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001f060c-6b7e-4705-a37d-f740a61334a2 | 4 | 0 | 0 | 2 | 1 | 2 | 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.
  $.
\nWhat is the difference between the values in parts 5b and 5a? Why is t... | 4 | 0.666667 | 3 | 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.
\nWe cannot simply calculate the hydrostatic pr... | 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.
\nWe cannot simply calculate the hydrostatic pressure difference between the two points. This is because the water is flowing, meaning that the pressure is not... | 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... | $$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
x+y&=4\, .
\end{aligned}
$$
\n$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
4x+3y+z&=1\, .
\end{aligned}
$$
\n$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
2x+2y+2z&=12\, .
\end{aligned}
$$
\n$$
\begin{aligned}
-x+2y-2z&=1, \\
4x-y+6z&=2\, ,\\
2x+3y+2z&=4\, ,\\
... | 4 | 0.666667 | 3 | Before starting Gaussian Elimination, ensure you have answered *Question 1* to identify the type of solution that you will expect.
***
Set-up the augmented matrix (**section 2.8**)...
***
... Attempt to manipulate the matrix into triangular form, e.g. :
$$
\left( \begin{array}{ccc|r} a & b & c & \ d\\ 0 & e & f ... | 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:
$$
\left( \begin{array}{ccc|r} 1 & 0 & 0 & \ a\\ 0 & 1 & 0 & b\\ 0 & 0 & 1 & c \end{array} \right)
$$
... | **(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... | 77 | 4 | 27 | 27 | 350 | 44 | 7 | 4 | 0 | 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. \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... | 4 |
012cc4b4-bb47-4ac4-a702-370084cef4cd | 1 | 2 | 0 | 13 | 4 | 2 | 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 ... | Find the angular velocity $\Omega$ that the gyroscope rotor should spin at if the car is travelling with velocity $v = 15~\mathrm{m/s}$.
\nIn which direction should the gyroscope spin?
\nObserve the two diagrams below:
### **Diagram A**
***
Centripetal moment applied to the car:
$$
M=\cfrac{{mv}^2h}{R}~~\mathrm{(Equation~1)}
$$
***
Where:
$$
v=\omega_3R~~\mathrm{(Equation~2)}
$$
**... | 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 ... | 190 | 5 | 8 | 8 | 86 | 2 | 53 | 2 | 3 | The situation is modelled as follows: In the diagram, the coordinate system is at the centre of the car's circular motion. 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 tw... | 4 |
01e1b7d0-d39c-4cdf-8200-05e95de00232 | 4 | 0 | 0 | 6 | 4 | 1 | 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 | 0.666667 | 3 | \n\nRefer to your thermodynamics problem sheets.
| \nThe 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.
$ 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 $\left(\frac{\partial y}{\partial x}\right)_{z}$, $\left(\frac{\partial x}{\partial y}\right)_{z}$, $\left(\frac{\partial y}{\partial x}\right)_{r}$, $\left(\frac{\partial x}{\partial y}\right)_{r}$. 
