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47639b61-2491-48af-a060-e83958a1dabd | 6 | 3 | 0 | 14 | 4 | 2 | 13 | 0 | In the boundary layer equations (Eqns. 14.7), | What limiting value of Mach number is considered?\nWhat limiting value of Reynolds number is considered?\nWhat observation did we make about the boundary layer to start the approximation?\nWhat is the approximate value of $\partial p / \partial y$?\nHow many equations are there? (see worked solutions for a discussion)\... | 9 | 0.333333 | 2 | \n\n\n\n\n\n\n\n | \n\n\n\n\n\n\n\n | In the boundary layer equations (Eqns. 14.7),What limiting value of Mach number is considered?\nWhat limiting value of Reynolds number is considered?\nWhat observation did we make about the boundary layer to start the approximation?\nWhat is the approximate value of $\partial p / \partial y$?\nHow many equations are th... | 96 | 2 | 0 | 0 | 1 | 0 | 90 | 2 | 0 | 14.7,What limiting value of Mach number is considered? What limiting value of Reynolds number is considered? What observation did we make about the boundary layer to start the approximation? What is the approximate value of $\partial p / \partial y$? How many equations are there? see worked solutions for a discussion H... | 10 |
49bd88a1-8d46-4ec3-a6c1-65636eed2f3d | 0 | 1 | 0 | 9 | 4 | 2 | 6 | 18 | Establish if by choosing a capacitor $C_1 = 1~\mathrm{mF}$, the circuit below is underdamped or overdamped:
  
  
| The other component values are as follows:
  
$R_1 = 1~\mathrm{k\Omega}$\
$R_2 = 3~\mathrm{k\Omega}$\
$R_3 = 1~\mathrm{k\Omega}$\
$C_2 = 0.5~\mathrm{mF}$\
$C_3 = 3~\mathrm{mF}$
| 1 | 0.666667 | 2 | null | The circuit in the question is a band-pass filter, comprised of an active low-pass filter followed by an active high-pass filter.
***
The total gain of the circuit is the product of the high-pass filter gain and low-pass filter gain (see the lecture notes section 6.5.5). This can be written as follows:
  
... | Establish if by choosing a capacitor $C_1 = 1~\mathrm{mF}$, the circuit below is underdamped or overdamped:
  
  
The other component va... | 35 | 6 | 12 | 12 | 114 | 0 | 17 | 5 | 1 | Establish if by choosing a capacitor $C_1 = 1~\mathrm{mF}$, the circuit below is underdamped or overdamped: The other component values are as follows: $R_1 = 1~\mathrm{k\Omega}$ $R_2 = 3~\mathrm{k\Omega}$ $R_3 = 1~\mathrm{k\Omega}$ $C_2 = 0.5~\mathrm{mF}$ $C_3 = 3~\mathrm{mF}$ | 1 |
49be2315-31e5-477f-aa2d-6b3f71ee2e17 | 4 | 0 | 1 | 18 | 6 | 1 | 1 | 4 | Five street lamps A, B, C, D, and E are located on a straight line along the $x$ axis at an equal distance apart as shown in the figure below. They turn on at times $t_A$, $t_B$, $t_C$, $t_D$, and $t_E$ respectively, in the frame at rest relative to the ground. These five events are indicated in the spacetime diagram b... | What is the order in which the lamps turn on in the ground rest frame? Rank the events from 1 to 5, with 1 occurring the earliest. If two events are simultaneous give them the same ranking. \nWhat is the order in which the lamps turn on in the car's rest frame?\nWhat is the order in which the light of the lamps re... | 5 | 0.666667 | 3 | In the rest frame, you can just read the times off the graph like you normally would. \nIt might help to go back at look at the solution to 2.4b
***
   
In the stationary frame, simultaneous events are defined as those which lie on a given line parallel to the $x$ axis. How can you use t... | \
\n = t$ (for $t>0$). The transfer function of the process is $H(s) = \frac{3s}{s+4}$.  
| 
| 1 | 0.666667 | 2 | null | Find an expression for $Y(s)$. It can be helpful to label the nodes as in question (5):
  
  
$Y(s) = aH(s)$
  
where $a = U(s... | Determine the time domain response of the closed loop system below when subject to a ramp input $u(t) = t$ (for $t>0$). The transfer function of the process is $H(s) = \frac{3s}{s+4}$.  
 = t$ for $t>0$. | 1 |
4a58c3bc-e81b-4fd8-a811-ca9d3ece0aaa | 2 | 0 | 0 | 9 | 4 | 2 | 6 | 3 | By matching the transfer operators of the op-amp stage and its block diagram representation, derive expressions for A and B.
| 
| 1 | 0.333333 | 1 | null | First write an expression for the op-amp stage. In this case, the op-amp is a summing amp, for which the general expression is:
  
$ v_\mathrm{o} = -R_\mathrm{f}\sum\limits_{n = 1,2,..}^N\frac{v_{\mathrm{i}n}}{R_n} $
***
Applying the expression to this particular op-amp stage:
  
$y = -R_\mat... | By matching the transfer operators of the op-amp stage and its block diagram representation, derive expressions for A and B.
| 21 | 0 | 12 | 12 | 89 | 0 | 1 | 0 | 1 | By matching the transfer operators of the op-amp stage and its block diagram representation, derive expressions for A and B. | 1 |
4a8f7227-9e61-4234-8426-b9843d27761d | 3 | 0 | 0 | 15 | 5 | 0 | 3 | 4 | There is a relationship between the size of an island (or other defined area) and the number of species it contains. This relationship is called (surprisingly enough) the species/area relationship and it is modelled by the following equation:
$$
S=C\cdot A^{z}
$$
* $S$ Number of species
* $A$ ... | Using logs, convert this equation into a linear form showing clearly what the gradient and intercept represent, and what you would need to plot to calculate them. Write $\log(x)$ as 'log(x)'. n this question it doesn't matter what base you use at all.
