 | 0 | 13 | Gyroscopic Motion | Sanding Machine | 5 | 1 | A hand-held rotary sanding machine is shown in the diagram below.
It has a sanding disc which rotates at $2400~\mathrm{rev/min}$ in the direction shown.... | 0.666667 | 30 | The fixed x-axis at $0.5~\mathrm{rad/s}$ in the direction shown.
The fixed z axis at $1.0~\mathrm{rad/s}$ in the direction shown.
| 2 | 2 | The gyroscopic moment about $x$ is given by:
$$
M_x=I{\dot{\theta}}_y{\dot{\theta}}_z
$$
***
Where the angular velocity of the sanding disc is given by:
$$
{\dot{\theta}}_y=2400\left(\cfrac{2\pi}{60}\right)
$$
$$
{\dot{\theta}}_y=251.3~\mathrm{rad/s}
$$
***
And the moment of inertia of the sanding disc is:
$... | Start off by calculating your values of the mass moment of inertia and the angular velocity of the uniform disk in rad/s (or any other units you may feel comfortable with!).
***
Refer to Equation 5.1 in your notes to calculate the magnitude of the moment required, keeping in mind that in this case the angular veloci... | Sanding Machine | 2 | 4 | 2 | A hand-held rotary sanding machine is shown in the diagram below.
It has a sanding disc which rotates at $2400~\mathrm{rev/min}$ in the direction shown.... | Determine the magnitude and direction of the couple which a person holding the sander must apply in order to rotate the sander about: The fixed x-axis at $0.5~\mathrm{rad/s}$ in the direction shown. | 1 | 2 | 0 | 98 | 101 | 410 | 0 | 67 | 119 | 19 | 40 | 32 | 20 | 1 | 0 | 664 | 4 | 1.333333 | 442.666667 | 2.666667 | 29 | 106 | 4 | 693 |
1c8bdc12-43a5-4b6a-a13a-1129f7e73552 | 2 | 0 | 2 | 14 | Steady laminar flows | Falling film | 10 | 2 | A viscous oil flows steadily down an inclined plate under the action of gravity, forming a film of constant thickness $h$. This is shown in the image, below, where a Cartesian frame of reference with $y$-axis normal to the plate has been defined. The flow is uniform along the plate (i.e. along $x$ and $z$ coordinates a... | 1 | 40 | If $\vec{u}=\left[u~~v~~w\right]^T$, show that $v=0$.
Show that the shear stress $\tau_w$ acting on the wall is equal to $\tau_w=\rho g h \mathrm{sin}(\alpha)$.
Explain why $u$ is a function of $y$ only and find an expression for $u(y)$.
Show that the pressure, $p$, is a function of $y$ only and compute $p(y)$. | 4 | 0 | Falling film | 2 | 4 | 2 | A viscous oil flows steadily down an inclined plate under the action of gravity, forming a film of constant thickness $h$. This is shown in the image, below, where a Cartesian frame of reference with $y$-axis normal to the plate has been defined. The flow is uniform along the plate (i.e. along $x$ and $z$ coordinates a... | Using the Cartesian frame of reference shown: If $\vec{u}=\left[u~~v~~w\right]^T$ , show that $v=0$ . Show that the shear stress $\tau_w$ acting on the wall is equal to $\tau_w=\rho g h \mathrm{sin}(\alpha)$ . Explain why $u$ is a function of $y$ only and find an expression for $y$1 . Show that the pressure, ... | 4 | 3 | 0 | 159 | 121 | 0 | 0 | 0 | 0 | 54 | 107 | 64 | 98 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 12 | 0 | 4 | 822 | ||
1c94b03d-b0dd-4fce-b753-ca96a7c4d8c8 | 2 | 0 | 0 | 0 | Tutorial 2 | Prandtl-Meyer expansion | 5 | 4 | The flow downstream of the shock wave in the figure below is turned through a Prandtl-Meyer expansion at C, until it has returned to its original flow direction.
-(e)).
   
The beam is subjected to a load of $ 10 \text{ kN} $ acting vertic... | 0.666667 | 40 | There is no support (i.e. the end is free.)
The end just rests on a rigid roller, which is mounted on a rigid foundation.
Same as (b) but the roller is raised by $5 \text{ mm}$.
Same as (b) but the roller is lowered by $5 \text{ mm}$.
Same as (b) but the foundation has a vertical stiffness of $1 \text{ MNm}$
| 5 | 5 | Here, the worked solution for the general equation is included. The general equation will then be directly referenced in parts (b)-(e).
   
***
First draw a diagram of the system:
-(e)).
   
The beam is subjected to a load of $ 10 \text{ kN} $ acting vertic... | For each of the cases, determine the deflection of the beam at the point of application of the load and find the maximum stress in the beam after the load has been applied. | 1 | 0 | 7 | 165 | 114 | 4,320 | 0 | 593 | 0 | 58 | 43 | 33 | 0 | 0 | 0 | 7,194 | 20 | 3.333333 | 4,796 | 13.333333 | 15 | 1,220 | 10 | 783 | |
1d3e94e1-f697-45ed-a04c-b3e62ab9d8fa | 0 | 5 | 0 | 24 | Differentiation and Integration | (A) Exact Differential | 0 | 3 | Determine which of the following are exact differentials:
| 0.333333 | 15 | $$
(3x+2)y\,dx + x(x+1)\, dy
$$
$$
y\tan x\,dx + x\tan y\,dy
$$
$$
y^2(\ln x + 1)\,dx + 2xy\ln x\,dy
$$
$$
y^2(\ln x + 1)\,dy + 2xy\ln x\,dx
$$
$$
\left[ x/(x^2 + y^2)\right]\,dy - \left[ y/(x^2 + y^2 )\right]\,dx
$$
| 5 | 5 | See $\color{green}\textsf{Structured Tutorial}$ or the lecture notes for an explanation of exact differentials. 
***
Let $A(x,y) = (3x+2)y$ and $B(x,y) = x(x+1)$.
***
Differentiating $A$ with respect to $y$:
***
$$
\frac{\partial A}{\partial y} = 3x+2
$$
and $B$ with respect to $x$:
***
$$
\frac{\partial ... | The differential of a function $\Omega$ in Cartesian form is:
$$
d\Omega = \frac{\partial \Omega}{\partial x}dx + \frac{\partial \Omega}{\partial y}dy
$$
The differentials given in the question are of the form:
$$
A(x,y)dx+ B(x,y)dy
$$
If:
$$
\frac{\partial A}{\partial y} = \frac{\partial B}{\partial x}
$$
We can... | (A) Exact Differential | 1 | 6 | 1 | Determine which of the following are exact differentials:
$$
(3x+2)y\,dx + x(x+1)\, dy
$$
$$
y\tan x\,dx + x\tan y\,dy
$$
$$
y^2(\ln x + 1)\,dx + 2xy\ln x\,dy
$$
$$
y^2(\ln x + 1)\,dy + 2xy\ln x\,dx
$$
$$
\left[ x/(x^2 + y^2)\right]\,dy - \left[ y/(x^2 + y^2 )\right]\,dx
$$
| Determine which of the following are exact differentials: $ (3x+2)y\,dx + x(x+1)\, dy $ $ y\tan x\,dx + x\tan y\,dy $ $ y^2(\ln x + 1)\,dx + 2xy\ln x\,dy $ $ y^2(\ln x + 1)\,dy + 2xy\ln x\,dx $ $ \left[ x/(x^2 + y^2)\right]\,dy - \left[ y/(x^2 + y^2 )\right]\,dx $ | 1 | 2 | 11 | 13 | 174 | 1,292 | 1,275 | 248 | 475 | 5 | 212 | 13 | 164 | 0 | 0 | 2,257 | 25 | 1.666667 | 752.333333 | 8.333333 | 38 | 137 | 5 | 224 |
1da7744a-6a6b-44ca-ace2-0dafe58f0c8c | 2 | 0 | 0 | 9 | Introduction to Control Systems | PI Controller | 6 | 26 | 
$$
H(s) = \frac{4}{s+1}
$$
  