\nFor $z=x^2-y^2$ confirm the cyclic ru... | 2 | 0.666667 | 2 | 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... | 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:
$$
\left(\dfrac{\partial y}{\partial x}\right)_z = \frac{x}{(x^2-z)^{1/2}} = \frac{x}{y}
$$
***
$$
x(y,z)=(y^2+z)^{1/2}
$$
***
$$
\therefore\left(\dfrac{\par... | In ordinary derivatives a function $y(x)$ implicitly defines $x=x(y)$ and their derivatives obey:
$$
\frac{dx}{dy} = \frac{1}{\left(\dfrac{dy}{dx}\right)}
$$
(check e.g. for $y=x^2$). In higher dimensions a similar relation holds, but it is important to keep track of which variable is being kept constant. The correct... | 94 | 14 | 26 | 26 | 111 | 16 | 27 | 8 | 0 | Confirm the above for $z=x^2-y^2$, $r=\sqrt{x^2+y^2}$ by calculating $\left(\frac{\partial y}{\partial x}\right)_{z}$, $\left(\frac{\partial x}{\partial y}\right)_{z}$, $x=x(y)$0, $x=x(y)$1. | 1 |
0315f302-37e4-420c-89aa-1674824e3cf5 | 4 | 0 | 0 | 15 | 5 | 0 | 3 | 2 | The acid dissociation constant, $K_{a}$ , is a measure of the strength of an acid ($\mathrm{HA}$), and is defined as the equilibrium constant for the reaction:
$$
\text{HA} \rightleftharpoons \text{H}^{+}+\text{A}^{-}
$$
It has a value of:
$$
K_{a}=\frac{[H^+][A^-]}{[HA]}
$$
To create a buffer solution, we can mix ... | Rearrange the first equation to give an expression for $[H^+]$ in terms of $K_a$, $[A^-]$ and $[HA]$
\nTake the base-10 logarithm of this equation, and simplify it, so the right hand side contains one term in $K_{a}$ and a second in $[HA]$ and $[A^-]$ . Enter $\log_{10}(x)$ as 'log10(x)'.
\nUsing the definitions of ... | 4 | 1 | 2 | \n\n\n | $$
[H^+]=\frac{K_{a} \cdot [HA]}{[A^{-}]}
$$
\n$$
\log_{10}[H^+]=\log_{10}(\frac{K_a[HA]}{[A^-]})=\log_{10} K_a+\log_{10}(\frac{[HA]}{[A^{-}]})
$$
\nFrom the previous part:
$$
\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... | 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 ... | 202 | 23 | 11 | 11 | 34 | 0 | 111 | 18 | 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 together s... | 6 |
03568619-d5de-4cc4-bbca-ed5fca655d1a | 0 | 0 | 2 | 4 | 3 | 3 | 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-... | Assuming that there is only molecular diffusion: $D = 0.202\ \mathrm{cm^{2}/s}$.
\nAssuming the flow is turbulent: $D = 10^4\ \mathrm{cm^{2}/s}$.
| 2 | 1 | 3 | \n | 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... | 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-... | 80 | 4 | 3 | 3 | 151 | 0 | 16 | 2 | 1 | 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. Assuming that there is only molecular diffusion: $D = 0.202\ \mathrm{cm^{2}/s}$. Assuming the flow is turbulent: $D ... | 3 |
037b6b33-24e8-4efa-a376-2e2d9e5261a7 | 0 | 0 | 1 | 9 | 4 | 2 | 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.).
| 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 | 0.333333 | 1 | null | A non-inverting op-amp stage appears as follows:
  
***
For a non-inverting op-amp, the gain can be calculated as follows:
  
$A_... | 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... | 58 | 7 | 4 | 4 | 112 | 0 | 32 | 2 | 0 | $180~ \Omega$, $18 ~\mathrm{k}\Omega$, etc.. Design a non-inverting amplifier with a gain of $263$, using fixed E12 resistors. Remember that resistors can be connected in parallel. | 2 |
03baff5d-e329-4336-8cd5-a97e93fc94a9 | 2 | 0 | 0 | 11 | 4 | 2 | 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. 
| If the blade speed is $350 ~\mathrm{m/s}$, calculate the velocity and flow angle relative to the rotor blades.
| 1 | 0.333333 | 2 | null | Draw a velocity triangle, for which:
***
$C_2 = 250~\mathrm{m/s}$
  
$\alpha = 17^{\circ}$
  
$U = 350~\mathrm{m/s}$
***
where... | 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.