\nCalculate the values of $C$ and $z$ from this data set using line... | 3 | 1 | 1 | \n\n | Gradient : $z$
Y-intercept : $\log{C}$
Plot $\log{S}$ (Y-axis) against $\log{A}$ (X-axis)
$$
\log S=\log C+z \log A \newline Y=c+mX
$$
\nThe $Y$-intercept is $\log_{10}(C)=0.178$, so $C=10^{0.178} = 1.5$ . If you use a different base, you'll need a different antilog, but the result will be the same, *e.g.* if you ... | There is a relationship between the size of an island (or other defined area) and the number of species it contains. This relationship is called (surprisingly enough) the species/area relationship and it is modelled by the following equation:
$$
S=C\cdot A^{z}
$$
* $S$ Number of species
* $A$ ... | 265 | 11 | 13 | 13 | 64 | 0 | 199 | 6 | 0 | This relationship is called surprisingly enough the species/area relationship and it is modelled by the following equation: $ S=C\cdot A^{z} $ $S$ Number of species $A$ Area of the island $C$ Data specific constant $z$ Data specific constant Using logs, convert this equation into a linear form showing clearly what ... | 4 |
4b789b57-bdcc-4255-833e-d051b488dd5e | 2 | 0 | 0 | 15 | 5 | 0 | 3 | 0 | Most cells produce thousands of different types of mRNA. Some mRNAs are more stable than others. The mRNAs for most proteins possess a ‘life-preserving’ poly-A tail, but mRNAs for the proteins called histones lack this tail and are consequently much shorter-lived than most mRNAs. The total amount of mRNA in a culture o... | Use (natural) logs to derive an equation of the form $y=m \cdot x+c$ . This allows you to calculate the cell growth rate, $k$ , from a graph. (In lambda, you can use $\ln$ or $\log$ to represent natural log: both will work and are treated as synonymous).
\nIf the mass of mRNA quadruples in half an hour, what is the cel... | 2 | 1 | 0 | \n | Start with
$$
M=M_0 e^{kt}
$$
Take natural logs:
$$
\ln(M)=\ln(M_0 e^{kt})
$$
Expand product in RHS using equivalency $\log(xy)=\log(x)+\log(y)$:
$$
\ln(M)=\ln(M_0)+\ln(e^{kt})
$$
Expand power in RHS using equivalency $\log(x^y)=y\log(x)$:
$$
\ln(M)=\ln(M_0)+kt\ln(e)
$$
Simplify product in RHS using identity $\... | Most cells produce thousands of different types of mRNA. Some mRNAs are more stable than others. The mRNAs for most proteins possess a ‘life-preserving’ poly-A tail, but mRNAs for the proteins called histones lack this tail and are consequently much shorter-lived than most mRNAs. The total amount of mRNA in a culture o... | 148 | 10 | 11 | 11 | 42 | 0 | 67 | 5 | 3 | The total amount of mRNA in a culture of rapidly dividing cells increases exponentially: $ M=M_{0}e^{kt} $ $M$ Mass of mRNA embedded $M_{0}$ Initial mass of mRNA embedded $k$ Cell growth rate embedded $t$ time Use natural logs to derive an equation of the form $y=m \cdot x+c$ . This allows you to calculate the cell... | 4 |
4b9be5bc-9941-404b-a653-681eec1b5108 | 1 | 0 | 0 | 24 | 6 | 1 | 0 | 13 | Consider the integral
$$
\ {\cal I} = \iint_R 8xy\left(x^2-y^2\right) e^{x^2 + y^2}\,dx\,dy
$$
where $R$ is the wedge of the circle bounded by $y=0$, $y=x$, and $x^2 + y^2 = 1$. Change to new variables $(u,v)$ given by $u=x^2 - y^2$, $v=x^2 + y^2$ and show that the integral becomes:
$$
{\cal I} = \int_{u=0}^1 \int_... | Consider the integral
$$
\ {\cal I} = \iint_R 8xy\left(x^2-y^2\right) e^{x^2 + y^2}\,dx\,dy
$$
where $R$ is the wedge of the circle bounded by $y=0$, $y=x$, and $x^2 + y^2 = 1$. Change to new variables $(u,v)$ given by $u=x^2 - y^2$, $v=x^2 + y^2$ and show that the integral becomes:
$$
{\cal I} = \int_{u=0}^1 \int_... | 1 | 0.666667 | 2 | Changing variables in integration involves the following:
1. Changing limits
2. Changing the integrand
3. Changing the differential using the Jacobian. 
***
(Step 1) Sketch the region in the $xy$ plane and identify 3 boundaries that describe the region...
***
(Step 1) ... Change these boundaries into $(u... | The region of integration in the $xy$ pane is:
***
Changing each limit into the new variables, starting with $y=x$:
***
$$
\underline{y=x}\implies u=0
$... | Consider the integral
$$
\ {\cal I} = \iint_R 8xy\left(x^2-y^2\right) e^{x^2 + y^2}\,dx\,dy
$$
where $R$ is the wedge of the circle bounded by $y=0$, $y=x$, and $x^2 + y^2 = 1$. Change to new variables $(u,v)$ given by $u=x^2 - y^2$, $v=x^2 + y^2$ and show that the integral becomes:
$$
{\cal I} = \int_{u=0}^1 \int_... | 57 | 11 | 22 | 22 | 121 | 10 | 57 | 11 | 0 | Consider the integral $ \ {\cal I} = \iint_R 8xy\left(x^2-y^2\right) e^{x^2 + y^2}\,dx\,dy $ where $R$ is the wedge of the circle bounded by $y=0$, $y=x$, and $x^2 + y^2 = 1$. Change to new variables $(u,v)$ given by $u=x^2 - y^2$, $v=x^2 + y^2$ and show that the integral becomes: $ {\cal I} = \int_{u=0}^1 \int_{v=u}^1... | 3 |
4c553593-e1b6-4c58-8d58-789fb45a411e | 0 | 0 | 2 | 24 | 6 | 1 | 3 | 6 | For $\Omega = 1/r$ where $r$ is the distance from the origin:
| Show that $\nabla\Omega = -\dfrac{1}{r^2}\mathbf{\hat{r}}$.