For the closed-loop system shown above, apply a PI controller to achieve a damping of $0.5$ and a natural freque... | 0.666667 | 20 | 
$$
H(s) = \frac{4}{s+1}
$$
  
For the closed-loop system shown above, apply a PI controller to achieve a damping of $0.5$ and a natural freque... | 1 | 1 | By converting $u(t)$ and $y(t)$ to the Laplace domain, find an expression for $Y(s)$:
  
$Y(s) = E(s)K(s)H(s)$
  
where $E(s) = U(s)-Y(s)$.
***
Hence:
  
$Y(s) = U(s)K(s)H(s)-Y(s)K(s)H(s)$
***
Find an expression for the transfer function:
  
$G(s) = \frac{Y(s)}{U(s)} ... | null | PI Controller | 1 | 4 | 2 | 
$$
H(s) = \frac{4}{s+1}
$$
  
For the closed-loop system shown above, apply a PI controller to achieve a damping of $0.5$ and a natural freque... | $ H(s) = \frac{4}{s+1} $ For the closed-loop system shown above, apply a PI controller to achieve a damping of $0.5$ and a natural frequency of $2~\mathrm{rad/s}$ . | 1 | 1 | 1 | 52 | 90 | 987 | 0 | 134 | 1 | 26 | 49 | 24 | 43 | 1 | 0 | 1,611 | 11 | 0.666667 | 1,074 | 7.333333 | 22 | 110 | 2 | 630 |
1e43fcca-b71a-43ba-9b27-a149926cdc75 | 0 | 0 | 2 | 19 | Cross Product | Tangents | 2 | 6 | **(L5)**: A curve is defined by the parametric equation,
$$
{\bf\vec{r}} = (x,y,z) = (\cos t, \sin t, t)
$$
| 0.666667 | 10 | Describe the curve
Find the derivative and the second derivative of the curve with respect to $t$. Show that the first and second derivatives are always perpendicular to one another. Describe how the vector describing the first derivative varies with $t$.
| 2 | 2 | In the $xy$ plane $|{\bf \vec{r}_{⊥}}| = \sqrt{\cos ^2 t +\sin ^2 t} = 1$ and $\tan θ = \sin t/\cos t → θ = t$. Hence the curve traces out a circle with angle varying linearly with $t$. 
***
  
However z also increases linearly with t, so this curve is a helix
The position vector ${\bf\vec{r}}$ to a ... | What does the curve look like in the x-y plane?
***
Continuing in the x-y plane, what is the angle made by ${\bf{\vec{r}}}$ to the x axis? How can you relate this angle to $t$? Thus, what is the shape traced out by the curve in the x-y pane as a function of $t$?
***
Remember that z also changes linearly with $t$ ... | Tangents | 0 | 6 | 1 | **(L5)**: A curve is defined by the parametric equation,
$$
{\bf\vec{r}} = (x,y,z) = (\cos t, \sin t, t)
$$
Describe the curve
Find the derivative and the second derivative of the curve with respect to $t$. Show that the first and second derivatives are always perpendicular to one another. Describe how the vector de... | L5: A curve is defined by the parametric equation, $ {\bf\vec{r}} = (x,y,z) = (\cos t, \sin t, t) $ Describe the curve Find the derivative and the second derivative of the curve with respect to $t$ . Show that the first and second derivatives are always perpendicular to one another. Describe how the vector describin... | 3 | 3 | 0 | 53 | 46 | 4 | 25 | 32 | 158 | 43 | 6 | 53 | 44 | 0 | 0 | 1,055 | 4 | 1.333333 | 703.333333 | 2.666667 | 8 | 0 | 2 | 303 |
1ea69f1c-ae9b-438f-8dc4-b7a7077b9c66 | 1 | 1 | 0 | 15 | Maths for Life Scientists 01 - Algebra | Photosystems | 0 | 2 | Photosynthesis relies on electrons in chlorophyll molecules absorbing photons of light energy. This energy promotes electrons from a low-energy ground state to a high-energy excited state. For the chlorophyll-a dimer in the reaction centre of photosystem I (PS-I), the difference in energy between the ground and excited... | 1 | 5 | By rearranging the equation below, calculate the wavelength of maximum absorbance of PS-I in nanometres $(1 \,\mathrm{nm} = 10^{-9} \, \mathrm{m})$.
$$
E=\frac{N_{a}\, \cdot \, h \, \cdot \, c}{\lambda}
$$
* $E$ Difference in energy between ground and excited states ($171 \, \mathrm{ kJ \cdot mol^{-1}}$)
* $N... | 2 | 2 | $$
\lambda=\frac{N_Ahc}{E} = \frac{ 6.022 \times 10^{23}\,\mathrm{mol}^{-1} \times 6.626\times10^{-34}\,\mathrm{J \cdot s} \times 3\times10^{8} \, \mathrm{m \cdot s^{-1}}}{171000 \, \mathrm{ J \cdot mol^{-1}}} = 7.00 \times 10^{-7} \,\mathrm{m} = 700 \times 10^{-9} \,\mathrm{m} = \ 700\ \mathrm{nm}
$$
Lower ($680 \mat... | Photosystems | 0 | 5 | 0 | Photosynthesis relies on electrons in chlorophyll molecules absorbing photons of light energy. This energy promotes electrons from a low-energy ground state to a high-energy excited state. For the chlorophyll-a dimer in the reaction centre of photosystem I (PS-I), the difference in energy between the ground and excited... | By rearranging the equation below, calculate the wavelength of maximum absorbance of PS-I in nanometres $(1 \,\mathrm{nm} = 10^{-9} \, \mathrm{m})$ . Would this absorb photons of higher or lower wavelength than PS-I, given that the energy difference for its reaction centre is about $(1 \,\mathrm{nm} = 10^{-9} \, \mat... | 2 | 2 | 0 | 155 | 326 | 294 | 0 | 25 | 0 | 102 | 331 | 42 | 76 | 0 | 0 | 394 | 0.5 | 2 | 394 | 0.5 | 69 | 82 | 2 | 1,027 | |
1f35e909-d901-43a0-840f-cebe806dd2ec | 2 | 0 | 0 | 11 | Entropy Generation, Entropy Balance and Exergy | Heat Exchanger | 6 | 3 | Carbon dioxide, which can be treated as a perfect gas, flows steadily at a rate of $10 ~\mathrm{kg/s}$ through a heat exchanger where it is cooled at constant pressure (i.e. with negligible friction) from $600 ~\mathrm{K}$ to $400 ~\mathrm{K}$. The ambient temperature of the environment surrounding the heat exchanger i... | 0.666667 | 15 | Via the flow stream exergy change: 
$\hspace{5 cm}\Delta\dot{B}_\mathrm{flowing \space fluid} = \dot{m}(\Delta h - T_0\Delta s)$  
Via the differential version of the expression for thermal exergy (or work potential of heat transferred), $\mathrm{d}\dot{B}_\mathrm{heat} =(1-\frac{T_0}{T})\mathrm{d}\dot{Q}$... | 2 | 2 | For a perfect gas:
  
$\Delta h = c_\mathrm{p}\Delta T$
***
and:
  
$\Delta s = c_\mathrm{p} \mathrm{ln} ( \frac{T_2}{ T_1} ) − R \mathrm{ln} ( \frac{P_2}{ P_1} )$
***
Hence:
  