| 42 | 3 | 14 | 14 | 126 | 0 | 18 | 1 | 0 | If the blade speed is $350 ~\mathrm{m/s}$, calculate the velocity and flow angle relative to the rotor blades. | 1 |
0592be77-e032-469d-bfca-4d9408f3632b | 1 | 1 | 0 | 9 | 4 | 2 | 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... | 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 | 3 | null | 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:
  
$K = \frac{K_\mathrm{t}}{K_\mathrm{f}R_\mathrm{a}+K_\mathrm{e}K_\mathrm{t}}$
  
$\tau = \frac{JR_\mathrm{a}}{K_\mathrm{f}R_\mathrm{a}+K_\mathrm{e}K_\mathrm{t}}$
*... | 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... | 72 | 9 | 45 | 45 | 298 | 0 | 16 | 5 | 1 | What is the value of $K_\mathrm{P}$ needed to sustain the speed within $1\%$ of the desired value $100~\mathrm{rad/s}$? | 1 |
05b6833a-94f9-46a4-8e93-018225fb5cd0 | 0 | 0 | 1 | 9 | 4 | 2 | 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.
| 
| 1 | 0.666667 | 2 | null | 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:... | 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... | 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.
\nThe specific humidity of the air.
| 30 | 3 | 10 | 10 | 102 | 0 | 13 | 0 | 0 | Determine: The partial pressure of the dry-air component. | 1 |
061a2ba6-c482-4891-be40-1a18f21b89bf | 6 | 0 | 0 | 15 | 5 | 0 | 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 ($\mathrm{mM}$)
* $[S]$ ... | 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?
\nNext, we will prove an important property of the Michalis-Menten equation. Find the value of $v$ when $[S]$=$K_{M}$.
\nThis graph is not very user-friendly as far as extractin... | 6 | 1 | 2 | \n\n\n\n\n | 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 ($\mathrm{mM}$)
* $[S]$ ... | 253 | 27 | 0 | 0 | 158 | 0 | 209 | 18 | 0 | The Michaelis-Menten equation describes enzyme kinetics. What is the horizontal limit in terms of its variables? Find the value of $v$ when $[S]$=$K_{M}$. Convert the Michaelis-Menten equation into the Lineweaver-Burk equation by taking reciprocal of both sides, so that the $\mathrm{mmol \cdot s^{-1}}$0-variable become... | 13 |
06696572-f255-45b9-bafa-cabbe184e559 | 2 | 1 | 0 | 4 | 3 | 3 | 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... | Compute the bed shear stress, $\tau_0$ $\mathrm{[N/m^2]}$.
\nWhat is the critical bed shear stress, $\tau_{cr}\ \mathrm{[N/m^2]}$, for this channel?
\nIs the channel bed stable in terms of initiation of motion for this flow?
| 3 | 0.333333 | 1 | \n\n | The bed sear stress is computed as: $\tau_0 = \rho g h S = 39.240\ \mathrm{N/m^2}$.
\nThe 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
$$
\mathrm{Re}_* = \dfrac{u_* d_{50}}{\nu} = \dfrac{0.198\cdot 2 \ti... | A wide freshwater stream has a smooth granular bed with a bed slope of $S = 0.002$, a uniform flow depth of $h =2.0\ \mathrm{m}$ and a median grain size of $d_{50} =2\ \mathrm{mm}$.
You may wish to use [the Shields diagram](https://bb.imperial.ac.uk/webapps/blackboard/execute/content/file?cmd=view\&content_id=_2543151... | 70 | 6 | 10 | 10 | 104 | 0 | 34 | 3 | 0 | You may wish to use . Compute the bed shear stress, $\tau_0$ $\mathrm{[N/m^2]}$. What is the critical bed shear stress, $\tau_{cr}\ \mathrm{[N/m^2]}$, for this channel? Is the channel bed stable in terms of initiation of motion for this flow? | 4 |
071be152-c0bf-4be1-8a82-3514e08b18ea | 1 | 0 | 0 | 19 | 6 | 1 | 0 | 4 | null | (L2) A weight of mass 10 kg is attached to a wall with a string, and is pulled horizontally with a force $\vec{F}$ so that it is in equilibrium, as in the diagram. Find the magnitude of the force $F$ required for the string to make an angle of $\theta=60^\circ$ to the *normal* to the wall.