\nShow that $\nabla^2 \Omega = 0$
| 2 | 0.333333 | 1 | The gradient in spherical coordinates, ignoring $\theta$ and $\phi$ components is:
$$
\nabla\Omega = \frac{\partial \Omega}{\partial r}\mathbf{\hat{r}} + \frac{1}{r}\frac{\partial \Omega}{\partial \theta}\mathbf{\hat{\theta}} + \frac{1}{r\sin\theta}\frac{\partial \Omega}{\partial \phi}\mathbf{\hat{\phi}}
$$
***
Wha... | $$
\nabla\Omega = \frac{\partial \Omega}{\partial r}\mathbf{\hat{r}} + \frac{1}{r}\frac{\partial \Omega}{\partial \theta}\mathbf{\hat{\theta}} + \frac{1}{r\sin\theta}\frac{\partial \Omega}{\partial \phi}\mathbf{\hat{\phi}}
$$
***
The $\mathbf{\hat{\theta}}$ and $\mathbf{\hat{\phi}}$ are zero since $\Omega=\Omega(r)$.... | For $\Omega = 1/r$ where $r$ is the distance from the origin:
Show that $\nabla\Omega = -\dfrac{1}{r^2}\mathbf{\hat{r}}$.
\nShow that $\nabla^2 \Omega = 0$
| 17 | 4 | 14 | 14 | 51 | 9 | 7 | 2 | 0 | Show that $ abla^2 \Omega = 0$ | 1 |
4c57560c-2216-42ec-bdce-938f4ac71be0 | 0 | 0 | 1 | 4 | 3 | 3 | 1 | 10 | Construct the integral solution to the 1-D diffusion equation of a contaminant with initial concentration
$$
C_0 =
\begin{cases}
1 & \text{if }x<0 \\
0 & \text{if }x\geq 0\,.
\end{cases}
$$
Starting from the fundamental solution for an instantaneous point release, show that the integral can be evaluated to give
... | Construct the integral solution to the 1-D diffusion equation of a contaminant with initial concentration
$$
C_0 =
\begin{cases}
1 & \text{if }x<0 \\
0 & \text{if }x\geq 0\,.
\end{cases}
$$
Starting from the fundamental solution for an instantaneous point release, show that the integral can be evaluated to give
... | 1 | 1 | 2 | null | We approach this problem by building on the fundamental solution for 1-D diffusion in an initially unpolluted environment (i.e. the scenario where $C(x, t = 0) = 0$). It is necessary to account for the non-zero concentration for $x < 0$. Given that we consider diffusion in an unbounded environment, the initial concentr... | Construct the integral solution to the 1-D diffusion equation of a contaminant with initial concentration
$$
C_0 =
\begin{cases}
1 & \text{if }x<0 \\
0 & \text{if }x\geq 0\,.
\end{cases}
$$
Starting from the fundamental solution for an instantaneous point release, show that the integral can be evaluated to give
... | 47 | 3 | 34 | 34 | 281 | 0 | 47 | 3 | 0 | Construct the integral solution to the 1-D diffusion equation of a contaminant with initial concentration $ C_0 = \begin{cases} 1 & \text{if }x<0 \\ 0 & \text{if }x\geq 0\,. \end{cases} $ Starting from the fundamental solution for an instantaneous point release, show that the integral can be evaluated to give $ C = \df... | 3 |
4c93a2e7-0473-454e-bd2f-9deaa7b505b0 | 2 | 0 | 3 | 14 | 4 | 2 | 11 | 1 | Couette flows are the archetype of lubrication-related issues. This problem investigates wall-temperature effects on the friction force. Two infinitely-long and infinitely-wide parallel impermeable walls are separated by a distance $h$ (see the image below). The bottom wall (located at $y = 0$ in the Cartesian frame of... | Choose the appropriate set of equations to solve the problem and show that the component of the velocity normal to the wall is strictly zero, i.e. $v=0$.\nLet $\underline{\underline{\tau}}$ be the viscous stress tensor. Show that $\tau_{xy}$ is the only component of the viscous stress tensor different from zero and tha... | 5 | 1 | 4 | \n\n\n\n | \n\n\n\n | Couette flows are the archetype of lubrication-related issues. This problem investigates wall-temperature effects on the friction force. Two infinitely-long and infinitely-wide parallel impermeable walls are separated by a distance $h$ (see the image below). The bottom wall (located at $y = 0$ in the Cartesian frame of... | 282 | 20 | 0 | 0 | 1 | 0 | 101 | 7 | 1 | A liquid bath assumed to be Newtonian with constant and uniform density $\rho_0$ separates both walls. No pressure gradient is applied in the $x$ direction. The flow is assumed to be steady, fully-developed in $x$ and $z$ with no flow in $z$, i.e. No body force is applied to this flow it is simply driven by the motion ... | 11 |
4cc77538-a3b5-4773-80ca-bf9ede89dc4e | 6 | 0 | 1 | 21 | 6 | 1 | 1 | 1 | Here we will look at how the smoothness of a function affects its Fourier series.
| Find the trigonometric Fourier series for the following three functions.
**Note:** The last function is constructed from two parabolas stuck together. They... | 3 | 0.666667 | 4 | For all of the functions, start by considering whether $f(x)$ is odd or even about $x=0$. This will allow you to set either $a_n$ or $b_n$ to 0 (see **section 2.2**). 
***
Then, for the non-zero term, you may be able to find that it is zero for odd or even $n$. Start by reducing the integration interval to $[0,... | Worked solutions for each function. 
, First, we consider the even/odd nature of $f(x)$ to determine if $a_n$ and $b_n$ can be set to 0 and for which values of $n$.
***
$f(x)$ is even, so $f(x)\sin(nx)$ is odd. 