$\Delta \dot{B}_\mathrm{flowing\space fluid} = \dot{m}(c_\mathrm{p}\Delta T - T_0(c_\mathrm{p} \mathrm... | Heat Exchanger | 1 | 4 | 2 | Carbon dioxide, which can be treated as a perfect gas, flows steadily at a rate of $10 ~\mathrm{kg/s}$ through a heat exchanger where it is cooled at constant pressure (i.e. with negligible friction) from $600 ~\mathrm{K}$ to $400 ~\mathrm{K}$. The ambient temperature of the environment surrounding the heat exchanger i... | Find the change in exergy flow rate in units of $\mathrm{kW}$ by two different methods: Via the flow stream exergy change: $\hspace{5 cm}\Delta\dot{B}_\mathrm{flowing \space fluid} = \dot{m}(\Delta h - T_0\Delta s)$ Via the differential version of the expression for thermal exergy or work potential of heat transfer... | 2 | 2 | 1 | 128 | 338 | 871 | 0 | 89 | 0 | 63 | 265 | 66 | 216 | 0 | 0 | 1,284 | 8 | 1.333333 | 856 | 5.333333 | 29 | 119 | 2 | 917 | |
1f39acd9-b66a-4a21-9401-613381437d73 | 0 | 0 | 1 | 17 | Collisions and centre-of-mass frame | Rocket accelerating against gravity | 3 | 9 | Show that the velocity $v(t)$ of a rocket starting from rest and accelerating against the Earth’s gravity is given by
$$
v(t) = u\ln \left ( \frac{M_0}{M(t)} \right ) - gt.
$$
The exhaust speed is $u$, the initial mass is $M_0$, and the mass after time $t$ is $M(t)$. Neglect the dependence of $g$ on height.
| 1 | 20 | Show that the velocity $v(t)$ of a rocket starting from rest and accelerating against the Earth’s gravity is given by
$$
v(t) = u\ln \left ( \frac{M_0}{M(t)} \right ) - gt.
$$
The exhaust speed is $u$, the initial mass is $M_0$, and the mass after time $t$ is $M(t)$. Neglect the dependence of $g$ on height.
| 1 | 1 | 
***
The diagram below shows that the change in momentum of the rocket+fuel in time $dt$ can be found as follows:
$$
\begin{aligned}
\text{change in ... | After ejecting fuel of mass $dM$, the rocket's speed increases from $v$ to $v+dv$...
***
... Hence write down the change in momentum of the rocket in a time $dt$ (see the lectures if you are unsure about this).
***
A change of momentum is an impulse. What is the impulse produced by gravity on this system in the t... | Rocket accelerating against gravity | 1 | 6 | 1 | Show that the velocity $v(t)$ of a rocket starting from rest and accelerating against the Earth’s gravity is given by
$$
v(t) = u\ln \left ( \frac{M_0}{M(t)} \right ) - gt.
$$
The exhaust speed is $u$, the initial mass is $M_0$, and the mass after time $t$ is $M(t)$. Neglect the dependence of $g$ on height.
Show t... | Show that the velocity $v(t)$ of a rocket starting from rest and accelerating against the Earth’s gravity is given by $ v(t) = u\ln \left ( \frac{M_0}{M(t)} \right ) - gt. | 1 | 1 | 0 | 98 | 144 | 691 | 39 | 78 | 137 | 49 | 83 | 31 | 6 | 0 | 0 | 1,208 | 6 | 1 | 1,208 | 6 | 14 | 64 | 1 | 504 |
1f3a8567-d040-4746-92b2-98b2caf040bf | 3 | 0 | 1 | 10 | Creep | Creep in a Beam | 8 | 0 | The following data apply to extruded and cold rolled Nimonic 80A at $750^\circ \text{C}$. 
   
* Elastic (Young’s) Modulus $= 140 \times 10^3 \text{ MN/m}^2$ 
* $0.2\%$ Proof Stress $= 450\text{ MN/m}^2$
* Elongation to fracture = $25\%$ (short term tensile test)
* Mean Coefficient of The... | 1 | 30 | Neglecting primary creep, calculate the secondary (steady state) creep strain rate at each stress level.
Estimate the coefficient $n$ in a power law representation between stress and strain rate.
What would be the total change in length of a bar of $50\text{ mm}$ initial length at $20^\circ\text{C}$, which is held at... | 4 | 3 | Secondary creep strain is the **slope** of the **linear** region of the creep strain vs. time curve (see figure below.)
   
$$
\footnotesize
... | Creep in a Beam | 2 | 4 | 2 | The following data apply to extruded and cold rolled Nimonic 80A at $750^\circ \text{C}$. 
   
* Elastic (Young’s) Modulus $= 140 \times 10^3 \text{ MN/m}^2$ 
* $0.2\%$ Proof Stress $= 450\text{ MN/m}^2$
* Elongation to fracture = $25\%$ (short term tensile test)
* Mean Coefficient of The... | Elastic Young’s Modulus $= 140 \times 10^3 \text{ MN/m}^2$ $0.2\%$ Proof Stress $= 450\text{ MN/m}^2$ Elongation to fracture = $25\%$ short term tensile test Mean Coefficient of Thermal Expansion $20~–~750^\circ \text{C}$ range $= 15.8 \times 10^{-6}$ The stress to cause a creep strain in 3000 hours is: ... | 4 | 6 | 2 | 159 | 259 | 1,000 | 0 | 190 | 0 | 99 | 141 | 140 | 305 | 0 | 1 | 2,272 | 6 | 4 | 2,272 | 6 | 68 | 182 | 4 | 1,021 | |
1f7df9e2-c4c9-49fa-851b-5dbb51dea7b2 | 3 | 0 | 0 | 13 | Momentum and impulse 2 | Record Turntable | 3 | 0 | The figure below shows a record turntable.
The electric drive of a record turntable develops a constant torque of $0.13~\mathrm{Nm}$ and takes $2$ secon... | 0.666667 | 35 | What is the immediate reduction in the speed of the turntable?
How long does it take before the correct speed is regained?
How much energy is dissipated in friction as the record slips on the turntable?
| 3 | 3 | Kinetic diagram:
***
The turntable and vinyl rotate about the same axis, so apply conservation of momentum:
$$
I_{G,\text{table}}~\dot\theta_{\tex... | This problem is analogous to linear momentum, where the record landing on the table can be taken as the impact event.
***
The moment of inertia of the turntable can be worked out using the information given about the electric drive torque and the time taken to reach the operating rotational speed (think of the rotat... | Record Turntable | 2 | 4 | 2 | The figure below shows a record turntable.
The electric drive of a record turntable develops a constant torque of $0.13~\mathrm{Nm}$ and takes $2$ secon... | Optional: See the video below for an explanation of how a record turntable works What is the immediate reduction in the speed of the turntable? How long does it take before the correct speed is regained? How much energy is dissipated in friction as the record slips on the turntable? | 3 | 5 | 0 | 111 | 71 | 2,485 | 20 | 234 | 177 | 36 | 0 | 50 | 0 | 2 | 0 | 3,672 | 15 | 2 | 2,448 | 10 | 17 | 304 | 3 | 879 |
207213eb-f3c2-4640-90d5-e428a440c347 | 5 | 0 | 0 | 17 | Kinematics and N2 | White City roller coaster | 0 | 10 | Imperial College, in trying to maximise revenue from its new White City campus, decides to build a roller coaster using new, frictionless technology developed in the Mechanical Engineering department. 
 for the speed of the carriage when it has dropped a height $h$ from point B.
The acceleration of an object moving around a circle of radius $r$ at speed $v$ is $a = v^2/r$. In order to maximise excitement, passengers should feel three times their normal weight at point C. How far ... | 5 | 5 | The potential energy lost is the kinetic energy gained, so $\frac{1}{2}mv^2 = mgh$ and $v = \sqrt{2gh}$.
For passengers to feel three times their normal weight at C, a passenger of mass $m$ should experience a reaction force of $3mg$ instead of the usual $mg$ when in equilibrium. This means that their centripetal f... | Use conservation of energy.
At point $C$, a person has a reaction force acting upwards and weight acting downwards...
***
... The reaction should be $3g$. What is the resulting centripetal acceleration $a$?
***
Hence use the result from part (a) to find the required height the rollercoaster must drop to achieve ... | White City roller coaster | 1 | 6 | 1 | Imperial College, in trying to maximise revenue from its new White City campus, decides to build a roller coaster using new, frictionless technology developed in the Mechanical Engineering department. 
 = \frac{4}{s+2}$
$H(s) = \frac{8}{4s+2}$
 
| 2 | 2 | Convert the expression for $H(s)$ to canonical form:
  