***
From the triangle, we see that $\tan \theta = \dfrac{W}{F} \q... | null | 1 | 0 | 9 | 9 | 50 | 1 | 56 | 3 | 0 | (L2) A weight of mass 10 kg is attached to a wall with a string, and is pulled horizontally with a force $\vec{F}$ so that it is in equilibrium, as in the diagram. Find the magnitude of the force $F$ required for the string to make an angle of $\theta=60^\circ$ to the *normal* to the wall.
**: The matrices below represent rotations in $\mathbb{R}^3$ about the $x$-axis ($\text{R}_1$) and about the $y$-axis ($\text{R}_2$), each by 90$^\circ$ in the counter-clockwise direction:
$$
\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 $\lambda_1$, $\lambda_2$ respectively). Determine the normalised eigenvectors corresponding to the eigenvalues.
\nFind the products ${\mathbf{\text{R}}}_1{\mathbf{\text{R}}}_2$ and ${\mathbf{\text{R}}}_2{\mathbf{\text{R}}}_1$ and show they do not commu... | 3 | 0.666667 | 3 | 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 $\mathbf{\underline{x}}=(x,y,z)$... | 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 ... | **(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:
$$
\text{R}_1=
\left(\begin{array}{ccr}
1&\hskip12pt 0&\hskip3pt 0\\
0&\hskip12pt 0&\hskip3pt -1\\
0&\hskip12pt 1&\hskip3pt... | 93 | 16 | 38 | 38 | 278 | 13 | 61 | 9 | 0 | $ Find the real eigenvalues of $\text{R}_1$ and $\text{R}_2$ denoted $\lambda_1$, $x$0 respectively. Determine the normalised eigenvectors corresponding to the eigenvalues. Find the products $x$1 and $x$2 and show they do not commute. Determine the rotation axes of $x$1 and $x$2 represent these as normalised eigenvecto... | 4 |
0b9fe9b9-efd5-40d5-a797-1efe86da2328 | 1 | 0 | 4 | 14 | 4 | 2 | 8 | 1 | Using three-dimensional Cartesian coordinates write all the components of: | $$
\vec{\mathcal{T}}^{\left(\hat{n}\right)}=\underline{\underline{{\sigma}}}^T\hat{n}
$$
\n$$
\vec{\mathcal{T}}^{\left(\hat{\mathrm{e}}_y\right)}=\underline{\underline{\sigma}}^T\hat{\mathrm{e}}_y
$$
\n$$
\mathrm{div}\left(\vec{\mathcal{T}}^{\left(\hat{\mathrm{e}}_y\right)}\right)
$$\n$$
\overrightarrow{\mathrm{div}}\... | 5 | 0.5 | 2 | \n\n\n\n | \n\n\n\n | Using three-dimensional Cartesian coordinates write all the components of:$$
\vec{\mathcal{T}}^{\left(\hat{n}\right)}=\underline{\underline{{\sigma}}}^T\hat{n}
$$
\n$$
\vec{\mathcal{T}}^{\left(\hat{\mathrm{e}}_y\right)}=\underline{\underline{\sigma}}^T\hat{\mathrm{e}}_y
$$
\n$$
\mathrm{div}\left(\vec{\mathcal{T}}^{\le... | 18 | 5 | 0 | 0 | 1 | 0 | 9 | 5 | 0 | 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(\hat{\m... | 1 |
0ba3880d-0abb-480d-a028-280c196df8ca | 1 | 0 | 0 | 17 | 6 | 1 | 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... | Distance $x_\text{w}$ travelled by woman relative to lake:
| 1 | 0.666667 | 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 $v_{\text{p}}$.
