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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, : | 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 that the gyroscope rotor should spin at if the car is travelling with velocity . 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 gyroscope? | 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)}
$$
(check e.g. for $y=x^2$). In higher dimensions a similar relation holds, but it is important to keep track of which variable is being kept constant. The correct... | Confirm the above for , by calculating , , 0 , 1 . For confirm the cyclic rule for partial derivatives 3 | 2 | 1 | 0 | 94 | 650 | 1,223 | 146 | 111 | 113 | 27 | 386 | 30 | 156 | 0 | 0 | 1,685 | 13 | 1.333333 | 1,123.333333 | 8.666667 | 45 | 150 | 2 | 1,044 |
0315f302-37e4-420c-89aa-1674824e3cf5 | 4 | 0 | 0 | 15 | Maths for Life Scientists 04 - Logarithms | Henderson-Hasselbalch derivation | 3 | 2 | The acid dissociation constant, $K_{a}$ , is a measure of the strength of an acid ($\mathrm{HA}$), and is defined as the equilibrium constant for the reaction:
$$
\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, , is a measure of the strength of an acid , and is defined as the equilibrium constant for the reaction: It has a value of: To create a buffer solution, we can mix together solutions of a weak acid and its conjugate base usually supplied using a salt of the weak acid. To calculat... | 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’: and and factors of larger or smaller e.g , , etc.. Design a non-inverting amplifier with a gain of , 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 , 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 needed to sustain the speed within of the desired value ? | 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 ($\mathrm{mM}$)
* $[S]$ ... | Velocity of enzyme-catalysed reaction Maximum rate of the reaction Michaelis constant Concentration of substrate Sketch the graph of this equation: is the 0 -variable, is the 2 -variable. What is the horizontal limit in terms of its variables? Next, we will prove an important property of the Mic... | 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
$$
\mathrm{Re}_* = \dfrac{u_* d_{50}}{\nu} = \dfrac{0.198\cdot 2 \tim... | Question 1 | 0 | 3 | 3 | A wide freshwater stream has a smooth granular bed with a bed slope of $S = 0.002$, a uniform flow depth of $h =2.0\ \mathrm{m}$ and a median grain size of $d_{50} =2\ \mathrm{mm}$.
You may wish to use [the Shields diagram](https://bb.imperial.ac.uk/webapps/blackboard/execute/content/file?cmd=view\&content_id=_2543151... | Compute the bed shear stress, . What is the critical bed shear stress, , for this channel? Is the channel bed stable in terms of initiation of motion for this flow? | 3 | 2 | 0 | 70 | 103 | 391 | 0 | 104 | 0 | 34 | 55 | 34 | 54 | 0 | 0 | 827 | 4.5 | 1 | 275.666667 | 1.5 | 22 | 111 | 3 | 490 | |
071be152-c0bf-4be1-8a82-3514e08b18ea | 1 | 0 | 0 | 19 | Vector Algebra | Force Balance | 0 | 4 | null | 0.333333 | 5 | (L2) A weight of mass 10 kg is attached to a wall with a string, and is pulled horizontally with a force $\vec{F}$ so that it is in equilibrium, as in the diagram. Find the magnitude of the force $F$ required for the string to make an angle of $\theta=60^\circ$ to the *normal* to the wall.
 for $T$:
$$
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 axis, then the velocity transformation formula needs to be written as Use the same method as in the lecture to work out the transformation rule for one of the perpendicular velocity components, e.g . | 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... | \text{R}_1 \text{R}_2 \lambda_1 x x x x x x$5 , and comment on your results in light of the result in part b. | 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: | 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 is approxim... | 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: . | 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 to radians. Using energy conservation and the concept of centripetal acceleration, find a formula for the minimum angle of release required for a smalle... | 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, 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 consumed and the heart... | 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 for about the -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.
$$
Show that the velocity fluctuations also satisfy ... | Consider a turbulent and incompressible flow.Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that: Show that the velocity fluctuations also satisfy the continuity equation, that is: | 3 | 3 | 0 | 34 | 219 | 0 | 0 | 0 | 0 | 29 | 243 | 34 | 215 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 12 | 0 | 2 | 421 | ||
0d2632f2-1c09-48e8-b0ec-13b3bf89315c | 1 | 0 | 0 | 15 | Maths for Life Scientists 07 - Trigonometry | Clinometry | 6 | 4 | An ecologist is using a clinometer to estimate the height (and therefore the biomass) of a tree. A clinometer is a tube attached to a protractor with a weighted string (see the diagram below), which can be used to measure angles between the ground and a distant point. The ecologist stands exactly $10 \, \mathrm{m}$ fro... | 1 | 7 | What is the *overall* height of the tree to the closest metre?
| 1 | 1 | $$
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 or irreversibility can be expressed in dimensionless form by: where and 0 | 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 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 , spinning at at a forward velocity of , while operating at an altitude where the density is ha... | 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 $150\text{ MPa}$:
   
![](https://lambda-feedb... | null | Larson-Miller Parameter | 1 | 4 | 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. 
   
![](https://lambda-feedback... | 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 for a duration of hours. Stress} 40^\circ\text{C}$ hotter? | 2 | 2 | 0 | 62 | 114 | 372 | 0 | 85 | 1 | 13 | 18 | 36 | 67 | 1 | 0 | 900 | 3 | 0.666667 | 600 | 2 | 17 | 93 | 1 | 547 |
0ea1717d-dcaa-4eec-a76f-2f7a350b1878 | 0 | 1 | 0 | 2 | Fluid Properties | Fluid Properties | 0 | 13 | Select all of the features that describe a turbulent flow.
| 0.333333 | 2 | Select all of the features that describe a turbulent flow.
| 1 | 0 | null | null | Fluid Properties | 0 | 1 | 2 | Select all of the features that describe a turbulent flow.
Select all of the features that describe a turbulent flow.
| Select all of the features that describe a turbulent flow. | 1 | 2 | 0 | 20 | 0 | 0 | 0 | 1 | 1 | 10 | 0 | 10 | 0 | 0 | 0 | 3 | 0 | 0.333333 | 1 | 0 | 0 | 0 | 1 | 98 |
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