$H(s) = \frac{K}{1+\tau s} = \frac{2}{1+0.5s}$
***
Hence:
  
Gain: $K = 2$
  
Time constant: $\tau = 0.5$
Convert the expression for $H(s)$ to canonical form:
  
$H(s) = \frac{K}{1+\tau s} = \frac{4}{1+2s}$
***
... | First Order Gain and Time Constant | 0 | 4 | 2 | What are the gain and the time constant of the following first order systems:
$H(s) = \frac{4}{s+2}$
$H(s) = \frac{8}{4s+2}$
 
| What are the gain and the time constant of the following first order systems: $H(s) = \frac{4}{s+2}$ $H(s) = \frac{8}{4s+2}$ | 1 | 1 | 1 | 17 | 41 | 120 | 0 | 38 | 0 | 3 | 45 | 16 | 41 | 0 | 0 | 324 | 2 | 0.666667 | 108 | 0.666667 | 11 | 36 | 4 | 111 | |
2187fc1b-a767-4956-a1de-b540b049a89a | 6 | 0 | 1 | 11 | Refrigeration, Heat Pumps and Gas Liquefaction | P-h Chart Practice | 4 | 6 | A vapour-compression cycle using R134a has the following conditions: 
  
Evaporator pressure: $\hspace{2cm}$ $1.5$ bar 
Compressor inlet state: $\hspace{1.6cm}$ Superheated
Compressor inlet temperature: $\hspace{0.0cm}$ $-10^{\circ}\mathrm{C}$ 
Condenser pressure: $\hspace{1.96 cm}... | 0.666667 | 25 | Plot the cycle on the $P-h$ chart (found on Blackboard).
Using enthalpies read from the $P - h$ chart, except at the expansion valve inlet where the usual approximation for subcooled liquid properties should be used, calculate the heat transfer in the evaporator, per $\mathrm{kg}$ of R134a. 
Calculate the work t... | 7 | 6 |
The heat transfer in the evaporator is represented by process $4-1$. Applying the SFEE and bearing in mind that there is no shaft work and kinetic and potential energy changes can be neglected:
  
$\frac{\dot{Q}_{41}}{\dot{m}} = h_1 - h_4$
***
Process $3-4$ is isenthalpic so $h_4 = h_3$. 
***
At s... | P-h Chart Practice | 2 | 4 | 2 | A vapour-compression cycle using R134a has the following conditions: 
  