***
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... | 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... | 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... | 105 | 1 | 14 | 14 | 194 | 12 | 8 | 1 | 0 | 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 centre of mass may see an easier way to do this problem, but it is good practice to try it using conservation of momentum only. See Problem Sheet 4, Question 3 for the cen... | 3 |
0ba7ab1f-f9af-439a-b379-e4b2a1fb959f | 1 | 0 | 0 | 9 | 4 | 2 | 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 $R_2 = 4~\mathrm{k\Omega}$.
| 
| 1 | 0.666667 | 1 | null | 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 $s = j\omega$:
&... | 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 $10^{11}$.
| 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?
\nGiven that the tower is 60 m high, what is the separation of the two balls when... | 3 | 0.666667 | 3 | 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.... | 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.
\nT... | 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... | 119 | 6 | 25 | 25 | 312 | 10 | 87 | 3 | 0 | Using a torsion balance, Roll, Krotkov and Dicke Ann. 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... | 7 |
0bf0e821-13b2-4de7-9c7c-2c2cf4b5c13b | 6 | 0 | 0 | 8 | 4 | 2 | 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 $=0$.
| What are the maximum hoop, radial and shear stresses, if the axial stress is zero **when treated as a thin cylinder**.
\nWhat 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_\t... | 2 | 0.333333 | 3 | \n | 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:
   
$$
\sigma_r=\boxed{0 \text{ MPa}}
$$
... | 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... | 94 | 7 | 27 | 27 | 215 | 0 | 55 | 1 | 0 | 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, 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 a... | 4 |
0c44ba33-2e71-4e4f-9ea3-d60b94200685 | 1 | 1 | 0 | 17 | 6 | 1 | 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... | 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 | 0.666667 | 4 | 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... | 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... | 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... | 145 | 9 | 38 | 38 | 354 | 10 | 145 | 9 | 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 circumference ... | 5 |
0c714910-48e8-4c59-bed0-de230105e923 | 3 | 0 | 0 | 15 | 5 | 0 | 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... | Rearrange the second equation, $Q = f S$ so that we could work out stroke volume when cardiac output and heartbeat frequency are known.
\nRearrange 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 ... | 3 | 1 | 1 | \n\n | $$
Q=S\cdot f
$$
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
$$
S=\frac{Q}{f}
$$
\n$$
S=\frac{Q}{f}
$$
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 f... | 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... | 197 | 20 | 14 | 14 | 46 | 0 | 100 | 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 blood $\mathrm{mL \, O_{2} ... | 5 |
0c82de4f-4a34-49dd-9d48-1605e94b1af5 | 1 | 1 | 0 | 2 | 1 | 2 | 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... | 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 | 2 | 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... | 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 ... | 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... | 105 | 3 | 6 | 6 | 179 | 2 | 105 | 3 | 0 | Can you help Damien to resolve this apparent paradox? | 1 |
0ce1cf22-a410-4a4f-8800-780ed360edeb | 0 | 0 | 2 | 14 | 4 | 2 | 16 | 4 | Consider a turbulent and incompressible flow. | Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that:
$$
\frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}+\frac{\partial \bar{w}}{\partial z} = 0.
$$\nShow that the velocity fluctuations also satisfy the continuity equation, that is:
$$
\frac{... | 2 | 0.5 | 2 | \n | \n | Consider a turbulent and incompressible flow.Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that:
$$
\frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}+\frac{\partial \bar{w}}{\partial z} = 0.
$$\nShow that the velocity fluctuations also satisfy... | 34 | 2 | 0 | 0 | 1 | 0 | 29 | 2 | 0 | Consider a turbulent and incompressible flow.Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that: $ \frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}+\frac{\partial \bar{w}}{\partial z} = 0. | 1 |
0d2632f2-1c09-48e8-b0ec-13b3bf89315c | 1 | 0 | 0 | 15 | 5 | 0 | 6 | 4 | An ecologist is using a clinometer to estimate the height (and therefore the biomass) of a tree. A clinometer is a tube attached to a protractor with a weighted string (see the diagram below), which can be used to measure angles between the ground and a distant point. The ecologist stands exactly $10 \, \mathrm{m}$ fro... | What is the *overall* height of the tree to the closest metre?