***
This means that $b_n=0$ for all $n$. We also see that the average is zero, so $a_0=0$ (... | Here we will look at how the smoothness of a function affects its Fourier series.
Find the trigonometric Fourier series for the following three functions.
... | 115 | 6 | 47 | 47 | 647 | 15 | 100 | 6 | 0 | Find the trigonometric Fourier series for the following three functions. Note: The last function is constructed from two parabolas stuck together. How quickly a Fourier series converges depends on the properties of the function. Try to identify the relationship between $p$ and which derivative of the function is discon... | 4 |
4ce870e1-9bbd-4bd9-8c6d-b2afec54d038 | 1 | 0 | 1 | 16 | 6 | 1 | 0 | 3 | For the function $f(x) = x^2(1 - x)^3$ ,
| Find the stationary points of $f(x)$ and determine their nature
\nSketch the graph of $y=f(x)$.
| 2 | 0.666667 | 2 | The stationary points are when $\displaystyle \frac{\mathrm{d}f}{\mathrm{d}x} = 0$. Now can you try finding them?
***
You can use the product rule to find $\displaystyle \frac{\mathrm{d}f}{\mathrm{d}x}$. Then set this to $0$ and factorise to find $x$.
Can you find the stationary points now? How can you determine th... | The stationary points are when $\displaystyle \frac{\mathrm{d}f}{\mathrm{d}x} = 0$. Now can you try finding them?
***
You can use the product rule to find $\displaystyle \frac{\mathrm{d}f}{\mathrm{d}x}$. Then set this to $0$ and factorise to find $x$.
***
$$
\displaystyle \frac{\mathrm{d}f}{\mathrm{d}x} = 2x(1-x)^3... | For the function $f(x) = x^2(1 - x)^3$ ,
Find the stationary points of $f(x)$ and determine their nature
\nSketch the graph of $y=f(x)$.
| 21 | 3 | 11 | 11 | 117 | 10 | 16 | 2 | 0 | For the function $f(x) = x^2(1 - x)^3$ , Find the stationary points of $f(x)$ and determine their nature Sketch the graph of $y=f(x)$. | 1 |
4ceb0f0d-4247-470d-b85d-766f50491227 | 0 | 0 | 1 | 16 | 6 | 1 | 7 | 6 | Locate the stationary points of $f(x, y) = xy(x + y -1)$ and deduce their nature
(i) from a contour sketch \[Sketch contours of the function and indicate regions where $f$ is respectively zero, positive and negative.]
(ii) from the criterion for the second partial derivatives of $f$ with respect to $x$ and $y$.
| Locate the stationary points of $f(x, y) = xy(x + y -1)$ and deduce their nature
(i) from a contour sketch \[Sketch contours of the function and indicate regions where $f$ is respectively zero, positive and negative.]
(ii) from the criterion for the second partial derivatives of $f$ with respect to $x$ and $y$.
| 1 | 1 | 3 | From section 5.5 of the notes, stationary points of a function $f(x,y)$ are where
$$
\begin{aligned}
{\partial f\over\partial x} = 0 = {\partial f\over\partial y}
\end{aligned}
$$
***
You should have found 4 stationary points, which you can find by solving the simultaneous equations that arise from the above equa... | From section 5.5 of the notes, stationary points of a function $f(x,y)$ are where
$$
\begin{aligned}
{\partial f\over\partial x} = 0 = {\partial f\over\partial y}
\end{aligned}
$$
***
Finding the derivatives,
$$
\begin{aligned}
{\partial f\over\partial x} &= 0 \\
2xy+y^2-y &= 0 \\
y(2x+y-1) &=0
\end{aligned}
$$... | Locate the stationary points of $f(x, y) = xy(x + y -1)$ and deduce their nature
(i) from a contour sketch \[Sketch contours of the function and indicate regions where $f$ is respectively zero, positive and negative.]
(ii) from the criterion for the second partial derivatives of $f$ with respect to $x$ and $y$.
| 49 | 5 | 38 | 38 | 354 | 16 | 49 | 5 | 0 | Locate the stationary points of $f(x, y) = xy(x + y -1)$ and deduce their nature i from a contour sketch Sketch contours of the function and indicate regions where $f$ is respectively zero, positive and negative. | 1 |
4d004959-68eb-4c1e-9665-d7cf0c0fe704 | 2 | 0 | 0 | 14 | 4 | 2 | 0 | 2 | A submerged submarine is towed horizontally at a steady speed $U$ in deep still water. An axially-symmetrical wake is formed behind the submarine in which the water velocity may be assumed to vary linearly from $U$ on the axis to zero at a radius of $R$. The variation of the water pressure with depth may be assumed to ... | The drag force $F$ of the submarine.\nThe power $P$ required to tow the submarine. | 2 | 0.5 | 2 | \n | \n | A submerged submarine is towed horizontally at a steady speed $U$ in deep still water. An axially-symmetrical wake is formed behind the submarine in which the water velocity may be assumed to vary linearly from $U$ on the axis to zero at a radius of $R$. The variation of the water pressure with depth may be assumed to ... | 121 | 6 | 0 | 0 | 1 | 0 | 14 | 2 | 1 | An axially-symmetrical wake is formed behind the submarine in which the water velocity may be assumed to vary linearly from $U$ on the axis to zero at a radius of $R$. The variation of the water pressure with depth may be assumed to be unaffected by the presence of the submarine. Using a control-volume analysis, we wan... | 4 |
4db11d69-3efb-44cc-a481-ee3120658046 | 3 | 0 | 0 | 24 | 6 | 1 | 2 | 7 | Find the work done by the force $\vec{F} = (2xy -3)\mathbf{\hat{i}} +x^2\mathbf{\hat{j}}$ in moving an object \[in the $x-y$ plane] from $(1,0)$ to \$(0,1) along each of the following paths:
| The circular arc of radius 1, centre at the origin, from (1,0) to (0,1) \[**Hint:** parameterise this arc in terms of the plane polar angle $\phi$]
\nFrom $(1,0)$ to $(1,1)$ to $(0,1)$, i.e., along two line segments parallel to the axes.