Evaporator pressure: $\hspace{2cm}$ $1.5$ bar 
Compressor inlet state: $\hspace{1.6cm}$ Superheated
Compressor inlet temperature: $\hspace{0.0cm}$ $-10^{\circ}\mathrm{C}$ 
Condenser pressure: $\hspace{1.96 cm}... | A vapour-compression cycle using R134a has the following conditions: Evaporator pressure: $\hspace{2cm}$ $1.5$ bar Compressor inlet state: $\hspace{1.6cm}$ Superheated Compressor inlet temperature: $\hspace{0.0cm}$ $-10^{\circ}\mathrm{C}$ Condenser pressure: $\hspace{1.96 cm}$ $10.0$ bar Expansion valve i... | 7 | 7 | 3 | 220 | 300 | 835 | 0 | 244 | 0 | 160 | 158 | 222 | 261 | 0 | 0 | 2,755 | 21 | 4.666667 | 1,836.666667 | 14 | 46 | 251 | 7 | 1,517 | |
219c6815-48ed-4752-b235-de33874ef529 | 0 | 1 | 7 | 14 | Introduction to turbulence | Turbulence kinetic energy and scales for large eddies | 16 | 2 | A fast response probe with sampling frequency equal to $5$ kHz has been placed at a given location (fixed in space) in the wake formed by an air flow ($\nu_{air}=1.8\times 10^{-5}$ $\mathrm{m^2/s}$) around a square bluff body with characteristic size $L=0.1$ m. The time history of the three components of velocity has b... | 0.666667 | 30 | Compute the turbulence kinetic energy, $k$, of the flow at that location
Estimate the velocity scale of the large eddies, $U_\mathrm{t}$.
Find an estimate of the large-eddies time scale, $T_\mathrm{t}$.
Considering the flow to be statistically stationary, discuss whether the averaging period is sufficient to have conve... | 8 | 0 | Turbulence kinetic energy and scales for large eddies | 2 | 4 | 2 | A fast response probe with sampling frequency equal to $5$ kHz has been placed at a given location (fixed in space) in the wake formed by an air flow ($\nu_{air}=1.8\times 10^{-5}$ $\mathrm{m^2/s}$) around a square bluff body with characteristic size $L=0.1$ m. The time history of the three components of velocity has b... | Post-process of the data shows that the mean velocity at that location is equal to zero and the RMS value of the velocity fluctuations is equal to $7$ $\mathrm{m/s}$ for all the velocity components. Compute the turbulence kinetic energy, $k$ , of the flow at that location Estimate the velocity scale of the large e... | 8 | 5 | 0 | 208 | 105 | 0 | 0 | 0 | 0 | 107 | 31 | 141 | 48 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 21 | 0 | 8 | 1,077 | ||
21bfda15-b80c-451e-9f5d-f3e4d3a1ca77 | 2 | 0 | 1 | 21 | Truncations and Alternate Forms | (Challenge): Taylor vs. Fourier Series | 1 | 4 | (*Challenge*): The sine function can be approximated on the interval $[0,\pi]$ by a second-order polynomial, $p_2(x)$, that satisfies
$$
p_2(0) = 0, \qquad p_2(\pi/2) = 1, \qquad p_2(\pi)=0 \; .
$$
For the interval $[-\pi,0]$ the function is just defined as the *odd continuation* of the positive interval.
| 0.666667 | 40 | Find the second-order polynomial that satisfies the criteria above.
Find the trigonometric Fourier series for $p_2(x)$ defined on the interval $[-\pi,\pi]$. 
By plotting, compare this approximation of the sine function to the one obtained from a Taylor series expansion around $x=0$ and around $x=\pi/2$. Is the b... | 3 | 3 | Let
$$
p_2(x) = A x^2 + B x + C \; ,
$$
***
Then, we write down the three equations and solve for $A$, $B$ and $C$ using the given conditions: 
$$
\begin{align}
&p_2(0) = C = 0\\
&p_2(\pi/2) = A\left(\frac{\pi^2}{4}\right) + B\left(\frac{\pi}{2}\right) =1\\
&p_2(\pi) = A\pi^2+B\pi=0
\end{align}
$$
Solving ... | Write the polynomial in the general form $p_2(x) = Ax^2 + Bx + C$.
***
Set up 3 equations using the given conditions, and use them to solve for $A,B,C$.
Sketch the function (**remember** that for the interval $[-\pi,0]$ the function is just defined as the odd continuation of the positive interval). 
***
Is ... | (Challenge): Taylor vs. Fourier Series | 2 | 6 | 1 | (*Challenge*): The sine function can be approximated on the interval $[0,\pi]$ by a second-order polynomial, $p_2(x)$, that satisfies
$$
p_2(0) = 0, \qquad p_2(\pi/2) = 1, \qquad p_2(\pi)=0 \; .
$$
For the interval $[-\pi,0]$ the function is just defined as the *odd continuation* of the positive interval.
Find the se... | Find the second-order polynomial that satisfies the criteria above. Find the trigonometric Fourier series for $p_2(x)$ defined on the interval $[-\pi,\pi]$ . By plotting, compare this approximation of the sine function to the one obtained from a Taylor series expansion around $x=0$ and around $x=\pi/2$ . Is the b... | 5 | 5 | 1 | 91 | 112 | 1,140 | 172 | 300 | 235 | 54 | 34 | 54 | 34 | 0 | 0 | 2,475 | 14 | 2 | 1,650 | 9.333333 | 35 | 194 | 3 | 562 |
229aa45a-b762-4ed2-8497-9da06715bfb8 | 2 | 0 | 0 | 8 | Complex Strains and Strain Gauges | Strain Gauge on Shaft 1 | 3 | 0 | A shaft, $25\text{ mm}$ diameter, $1500\text{ mm}$ long, is fitted with a strain gauge, the axis of which is inclined at $45^{\circ}$ to the axis of the shaft. When calibrated with the shaft in pure torsion, the gauge records a strain of $0.0008$. In service, there is an end load which causes the shaft to extend $2.3\t... | 0.666667 | 25 | What would the strain gauge record in service if subjected to the same torque as before? 
What is the Poisson's ratio of the shaft material? 
| 2 | 2 | See the drawing of the strain gauge as it would look on the shaft:
***
Using the data book, the strain in torsion can be calculated, **prior to the load... | Strain Gauge on Shaft 1 | 2 | 4 | 2 | A shaft, $25\text{ mm}$ diameter, $1500\text{ mm}$ long, is fitted with a strain gauge, the axis of which is inclined at $45^{\circ}$ to the axis of the shaft. When calibrated with the shaft in pure torsion, the gauge records a strain of $0.0008$. In service, there is an end load which causes the shaft to extend $2.3\t... | What would the strain gauge record in service if subjected to the same torque as before? What is the Poisson's ratio of the shaft material? | 2 | 2 | 1 | 107 | 78 | 595 | 0 | 126 | 0 | 25 | 0 | 25 | 0 | 0 | 0 | 1,335 | 4 | 1.333333 | 890 | 2.666667 | 22 | 158 | 2 | 540 | |
22b6b30c-24b4-4094-affd-b44b8e139021 | 1 | 0 | 0 | 19 | Dot Product | Fruit | 1 | 5 | **(L3)** A greengrocer sells the fruits $\text{(banana, apple, orange, kumquat)}$ at a price $\vec{p} = (40, 55, 50, 12)$ (in pence). On a certain day he sells $\vec{N} = (32, 14, 20, 40)$ of each respectively. Calculate $\vec{p}\cdot\vec{N}$ and give an interpretation of this quantity.
| 0.333333 | 5 | **(L3)** A greengrocer sells the fruits $\text{(banana, apple, orange, kumquat)}$ at a price $\vec{p} = (40, 55, 50, 12)$ (in pence). On a certain day he sells $\vec{N} = (32, 14, 20, 40)$ of each respectively. Calculate $\vec{p}\cdot\vec{N}$ and give an interpretation of this quantity.
| 1 | 1 | $$
\qquad \vec{p}\cdot\vec{N} = \begin{pmatrix}
40\\55\\50\\12
\end{pmatrix} \cdot \begin{pmatrix}
32\\14\\20\\40
\end{pmatrix}
$$
***
$$
=1280+770+1000+480 = \pounds 35.30
$$
The dot product is the sum of money the greengrocer made.
| See **section 1.9** for how to evaluate a dot product. 
***
Writing out the full dot product algebraically may help you understand the physical interpretation of this quantity. 
| Fruit | 0 | 6 | 1 | **(L3)** A greengrocer sells the fruits $\text{(banana, apple, orange, kumquat)}$ at a price $\vec{p} = (40, 55, 50, 12)$ (in pence). On a certain day he sells $\vec{N} = (32, 14, 20, 40)$ of each respectively. Calculate $\vec{p}\cdot\vec{N}$ and give an interpretation of this quantity.
**(L3)** A greengrocer sells th... | Calculate $\vec{p}\cdot\vec{N}$ and give an interpretation of this quantity. | 1 | 1 | 0 | 64 | 210 | 154 | 0 | 14 | 28 | 32 | 118 | 9 | 21 | 0 | 0 | 204 | 1 | 0.333333 | 68 | 0.333333 | 60 | 46 | 1 | 486 |
231198e3-e44c-4104-9748-51c0d8906a1b | 0 | 0 | 1 | 17 | N3 and momentum | Vector impulse | 2 | 4 | In lectures we showed that the change in momentum of an object is equal to the impulse applied:
$$
mv(t_f) - mv(t_i) = \int_{t_i}^{t_f} F(t) \, dt .
$$
Derive the three-dimensional vector equivalent:
$$
m\boldsymbol{\vec{v}}(t_f) - m\boldsymbol{\vec{v}}(t_i) = \int_{t_i}^{t_f} \boldsymbol{\vec{F}}(t) \, dt .
$$
&#x... | 0.333333 | 10 | In lectures we showed that the change in momentum of an object is equal to the impulse applied:
$$
mv(t_f) - mv(t_i) = \int_{t_i}^{t_f} F(t) \, dt .
$$
Derive the three-dimensional vector equivalent:
$$
m\boldsymbol{\vec{v}}(t_f) - m\boldsymbol{\vec{v}}(t_i) = \int_{t_i}^{t_f} \boldsymbol{\vec{F}}(t) \, dt .
$$
&#x... | 1 | 1 | Since you already know how to differentiate a vector with respect to $t$, you can integrate one too. An integral such as $\int \boldsymbol{\vec{F}}(t) , dt$ is really three integrals, one for each component:
 $\int \boldsymbol{\vec{F}}(t) , dt = \left ( \int F_x(t) , dt, \int F_y(t) , dt, \int F_z(t) , dt \ri... | Since you already know how to differentiate a vector with respect to $t$, you can integrate one too. An integral such as $\int \boldsymbol{\vec{F}}(t) , dt$ is really three integrals, one for each component:
 $\int \boldsymbol{\vec{F}}(t) , dt = \left ( \int F_x(t) , dt, \int F_y(t) , dt, \int F_z(t) , dt \ri... | Vector impulse | 0 | 6 | 1 | In lectures we showed that the change in momentum of an object is equal to the impulse applied:
$$
mv(t_f) - mv(t_i) = \int_{t_i}^{t_f} F(t) \, dt .
$$
Derive the three-dimensional vector equivalent:
$$
m\boldsymbol{\vec{v}}(t_f) - m\boldsymbol{\vec{v}}(t_i) = \int_{t_i}^{t_f} \boldsymbol{\vec{F}}(t) \, dt .
$$
&#x... | $ Derive the three-dimensional vector equivalent: $ m\boldsymbol{\vec{v}}(t_f) - m\boldsymbol{\vec{v}}(t_i) = \int_{t_i}^{t_f} \boldsymbol{\vec{F}}(t) \, dt . $ | 2 | 0 | 2 | 52 | 294 | 892 | 142 | 104 | 63 | 26 | 167 | 11 | 45 | 0 | 0 | 1,305 | 6 | 0.333333 | 435 | 2 | 24 | 43 | 1 | 560 |
237c59c3-85aa-4c7d-b311-3f23d5397b37 | 3 | 0 | 0 | 11 | Power Cycles and Plants 2 | CCP & CHP | 2 | 4 | This question relates to combined cycle plants and combined heat and power plants.
| 0.666667 | 15 | A gas turbine (GT) with an efficiency of $36\%$ is used in a conventional gas-steam combined cycle. The energy lost in the exhaust duct from the heat recovery steam generator (HRSG), as a result of the exhaust gas being above ambient temperature, is $15\%$ of the energy supplied in the gas turbine fuel. Other losses fr... | 2 | 2 | It can be helpful to draw what is happening as a diagram:
***
  