| 1 | 1 | 1 | null | $$
1\,\mathrm{m} + 10\,\mathrm{m} \times \tan(90\degree -42\degree) = 12.1\,\mathrm{m}
$$
| An ecologist is using a clinometer to estimate the height (and therefore the biomass) of a tree. A clinometer is a tube attached to a protractor with a weighted string (see the diagram below), which can be used to measure angles between the ground and a distant point. The ecologist stands exactly $10 \, \mathrm{m}$ fro... | 110 | 3 | 1 | 1 | 1 | 0 | 10 | 0 | 1 | An ecologist is using a clinometer to estimate the height and therefore the biomass of a tree. A clinometer is a tube attached to a protractor with a weighted string see the diagram below, which can be used to measure angles between the ground and a distant point. What is the overall height of the tree to the closest m... | 3 |
0d36ee3e-25fd-49a8-9452-894589501cb0 | 0 | 4 | 0 | 19 | 6 | 1 | 5 | 0 | **(L7/8)**: For each of the following sets of simultaneous equations, determine if there is:
1. A unique solution
2. No solutions
3. An infinity of solutions
| $$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
x+y&=4\, .
\end{aligned}
$$
\n$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
4x+3y+z&=1\, .
\end{aligned}
$$
\n$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
2x+2y+2z&=12\, .
\end{aligned}
$$
\n$$
\begin{aligned}
-x+2y-2z&=1, \\
4x-y+6z&=2\, ,\\
2x+3y+2z&=4\, ,\\
... | 4 | 0.333333 | 3 | 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... | (**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. 
***
... | **(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
$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
x+y&=4\, .
\end{aligned}
$$
\n$$
\begin{aligned}
x+y+z&=6\, ,\\
2x+y-z&=3\, ,\\
4x+3y+z&=1\, .
\end{aligned}... | 33 | 4 | 57 | 57 | 650 | 16 | 7 | 4 | 0 | L7/8: For each of the following sets of simultaneous equations, determine if there is: 1. \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\, ,\\ 2... | 4 |
0d4b9dc3-b242-493c-982a-e08fefe374f5 | 0 | 0 | 1 | 11 | 4 | 2 | 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... | $\frac{\dot{B}_\mathrm{destroyed}}{\dot{m}c_\mathrm{p}T_0} = \frac{\dot{I}}{\dot{m}c_\mathrm{p}T_0} = \mathrm{ln}(1+K_1\Delta T + K_2(\Delta T)^2)$
  
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 | 2 | null | 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... | 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... | 125 | 11 | 15 | 15 | 81 | 0 | 9 | 3 | 0 | 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... | 3 |
0d81759e-b284-40f7-854b-fce603b34a62 | 1 | 0 | 0 | 13 | 4 | 2 | 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 $... | 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 | 0.666667 | 3 | 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... | Free body diagram and kinematic diagram:
***
Starting with the equation for angular momentum about a general point:
$$
\begin{aligned}
H_O=I_G\do... | 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 $... | 88 | 6 | 11 | 11 | 99 | 3 | 40 | 1 | 1 | 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 | 3 |
0da53862-6330-4f62-a563-840b35c837dd | 2 | 0 | 0 | 14 | 4 | 2 | 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 ... | Using dimensional analysis find the non-dimensional parameters which govern this observed behavior.\nUsing 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 | 0.5 | 2 | \n | \n | 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 ... | 105 | 7 | 0 | 0 | 1 | 0 | 61 | 3 | 0 | 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. Using dimensional analysis find the non-dimensional parameters which govern this observed behavior. Using experimental data given in the table, ... | 3 |
0df0dd8e-3185-427a-9f46-f4e28cddca81 | 1 | 0 | 0 | 10 | 4 | 2 | 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 $150\text{ MPa}$:
   
$ 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. 