\nShow that in fact the force $\vec{F}$ is conservative. Find the function $\Omega... | 3 | 0.666667 | 2 | First, write down $\vec{F}\cdot d\vec{r}$. 
***
Convert $\vec{F}\cdot d\vec{r}$ to polar coordinates ($\rho=1$)...
***
... This will require you to express $x$ and $y$ as a function of $\phi$. What are $dx$ and $dy$?
***
Convert to a more integrable form and evaluate over the limits of $\phi$.
\nEvaluate ... | 
***
$$
\vec{F}\cdot d\vec{r} = F_x\,dx + F_y\,dy = (2xy-3)dx+x^2dy
$$
Converting to polar coordinates, with $\rho=1$:
***
$$
x=\cos\phi \implies dx = ... | Find the work done by the force $\vec{F} = (2xy -3)\mathbf{\hat{i}} +x^2\mathbf{\hat{j}}$ in moving an object \[in the $x-y$ plane] from $(1,0)$ to \$(0,1) along each of the following paths:
The circular arc of radius 1, centre at the origin, from (1,0) to (0,1) \[**Hint:** parameterise this arc in terms of the plane p... | 35 | 11 | 6 | 6 | 284 | 26 | 59 | 7 | 0 | Find the work done by the force $\vec{F} = (2xy -3)\mathbf{\hat{i}} +x^2\mathbf{\hat{j}}$ in moving an object in the $x-y$ plane from $(1,0)$ to $(0,1) along each of the following paths: The circular arc of radius 1, centre at the origin, from (1,0) to (0,1) \[**Hint:** parameterise this arc in terms of the plane polar... | 3 |
4de7d660-9a1c-486a-9cc3-9f8d98511e4e | 1 | 0 | 0 | 2 | 1 | 2 | 2 | 5 | Acceleration due to gravity on the moon is 1.62 m/s$^2,$and atmospheric pressure is $3\times10^{-15}$ bar (at night). What is the pressure at the bottom of a (hypothetical) $1$ mm high column of mercury, $\rho_{mercury}=13600$ kg/$m^3$?
Note that $1\,$bar $=100,000\,$Pa.
| Acceleration due to gravity on the moon is 1.62 m/s$^2,$and atmospheric pressure is $3\times10^{-15}$ bar (at night). What is the pressure at the bottom of a (hypothetical) $1$ mm high column of mercury, $\rho_{mercury}=13600$ kg/$m^3$?
Note that $1\,$bar $=100,000\,$Pa.
| 1 | 0.333333 | 1 | We have that the pressure in the column obeys
$\frac{\delta p_{\sf mercury}}{\delta z}=-\rho_{\sf mercury} g_{\sf moon}.$
***
Solving the above equation to get an expression for $p_{mercury}$
***
We get $p_{\sf mercury}=-\rho_{\sf mercury} g_{\sf moon}z+p_0$. Setting $z=0$ at the bottom of the column, at a heigh... | We have that the pressure in the column obeys
$\frac{\delta p_{\sf mercury}}{\delta z}=-\rho_{\sf mercury} g_{\sf moon}.$
Solving this, we get
$p_{\sf mercury}=-\rho_{\sf mercury} g_{\sf moon}z+p_0.$
Setting $z=0$ at the bottom of the column, at a height of $h=1$ mm, we have $p=p_{\sf atmosphere}=3\times10^{-15}$ b... | Acceleration due to gravity on the moon is 1.62 m/s$^2,$and atmospheric pressure is $3\times10^{-15}$ bar (at night). What is the pressure at the bottom of a (hypothetical) $1$ mm high column of mercury, $\rho_{mercury}=13600$ kg/$m^3$?
Note that $1\,$bar $=100,000\,$Pa.
| 45 | 7 | 13 | 13 | 105 | 12 | 45 | 7 | 0 | What is the pressure at the bottom of a hypothetical $1$ mm high column of mercury, $\rho_{mercury}=13600$ kg/$m^3$? Note that $1\,$bar $=100,000\,$Pa. | 2 |
4dec19fe-441b-402d-b868-7aaa3b5bb185 | 4 | 0 | 1 | 9 | 4 | 2 | 5 | 5 | Rotational speed can be measured using a small generator producing a voltage which is proportional the rotational speed (as seen in MECH50004 - Mechatronics 2 - Lab 4). Due to the characteristics of the generator, there will always be a “ripple” on the voltage signal caused by construction of the coils (stator and roto... | Select a resistor which, in combination with a $36~\mu\mathrm{F}$ capacitor, will reduce the ripple by $20~\mathrm{dB}$. What will the peak-to-peak amplitude of the ripple voltage be after filtering?
\nSelect a resistor which, in combination with a $36~\mu\mathrm{F}$ capacitor, will reduce the ripple by $40~\mathrm{dB}... | 4 | 0.666667 | 3 | \n\n\n | Using the Bode plot for a normalised Low-Pass filter, one can estimate that when $|H|=-20~\mathrm{dB}$, $\frac{\omega}{\omega_\mathrm{c}}=10$ as shown:
  
. Due to the characteristics of the generator, there will always be a “ripple” on the voltage signal caused by construction of the coils (stator and roto... | 198 | 11 | 42 | 42 | 268 | 0 | 76 | 6 | 0 | Rotational speed can be measured using a small generator producing a voltage which is proportional the rotational speed as seen in MECH50004 - Mechatronics 2 - Lab 4. This ripple can be removed using a low pass filter. The speed sensor used in the lab generates $15\%$ ripple voltage peak to peak at $4$ times the freque... | 8 |
4e2e1aa7-7115-46b3-a00a-33c3be6adb8f | 0 | 1 | 0 | 17 | 6 | 1 | 3 | 1 | A ping-pong (table tennis) ball makes a head-on elastic collision with a much heavier basketball, which is initially stationary. Which of the following statements are true after the collision?
| A ping-pong (table tennis) ball makes a head-on elastic collision with a much heavier basketball, which is initially stationary. Which of the following statements are true after the collision?
| 1 | 0.666667 | 1 | The ping-pong ball of mass $m$ is so much lighter than the basketball of mass $M$ that its momentum almost reverses when it collides: $m\vec{\boldsymbol{u}}_m \rightarrow -m\vec{\boldsymbol{u}}_m$. 