(i) The CHP produces one unit of power and one unit of heat. Since the prime... | CCP & CHP | 1 | 4 | 2 | This question relates to combined cycle plants and combined heat and power plants.
A gas turbine (GT) with an efficiency of $36\%$ is used in a conventional gas-steam combined cycle. The energy lost in the exhaust duct from the heat recovery steam generator (HRSG), as a result of the exhaust gas being above ambient tem... | If the steam cycle efficiency is $34\%$ , what is the overall efficiency of the combined cycle? What will be: i the energy utilisation factor of the CHP plant? ii the percentage saving in fuel energy considering fuel consumed on site and in power stations? | 3 | 2 | 0 | 215 | 39 | 161 | 0 | 92 | 0 | 202 | 39 | 45 | 6 | 0 | 0 | 773 | 7 | 1.333333 | 515.333333 | 4.666667 | 13 | 90 | 3 | 1,078 | |
23d553e2-62ee-43d2-b402-6b798bbb4f6e | 2 | 0 | 2 | 11 | Power Cycles and Plants 2 | Rankine Cycle Calculations 2 | 2 | 6 | In a steam power plant, the boiler produces $40 ~\mathrm{kg/s}$ of steam in the saturated vapour state at a pressure of $20$ bar. In order to run the turbine at less than full load, the boiler exit steam is throttled to a pressure of $10$ bar before entering the turbine, by means of a well-insulated, partly-closed valv... | 1 | 30 | Sketch the processes in the valve and turbine on an enthalpy - entropy ($h - s$) diagram, including the relevant isobars. (Assume that the steam is wet at the turbine exhaust).
Show that the dryness fraction of the steam at the turbine exhaust is $0.856$.
Calculate the shaft power produced.
Calculate the required fl... | 4 | 4 | On an $h-s$ diagram, the bell shape is slightly skewed.
Throttling is an isenthalpic process (see Q3), so process $1-2$ is a straight horizontal line.
Process $2-3$, had it been isentropic, would have been represented by a straight vertical line. However, since the isentropic efficiency is not $1$, entropy increases... | Rankine Cycle Calculations 2 | 2 | 4 | 2 | In a steam power plant, the boiler produces $40 ~\mathrm{kg/s}$ of steam in the saturated vapour state at a pressure of $20$ bar. In order to run the turbine at less than full load, the boiler exit steam is throttled to a pressure of $10$ bar before entering the turbine, by means of a well-insulated, partly-closed valv... | Sketch the processes in the valve and turbine on an enthalpy - entropy $h - s$ diagram, including the relevant isobars. Show that the dryness fraction of the steam at the turbine exhaust is $0.856$ . Calculate the shaft power produced. Calculate the required flow cross-sectional area at the turbine exhaust. | 4 | 4 | 3 | 162 | 84 | 1,090 | 0 | 462 | 0 | 61 | 14 | 49 | 12 | 0 | 0 | 3,193 | 17 | 4 | 3,193 | 17 | 30 | 412 | 4 | 809 | |
24124ac6-4477-4d13-b8cc-1a192c099fec | 0 | 0 | 2 | 14 | Steady laminar flows | Laplacian of a vector field | 10 | 1 | Write the Laplacian, $\overrightarrow{\mathrm{Lap}}(\vec{u})$, of the velocity vector | 0.666667 | 15 | in a Cartesian frame of reference (consider $\vec{u}=\left[u~~v~~w \right]^T$).
in a cylindrical frame of reference (consider $\vec{u}=\left[u_r~~u_\theta~~u_z \right]^T$). | 2 | 0 | Laplacian of a vector field | 1 | 4 | 2 | Write the Laplacian, $\overrightarrow{\mathrm{Lap}}(\vec{u})$, of the velocity vectorin a Cartesian frame of reference (consider $\vec{u}=\left[u~~v~~w \right]^T$).
in a cylindrical frame of reference (consider $\vec{u}=\left[u_r~~u_\theta~~u_z \right]^T$). | Write the Laplacian, $\overrightarrow{\mathrm{Lap}}(\vec{u})$ , of the velocity vectorin a Cartesian frame of reference consider $\vec{u}=\left[u~~v~~w \right]^T$ . | 1 | 1 | 0 | 26 | 115 | 0 | 0 | 0 | 0 | 18 | 77 | 17 | 72 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 14 | 0 | 2 | 233 | ||
243d5396-590a-4bb1-9283-9c19269bfd79 | 0 | 1 | 1 | 19 | Matrices | Matrix Multiplication | 6 | 2 | **(L9)**: For matrices $\text{A}$, $\text{B}$, and $\text{C}$,
$$
\begin{aligned}
\text{A}=\left(
\begin{array}{cccr}
2&\hskip10pt 3&\hskip10pt 1& -4\\
2&\hskip10pt 1&\hskip10pt 0& 5
\end{array}\right)\, ,\qquad
\text{B}=\left(
\begin{array}{cr}
2&\hskip2pt 4\\
1&\hskip2pt -1\\
2&\hskip2pt -1
\end{array}\right)\, ,\... | 0.333333 | 15 | Select the matrix combinations for which the product is possible to calculate.
Calculate the matrix product for those combinations that *can* be computed.
| 2 | 2 | For two matrices of dimensions $n_A\times m_A$ and $n_B\times m_B$, to multiply the matrices in the given order, we require $m_A = n_B$.
***
$\text{A}$ is $(2\times4)$, $\text{B}$ is $(3\times2)$, and $\text{C}$ is $(3\times3)$, so for the different combinations:
***
* AB is $(2\times4)(3\times2)$: can't compute
... | For two matrices of dimensions $n_A\times m_A$ and $n_B\times m_B$, to multiply the matrices in the given order, we require $m_A = n_B$.
For two matrices of dimensions $n_A\times m_A$ and $n_B\times m_B$, to multiply the matrices in the given order, we require $m_A = n_B$.
***
See **section 3.3** for how to multipl... | Matrix Multiplication | 1 | 6 | 1 | **(L9)**: For matrices $\text{A}$, $\text{B}$, and $\text{C}$,
$$
\begin{aligned}
\text{A}=\left(
\begin{array}{cccr}
2&\hskip10pt 3&\hskip10pt 1& -4\\
2&\hskip10pt 1&\hskip10pt 0& 5
\end{array}\right)\, ,\qquad
\text{B}=\left(
\begin{array}{cr}
2&\hskip2pt 4\\
1&\hskip2pt -1\\
2&\hskip2pt -1
\end{array}\right)\, ,\... | \end{aligned} $ Select the matrix combinations for which the product is possible to calculate. Calculate the matrix product for those combinations that can be computed. | 2 | 3 | 1 | 34 | 410 | 1,406 | 74 | 152 | 51 | 23 | 0 | 25 | 0 | 0 | 0 | 1,916 | 6 | 0.666667 | 638.666667 | 2 | 60 | 246 | 2 | 568 |
248240cb-dee7-445a-b5fc-83649bff2f7b | 0 | 0 | 1 | 17 | Collisions and centre-of-mass frame | Inelastic collision | 3 | 6 | In one-dimensional inelastic collisions, the coefficient of restitution $e$ is defined by the equation
$$
v_2 - v_1 = -e (u_2 - u_1),
$$
where $v_1$ and $v_2$ are the final velocities of the two bodies and $u_1$ and $u_2$ are their initial velocities. Use this equation, plus momentum conservation, to derive the follo... | 0.333333 | 10 | In one-dimensional inelastic collisions, the coefficient of restitution $e$ is defined by the equation
$$
v_2 - v_1 = -e (u_2 - u_1),
$$
where $v_1$ and $v_2$ are the final velocities of the two bodies and $u_1$ and $u_2$ are their initial velocities. Use this equation, plus momentum conservation, to derive the follo... | 1 | 1 | Start by writing down the definition of the coefficient of restitution and the law of conservation of momentum in the case when $u_2 = 0$:
***
$$
\begin{align}
v_2 - v_1 &= eu_1 ,\\
m_1 v_1 + m_2 v_2 &= m_1 u_1 .
\end{align}
$$
***
Our job is to solve for $v_1$ and $v_2$ as functions of $u_1$. One way is to... | Write down momentum conservation alongside the equation given in the question (remembering that $u_2=0$)
***
You now have a pair of simultaneous equations that you must solve for $v_1$ and $v_2$ in terms of $u_1$.
***
There are many ways to approach this, but one approach is to multiply $v_2 - v_1 = eu_1$ by $m_1... | Inelastic collision | 0 | 6 | 1 | In one-dimensional inelastic collisions, the coefficient of restitution $e$ is defined by the equation
$$
v_2 - v_1 = -e (u_2 - u_1),
$$
where $v_1$ and $v_2$ are the final velocities of the two bodies and $u_1$ and $u_2$ are their initial velocities. Use this equation, plus momentum conservation, to derive the follo... | Use this equation, plus momentum conservation, to derive the following expressions for the final velocities of the two bodies of masses $m_1$ and $m_2$ in the case when the second body is initially at rest $u_2 = 0$ : $ \begin{aligned} v_1 &= \frac{(m_1 - e m_2) u_1}{m_1 + m_2}, & v_2 &= \frac{(1 + e)m_1 u_1}{m_1... | 1 | 2 | 1 | 150 | 330 | 303 | 51 | 84 | 70 | 75 | 207 | 57 | 17 | 0 | 0 | 607 | 5 | 0.333333 | 202.333333 | 1.666667 | 84 | 94 | 1 | 984 |
24a58bb8-afbf-45b5-8a21-baf48f42769c | 2 | 2 | 1 | 14 | Material Derivative | Straining flow | 1 | 1 | Consider a simple straining flow in Cartesian coordinates, $\left(x,y\right)$, given by
$$
\vec{u}=\begin{bmatrix}1+\alpha x\\-\alpha y\end{bmatrix},
$$
where $\alpha$ is a constant real number.
| 0.666667 | 20 | Plot the velocity field (by hand or computer).
In the channel flow pictured below, with water entering from the left, in which region (labelled A to F) is the flow field approximately the same as the 'straining flow' field described above?
$, given by
$$
\vec{u}=\begin{bmatrix}1+\alpha x\\-\alpha y\end{bmatrix},
$$
where $\alpha$ is a constant real number.
Plot the velocity field (by hand or computer).