\nFind the position at which the particle strikes the wall. 
\nFind the velocity of the particle after the collision with the wall, assuming the collision is elastic.
| 3 | 0.666667 | 2 | $\vec{r}(t) = \vec{r}_0 + \vec{v}t$
\nYour 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.
... | $$
\vec{r}(t) = \vec{r}_0 + \vec{v}t=\begin{pmatrix}0\\1\\0\end{pmatrix}+\begin{pmatrix}2\\1\\3\end{pmatrix}t=\begin{pmatrix}2t\\1+1t\\3t\end{pmatrix}
$$
\nFrom 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
& \Rightarro... | 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... | 89 | 3 | 12 | 12 | 144 | 4 | 39 | 0 | 0 | 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 the particle. Give an expression for the trajectory of the particle before the collision. Find th... | 5 |
10703359-bbd8-4c73-b85e-57c27e140d16 | 3 | 0 | 0 | 15 | 5 | 0 | 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):
... | Rearrange the left-hand equation for $\lambda$
\nCalculate 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=18... | 3 | 1 | 1 | \n\n | $\lambda=\frac{-\ln{\frac{L}{L_0}}}{t}=\frac{\ln{L_0}-\ln{L}}{t}$
There are lots of other ways of writing this!
\nHalf-lives are $10$ and $0.5$
$10/0.5= 20$
| mRNA | Calculation | $H$ |
| :--------- | :-----------------------------------------------------... | 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):
... | 225 | 16 | 10 | 10 | 47 | 0 | 131 | 9 | 3 | 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 $H$ Half-l... | 3 |
10cb0c12-9685-4f7f-9bc4-fe7f3022a872 | 3 | 0 | 0 | 8 | 4 | 2 | 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 ... | Calculate the transmitted torque.
\nCalculate the applied bending moment.
\nCalculate the end thrust.
| 3 | 1 | 4 | \n\n | First draw a diagram of the problem:
  
The strain gauges readings pre rotation:
   
$$
e_{45}=-0.00060 \hspace{30pt} e_{-45}=+0.00030
... | Two strain gauges are fitted at $\pm45^{\circ}$ to the axis of a $75\text{ mm}$ diameter steel shaft. The shaft is rotating, and in addition to transmitting power, it is subjected to an unknown bending moment and a direct thrust. The readings of the gauges are recorded, and it is found that the maxima or minima values ... | 122 | 8 | 58 | 58 | 491 | 0 | 13 | 0 | 0 | The readings of the gauges are recorded, and it is found that the maxima or minima values for each gauge occur at $180^{\circ}$ intervals of shaft rotation and are $-0.00060$ and $+0.00030$ for the two gauges at one instant, and $-0.00050$ and $+0.00040$ for the same gauges $180^{\circ}$ of rotation later. Assume that ... | 5 |
117dcbfd-a250-41f3-8eeb-4be2993a67c3 | 0 | 1 | 0 | 15 | 5 | 0 | 1 | 1 | The binding of calcium ions to the signalling protein calmodulin can be modelled by the equation:
$$
\theta=\frac{C^{n}}{K_{d}+C^n}
$$
Where:
* $\theta$ is the fraction of calmodulin molecules bound to calcium 
* $C$ is the concentration of calcium
* $n$ is a constant, greater than zero
* $K_{d}$ is... | Consider a plot of $\theta$ ($Y$-axis) against $C$ ($X$-axis).
Which one of the following is true?
| 1 | 1 | 0 | null | As $C$ becomes very large compared to $K_d$, $θ$ tends to $C^{n} / ( C^{n} ) = 1 $
| The binding of calcium ions to the signalling protein calmodulin can be modelled by the equation:
$$
\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... | 81 | 9 | 4 | 4 | 13 | 0 | 20 | 4 | 0 | Which one of the following is true? | 1 |
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