***
What is the momentum of the basketball after the collision? ...
***
... Use conservation of momentum. This... | The ping-pong ball of mass $m$ is so much lighter than the basketball of mass $M$ that its momentum almost reverses when it collides: $m\vec{\boldsymbol{u}}_m \rightarrow -m\vec{\boldsymbol{u}}_m$. 
***
Since the total momentum $m\vec{\boldsymbol{u}}_m$ is conserved, the momentum of the basketball after the coll... | A ping-pong (table tennis) ball makes a head-on elastic collision with a much heavier basketball, which is initially stationary. Which of the following statements are true after the collision?
| 29 | 0 | 9 | 9 | 138 | 4 | 29 | 0 | 0 | Which of the following statements are true after the collision? | 1 |
4e6a7b17-240f-482b-aea1-e38b59c303d4 | 15 | 0 | 0 | 19 | 6 | 1 | 8 | 0 | Find the *real* eigenvalues and *normalised* eigenvectors of the following transformation matrices in $\mathbb{R}^{2}$, and use the information to identify the transformations.
***
**Note:** In the response area, type the same answer twice for repeated solutions. Type $\begin{pmatrix}n\\n\end{pmatrix}$ for no eigenve... | $$
\text{A}=\left(
\begin{array}{cc}
0&\hskip6pt 1\\
1&\hskip6pt 0
\end{array}
\right)\quad
$$
\n$$
\text{B}=\left(
\begin{array}{cc}
\frac{{1}}{2}& \frac{\sqrt{3}}{2} \\
\frac{\sqrt{3}}{2}&\hskip4pt -\frac{{1}}{2}
\end{array}
\right)\quad
$$
\n$$
\text{C}=\left(
\begin{array}{cc}
-1&\hskip7pt 0\\
\hskip7pt 0& -1
\e... | 5 | 0.333333 | 3 | Set-up the characteristic equation $p(\lambda) = \det(\text{A}-\lambda\mathbb{I}_2)=0$ (**section 3.19**)
***
Hence solve for the eigenvalues $\lambda$ (you may find that none exist). 
***
Given $(\text{A}-\lambda\mathbb{I})\mathbf{\underline{x}} = 0$ and $\mathbf{\underline{x}}=(x,y)$, solve for $x$ and $y$... | Setting up and solving the characteristic equation $p(\lambda) = \det(\text{A}-\lambda\mathbb{I}_2)=0$ (**section 3.19**):
***
$$
p(\lambda) = \det\left(\begin{pmatrix}0&1\\1&0\end{pmatrix}-\lambda\begin{pmatrix}1&0\\0&1\end{pmatrix}\right)
$$
***
$$
=\left| \begin{array}{cc}
-\lambda & 1\\
1 & -\lambda
\end{arr... | Find the *real* eigenvalues and *normalised* eigenvectors of the following transformation matrices in $\mathbb{R}^{2}$, and use the information to identify the transformations.
***
**Note:** In the response area, type the same answer twice for repeated solutions. Type $\begin{pmatrix}n\\n\end{pmatrix}$ for no eigenve... | 51 | 7 | 54 | 54 | 360 | 40 | 9 | 5 | 0 | Find the real eigenvalues and normalised eigenvectors of the following transformation matrices in $\mathbb{R}^{2}$, and use the information to identify the transformations. | 1 |
4e709bc9-f407-49dc-8e88-2f20a9da288b | 1 | 0 | 0 | 16 | 6 | 1 | 5 | 5 | The horizontal range $R$ of a projectile is given by $\displaystyle R = \left({{U^2} {\sin{2\alpha}}} \over {g}\right)$, 
where $U$ is the projection speed, $\alpha$ is the angle of elevation and $g$ is the gravitational acceleration.
If $U, \alpha$ are each known to an accuracy of $\pm 0.1\%$ and $g$ is exact,... | The horizontal range $R$ of a projectile is given by $\displaystyle R = \left({{U^2} {\sin{2\alpha}}} \over {g}\right)$, 
where $U$ is the projection speed, $\alpha$ is the angle of elevation and $g$ is the gravitational acceleration.
If $U, \alpha$ are each known to an accuracy of $\pm 0.1\%$ and $g$ is exact,... | 1 | 0.666667 | 1 | Find $\delta R$ to first order in $\delta U, \delta\alpha$. Use the total differential of $R$.
***
Using the total differential of $R$,
$$
\begin{aligned}
\delta R &\simeq {\partial R\over\partial U}\delta U + {\partial R\over\partial\alpha}\delta\alpha
\end{aligned}
$$
Can you find $\delta R$ now to first order... | Find $\delta R$ to first order in $\delta U, \delta\alpha$. Use the total differential of $R$.
***
$$
\displaystyle R = {U^2 \sin(2\alpha) \over g}
$$
Using the total differential of $R$,
$$
\begin{aligned}
\delta R &\simeq {\partial R\over\partial U}\delta U + {\partial R\over\partial\alpha}\delta\alpha
\end{ali... | The horizontal range $R$ of a projectile is given by $\displaystyle R = \left({{U^2} {\sin{2\alpha}}} \over {g}\right)$, 
where $U$ is the projection speed, $\alpha$ is the angle of elevation and $g$ is the gravitational acceleration.