In the channel flow pictured below, with water entering from the left, in wh... | Plot the velocity field by hand or computer. In the channel flow pictured below, with water entering from the left, in which region labelled A to F is the flow field approximately the same as the 'straining flow' field described above? Simplify the expression for velocity, $\vec{u}$ , on $y=0$ . Derive an expression ... | 4 | 4 | 0 | 83 | 136 | 0 | 0 | 0 | 0 | 63 | 52 | 62 | 50 | 1 | 0 | 0 | 0 | 2 | 0 | 0 | 15 | 0 | 5 | 663 | ||
24e6c3bc-efcf-4212-98a2-42c23b645666 | 2 | 0 | 0 | 16 | Functions - Differentiation | Challenge problem: Snow-plough problem | 0 | 7 | At some time $(t = - T)$ in the morning, it starts snowing at a constant heavy rate and continues to do so. A snow plough starts at noon $(t = 0)$ and moves along a road removing snow at a constant rate. The velocity $\displaystyle {\mathrm{d}x\over \mathrm{d}t}$ of the plough is given by
$$
\displaystyle {\mathrm{d}x... | 0.666667 | 20 | At what time did it start snowing (i.e. find $T$)? 
Convert your answer to a decimal and type it in the answer box below
| 1 | 1 | (This is not *too* unreasonable a model in that snow is removed at a constant rate by *volume*. There is a drawback when the snow depth is very small..)
You can start by finding $x(t)$ from the equation of $\displaystyle {\mathrm{d}x\over \mathrm{d}t}$.
***
By integrating (and where $C$ is an arbitrary integration c... | You can start by finding $x(t)$ from the equation of $\displaystyle {\mathrm{d}x\over \mathrm{d}t}$.
***
Now, look at the conditions given.
Find $x$ at the three conditions for $t$, and solve the simultaneous equations to find $T$.
***
Remember that $T$ is positive as it starts snowing before noon ($t=0$) at $t=... | Challenge problem: Snow-plough problem | 1 | 6 | 1 | At some time $(t = - T)$ in the morning, it starts snowing at a constant heavy rate and continues to do so. A snow plough starts at noon $(t = 0)$ and moves along a road removing snow at a constant rate. The velocity $\displaystyle {\mathrm{d}x\over \mathrm{d}t}$ of the plough is given by
$$
\displaystyle {\mathrm{d}x... | At what time did it start snowing i.e find $T$ ? Convert your answer to a decimal and type it in the answer box below | 2 | 3 | 8 | 94 | 125 | 935 | 76 | 361 | 63 | 25 | 3 | 25 | 3 | 0 | 0 | 2,681 | 8 | 0.666667 | 1,787.333333 | 5.333333 | 9 | 199 | 2 | 473 |
250bd94a-2c36-42bd-a8a7-ce0360fc1620 | 2 | 0 | 0 | 22 | Problem Sheet 1 | Numerical exercises 2 | 0 | 1 | Pendulum clocks tick every half period of the pendulum’s swing.
| 0.333333 | 15 |  Calculate the length of the pendulum needed to make a clock that ticks once every second.
This pendulum is given an initial angular displacement of 6$^{\circ}$. Neglecting any dissipation, calculate the maximum speed attained by the pendulum.
| 2 | 2 | For a pendulum, $\omega_0 = \sqrt{g/l}$. 
***
The period is $T=2\pi/\omega_0$. Rearranging gives $l$:
***
 $l = T^2 g/(4\pi^2)$. 
***
To tick once per second, we need a period of $T=2$ s, so...
***
$l=99.4$ cm.
The linear displacement is $x = x_0 \cos(\omega_0 t)$.
***
The initial displacement... | Recall that $\omega_0 = \sqrt{g/l}$
***
The period of oscillation is related to the angular frequency: $T=2\pi/\omega_0$
***
Rearrange for $l$ and put the numbers into the formula.
Start from the solution for the displacement as a function of time: $x=x_0\cos(\omega_0 t)$
***
$x_0$ is given by the initial con... | Numerical exercises 2 | 1 | 6 | 1 | Pendulum clocks tick every half period of the pendulum’s swing.
 Calculate the length of the pendulum needed to make a clock that ticks once every second.
This pendulum is given an initial angular displacement of 6$^{\circ}$. Neglecting any dissipation, calculate the maximum speed attained by the pendulum.
| Calculate the length of the pendulum needed to make a clock that ticks once every second. Neglecting any dissipation, calculate the maximum speed attained by the pendulum. | 2 | 2 | 0 | 49 | 10 | 211 | 93 | 77 | 79 | 39 | 10 | 27 | 0 | 0 | 0 | 486 | 8 | 0.666667 | 162 | 2.666667 | 0 | 53 | 2 | 266 |
25481b22-b855-4627-8ea9-8788f6ed2f34 | 3 | 0 | 0 | 16 | Functions - Differentiation | Implicit Differentiation | 0 | 1 | Find $\displaystyle {\mathrm{d}y\over \mathrm{d}x}$ for each of these relationships:
| 0.333333 | 5 | $$
y^3 = x^3 - xy
$$
$$
xe^{y} = \cos(xy)
$$
$$
y =\sec^{-1}(x)
$$
| 3 | 3 | As this cannot simply be put into the form $y=f(x)$, use implicit differentiation. For implicit differentiation, take the derivative with respect to $x$ on both sides of the equation.
***
Take derivative $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}$ on both sides:
$$
\displaystyle {\mathrm{d}\over \mathrm{d}x}(y^3)... | As this cannot simply be put into the form $y=f(x)$, use implicit differentiation. For implicit differentiation, take the derivative with respect to $x$ on both sides of the equation.
***
Take derivative $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}$ on both sides:
$$
\displaystyle {\mathrm{d}\over \mathrm{d}x}(y^3... | Implicit Differentiation | 0 | 6 | 1 | Find $\displaystyle {\mathrm{d}y\over \mathrm{d}x}$ for each of these relationships:
$$
y^3 = x^3 - xy
$$
$$
xe^{y} = \cos(xy)
$$
$$
y =\sec^{-1}(x)
$$
| Find $\displaystyle {\mathrm{d}y\over \mathrm{d}x}$ for each of these relationships: $ y^3 = x^3 - xy $ $ xe^{y} = \cos(xy) $ $ y =\sec^{-1}(x) $ | 1 | 1 | 20 | 10 | 95 | 2,095 | 1,090 | 306 | 265 | 3 | 64 | 10 | 89 | 0 | 0 | 3,338 | 15 | 1 | 1,112.666667 | 5 | 11 | 100 | 3 | 127 |
256d9e06-c6aa-40e0-8b63-d39e54c73ffd | 3 | 1 | 0 | 4 | Solving advection-diffusion problems | Question 4 | 2 | 3 | An explosion on a ship $5\ \mathrm{km}$ away from the coast releases a quantity $M$ of contaminant. The wind velocity is $U$ towards the shoreline. The contaminant diffuses at a rate of $D$. The release occurs at the water surface, which is modelled as a perfect reflector.
Here, the wind velocity is chosen along the $... | 0.666667 | 20 | Give the equation for the three-dimensional time-evolution of contaminant concentration.
State the value of the maximum concentration $C_{\max}$ at time $t=10\ \mathrm{s}$. Repeat this calculation for every $10\ \mathrm{s}$ until $100\ \mathrm{s}$. From these values, sketch the maximum concentration and its location a... | 4 | 3 | Diffusion occurs in the vertical $z$ direction, horizontal $x$ direction (towards the shoreline) and span-wise $y$ direction. Advection only occurs in the $x$ direction. Therefore, the equation governing the concentration distribution is given by
$$
\dfrac{\partial C}{\partial t} + U\dfrac{\partial C}{\partial x} = D ... | Question 4 | 1 | 3 | 3 | An explosion on a ship $5\ \mathrm{km}$ away from the coast releases a quantity $M$ of contaminant. The wind velocity is $U$ towards the shoreline. The contaminant diffuses at a rate of $D$. The release occurs at the water surface, which is modelled as a perfect reflector.
Here, the wind velocity is chosen along the $... | Give the equation for the three-dimensional time-evolution of contaminant concentration. State the value of the maximum concentration $C_{\max}$ at time $t=10\ \mathrm{s}$ . Repeat this calculation for every $10\ \mathrm{s}$ until $100\ \mathrm{s}$ . From these values, sketch the maximum concentration and its loc... | 9 | 9 | 0 | 182 | 194 | 579 | 0 | 248 | 0 | 119 | 180 | 110 | 71 | 1 | 0 | 3,083 | 3 | 2.666667 | 2,055.333333 | 2 | 37 | 109 | 4 | 1,211 | |
25a0a40c-6c6f-4ff3-8520-ec42cb4636f8 | 3 | 0 | 0 | 4 | Boundary Layer Processes | Question 2 | 4 | 1 | Consider a river stream with the specific flow rate $q=Uh=1.60\ \mathrm{m^2/s}$, and a water depth of $h=2.0\ \mathrm{m}$. The sediment size at the river bed is described by $d_{50}=0.85\ \mathrm{mm}$.
| 0.333333 | 15 | Compute the roughness scale, $z_0$.
Compute the coefficient of friction, $C_f$.
Compute the bed shear stress, $\tau_0$.
| 3 | 3 | The roughness scale is given by
$$
z_0 = \begin{cases}
\dfrac{\nu}{9u_*} \qquad & \text{for smooth turbulent flow, } \mathrm{Re_{k}} = \dfrac{u_* k_s}{\nu} < 10 \\
\dfrac{k_s}{30} \qquad &\text{for rough turbulent flow } \mathrm{Re_k} = \dfrac{u_* k_s}{\nu} >10
\end{cases}
$$
***
We start assuming rough turbulent fl... | Question 2 | 1 | 3 | 3 | Consider a river stream with the specific flow rate $q=Uh=1.60\ \mathrm{m^2/s}$, and a water depth of $h=2.0\ \mathrm{m}$. The sediment size at the river bed is described by $d_{50}=0.85\ \mathrm{mm}$.
Compute the roughness scale, $z_0$.
Compute the coefficient of friction, $C_f$.
Compute the bed shear stress, $\tau_... | Compute the roughness scale, $z_0$ . Compute the coefficient of friction, $C_f$ . Compute the bed shear stress, $\tau_0$ . | 3 | 3 | 3 | 50 | 87 | 998 | 0 | 190 | 0 | 20 | 18 | 20 | 18 | 0 | 0 | 2,038 | 8 | 1 | 679.333333 | 2.666667 | 25 | 297 | 3 | 275 | |
25c603f5-111c-4cca-b597-0999c8615452 | 3 | 0 | 0 | 10 | Creep | Creep Damage Fraction | 8 | 3 | A steel is used in a steam turbine pipe which operates at $550^\circ\text{C}$. The creep rupture properties of the steel at this temperature are given by:
   