If $U, \alpha$ are each known to an accuracy of $\pm 0.1\%$ and $g$ is exact,... | 57 | 10 | 27 | 27 | 172 | 11 | 57 | 10 | 0 | The horizontal range $R$ of a projectile is given by $\displaystyle R = \left({{U^2} {\sin{2\alpha}}} \over {g}\right)$, where $U$ is the projection speed, $\alpha$ is the angle of elevation and $g$ is the gravitational acceleration. If $U, \alpha$ are each known to an accuracy of $\pm 0.1\%$ and $g$ is exact, find an ... | 2 |
4e891748-7f57-4df8-9ccb-8e5efb042ee1 | 3 | 0 | 0 | 13 | 4 | 2 | 2 | 0 | 
A weight of $3.2~\mathrm{kN}$ falls $2~\mathrm{m}$ onto a pile weighing $2.4~\mathrm{kN}$. Assuming that the weight and the pile move together after the ... | The velocity, $v$, with which they begin to move.
\nThe mean resistance of the earth to penetration, $F$, if the pile is driven $20~mm$ into the ground by the blow.
\nThe energy lost in the impact, $\Delta KE$.
| 3 | 0.333333 | 3 | This is a tough one – you’ll have to think back to your school physics lessons! If you happened to miss that particular lesson, check section 3.2 in your notes for the necessary explanation and equations you need. You may want to use the energy method to calculate the velocity of the falling weight right before impact.... | Employing the following SUVAT equation:
$$
v_1^2 = v_0^2 + 2as
$$
***
Rearranging:
$$
v_1 = \sqrt{v_0^2 + 2as}
$$
***
Substituting the values of the parameters in:
$$
v_1 = \sqrt{2g \times 2} = 2\sqrt{g}~~\mathrm{(Equation~4)}
$$
***
Substituting Equation 4 into Equation 3:
$$
v_2 = \cfrac{m_w}{m_w + m_p}\tim... | 
A weight of $3.2~\mathrm{kN}$ falls $2~\mathrm{m}$ onto a pile weighing $2.4~\mathrm{kN}$. Assuming that the weight and the pile move together after the ... | 66 | 7 | 47 | 47 | 444 | 0 | 40 | 4 | 1 | Assuming that the weight and the pile move together after the impact, find: The velocity, $v$, with which they begin to move. | 1 |
4f074b46-971b-4a1d-976c-cfd514e23116 | 2 | 0 | 3 | 22 | 6 | 1 | 0 | 4 | A damped harmonic oscillator has the equation of motion
$$
\ddot{\psi} + \gamma \dot{\psi} + \omega_0^2 \psi = 0.
$$
| When $\gamma/2 < \omega_0$, the damping is light. Show, by substitution, that for light damping the solution is
$$
\psi(t) = A e^{-\gamma t /2}\cos(\omega_{\rm d} t + \phi),
$$
where $\omega_{\rm d} = \sqrt{\omega_0^2 - \gamma^2/4}$.
\nThe amplitude decay time, $\tau_A$, is the time taken for the amplitude of the... | 4 | 0.666667 | 3 | Find the derivatives $\dot{\psi}$ and $\ddot{\psi}$.
***
You should get
$\dot{\psi} = A e^{-\gamma t /2}\left[-\omega_{\rm d}\sin(\omega_{\rm d} t + \phi) -\frac{\gamma}{2}\cos(\omega_{\rm d} t + \phi)\right]$
and
$\ddot{\psi} = A e^{-\gamma t/2}\left[-\omega_{\rm d}^2\cos(\omega_{\rm d} t + \phi) + 2 \frac{\gamm... | Find the derivatives $\dot{\psi}$ and $\ddot{\psi}$:
***
$\psi = A e^{-\gamma t /2}\cos(\omega_{\rm d} t + \phi)$
$\dot{\psi} = A e^{-\gamma t /2}\left[-\omega_{\rm d}\sin(\omega_{\rm d} t + \phi) -\frac{\gamma}{2}\cos(\omega_{\rm d} t + \phi)\right]$
$\ddot{\psi} = A e^{-\gamma t/2}\left[-\omega_{\rm d}^2\cos(\ome... | A damped harmonic oscillator has the equation of motion
$$
\ddot{\psi} + \gamma \dot{\psi} + \omega_0^2 \psi = 0.
$$
When $\gamma/2 < \omega_0$, the damping is light. Show, by substitution, that for light damping the solution is
$$
\psi(t) = A e^{-\gamma t /2}\cos(\omega_{\rm d} t + \phi),
$$
where $\omega_{\r... | 167 | 20 | 37 | 37 | 228 | 30 | 157 | 19 | 0 | Find expressions for $\tau_A$ and $\tau_E$ for light damping. For very light damping $\gamma/2 < \omega_0$1, show the following: i $\gamma/2 < \omega_0$2. | 2 |
500b8b88-75d7-4d91-8703-ec76a9cadce6 | 10 | 0 | 1 | 16 | 6 | 1 | 7 | 3 | \
$f(x, y) = {(x + 2y)} \cos {(2x + y)}$.
| Find all the partial derivatives required for a third-order Taylor-Maclaurin expansion of $f(x, y)$.
\nHence find the expansion of this function to cubic terms about the origin $(0, 0)$.
\nCheck your results using the usual small-angle expansion of the cosine function for a single variable.
| 3 | 0.666667 | 3 | For a third-order Taylor Maclaurin expansion, you will need to find the third-order derivatives, up to terms such as $\displaystyle {\partial ^3 f \over\partial x^3}$.
$$
f(x,y) = (x+2y)\cos(2x+y)
$$
First, find $\displaystyle f_x = {\partial f\over\partial x}$ and $\displaystyle f_y = {\partial f\over\partial y}$. Y... | For a third-order Taylor Maclaurin expansion, you will need to find the third-order derivatives, up to terms such as $\displaystyle {\partial ^3 f \over\partial x^3}$.
$$
f(x,y) = (x+2y)\cos(2x+y)
$$
First, find $\displaystyle f_x = {\partial f\over\partial x}$ and $\displaystyle f_y = {\partial f\over\partial y}$. Y... | \
$f(x, y) = {(x + 2y)} \cos {(2x + y)}$.