$$
t_r=1\times10^{22}~\sigma^{-6}
$$
   
Where $t_r$ is the rupture life in hours and $\sigma$ is the applied stress in $\text{MPa}$. Th... | 0.666667 | 15 | Assuming it operates continuously at a fixed pressure and temperature, calculate the maximum pressure that can be applied to the pipe for rupture to be avoided in 20 years.
The pressure in the pipe can fluctuate, such that in a given week it can spend the following times at the following pressures at a constant temper... | 2 | 2 | Starting from the equation provided in the question:
   
$$
t_r=1\times10^{22}\sigma^{-6}
$$
   
***
Converting 20 years into hours to find $t_r$, then solve to find $\sigma$.
   
$$
20\times365\times24=1\times10^{22}\times\sigma^{-6} \\
\sigma=620.5\text{ MPa}
$$
   
**... | Creep Damage Fraction | 1 | 4 | 2 | A steel is used in a steam turbine pipe which operates at $550^\circ\text{C}$. The creep rupture properties of the steel at this temperature are given by:
   
$$
t_r=1\times10^{22}~\sigma^{-6}
$$
   
Where $t_r$ is the rupture life in hours and $\sigma$ is the applied stress in $\text{MPa}$. Th... | Assuming it operates continuously at a fixed pressure and temperature, calculate the maximum pressure that can be applied to the pipe for rupture to be avoided in 20 years. $ table Using the life fraction rule, calculate the creep damage fraction accumulated in a week. Therefore, assuming that the pressure fluctuation... | 3 | 4 | 2 | 172 | 189 | 792 | 0 | 166 | 0 | 99 | 94 | 65 | 0 | 0 | 1 | 1,633 | 6 | 1.333333 | 1,088.666667 | 4 | 27 | 172 | 3 | 1,036 |
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