Find all the partial derivatives required for a third-order Taylor-Maclaurin expansion of $f(x, y)$.
\nHence find the expansion of this function to cubic terms about the origin $(0, 0)$.
\nCheck your results using the usual small-angle expansion of the cosine function for a si... | 48 | 3 | 27 | 27 | 244 | 18 | 45 | 2 | 0 | Find all the partial derivatives required for a third-order Taylor-Maclaurin expansion of $f(x, y)$. Hence find the expansion of this function to cubic terms about the origin $(0, 0)$. Check your results using the usual small-angle expansion of the cosine function for a single variable. | 3 |
50141b29-3919-4963-a5ed-1fb35af93870 | 1 | 0 | 0 | 24 | 6 | 1 | 1 | 2 | **\[Boas 5.5.3]:** Find the area of the paraboloid $x^2 + y^2 = z$ inside the cylinder $x^2 + y^2 = 9$. 
***
*Hint:* Don't use the surface of revolution formula here (instead, perform a double integral). Leave your answer in fractional form, or as a number to 1 decimal place.
| **\[Boas 5.5.3]:** Find the area of the paraboloid $x^2 + y^2 = z$ inside the cylinder $x^2 + y^2 = 9$. 
***
*Hint:* Don't use the surface of revolution formula here (instead, perform a double integral). Leave your answer in fractional form, or as a number to 1 decimal place.
| 1 | 0.666667 | 2 | The surface area of the paraboloid within the cylinder is given by $\iint_S{|d\vec{S}|}$, the sum of the magnitude of the surface area elements. Start by trying to understand the geometry of the problem...
***
... First, what is the shape of the paraboloid? ...
***
... If $x^2+y^2=z$, when $z=0$ the paraboloid is... | At the origin $(0,0,0)$, the paraboloid is a point. For a fixed $z$: $x^2 + y^2 = z$ $\implies$the paraboloid is a circle of radius $\sqrt{z}$. As $z$ increases, the paraboloid expands outwards in proportion to $\sqrt{z}$.
***
The boundary of the paraboloid is defined by the cylinder. This occurs at $z=9$, where the ... | **\[Boas 5.5.3]:** Find the area of the paraboloid $x^2 + y^2 = z$ inside the cylinder $x^2 + y^2 = 9$. 
***
*Hint:* Don't use the surface of revolution formula here (instead, perform a double integral). Leave your answer in fractional form, or as a number to 1 decimal place.
| 43 | 2 | 28 | 28 | 196 | 14 | 43 | 2 | 0 | Boas 5.5.3: Find the area of the paraboloid $x^2 + y^2 = z$ inside the cylinder $x^2 + y^2 = 9$. Hint: Don't use the surface of revolution formula here instead, perform a double integral. Leave your answer in fractional form, or as a number to 1 decimal place. | 3 |
50559ad7-6651-4857-94b5-b39e3334d5a6 | 1 | 0 | 0 | 9 | 4 | 2 | 0 | 5 | Calculate the total resistance as measured between points **a** and **b** of the network below. You will have to apply Kirchhoff’s laws and set-up a set of linear equations to solve this. It is also advantageous to apply a virtual voltage source across **ab**. This question is a bonus question for strong students.
|   
| 1 | 1 | 4 | null | Firstly, resolve any branches that are in series or parallel to simplify the network:
***
The highlighted branches are in parallel:
***
This becomes:
... | Calculate the total resistance as measured between points **a** and **b** of the network below. You will have to apply Kirchhoff’s laws and set-up a set of linear equations to solve this. It is also advantageous to apply a virtual voltage source across **ab**. This question is a bonus question for strong students.
... | 55 | 0 | 18 | 18 | 176 | 0 | 2 | 0 | 1 | Calculate the total resistance as measured between points a and b of the network below. You will have to apply Kirchhoff’s laws and set-up a set of linear equations to solve this. It is also advantageous to apply a virtual voltage source across ab. | 3 |
50a9bdb3-2c61-4406-aa69-0ebd5649ac30 | 1 | 0 | 0 | 8 | 4 | 2 | 2 | 2 | A helicopter drive shaft is $110\text{ mm}$ diameter, with a wall thickness of $4\text{ mm}$. It is made of titanium alloy ($E=110\text{ GPa}$, failure stress $=650\text{ MPa}$). The engine power output is $297\text{ kW}$ and the rotor speed $190 \text{ RPM}$. The helicopter weighs $86 \text{ kN}$. As it flies along, a... | What is the safety factor according to the shear-strain energy criterion, if fatigue considerations are ignored?
| 1 | 0.666667 | 3 | null | First the axial stress needs to be calculated:
   
$$
\begin{align*}
\sigma_z&=\frac{F}{A} \\
&=\frac{W}{2\pi rt} \\
&=\frac{86\times10^3}{2\pi\times0.055\times0.004} \\
&=62.21 \text{ MPa}
\end{align*}
$$
   
***
However, the bending **along the axial axis** induces $\pm100 \%$ of this value. ... | A helicopter drive shaft is $110\text{ mm}$ diameter, with a wall thickness of $4\text{ mm}$. It is made of titanium alloy ($E=110\text{ GPa}$, failure stress $=650\text{ MPa}$). The engine power output is $297\text{ kW}$ and the rotor speed $190 \text{ RPM}$. The helicopter weighs $86 \text{ kN}$. As it flies along, a... | 76 | 8 | 8 | 8 | 106 | 0 | 16 | 0 | 0 | What is the safety factor according to the shear-strain energy criterion, if fatigue considerations are ignored? | 1 |
50bcbdb1-5795-41dd-966b-7cf9abd9ad84 | 3 | 0 | 0 | 14 | 4 | 2 | 4 | 5 | Wind with speed $U$ blows perpendicularly to a suspension bridge of length $l$. The bridge deck of thickness $h$ disturbs the flow, leading to the periodic release of vortices, the frequency of which is called the vortex shedding frequency, and is denoted by $f$. The shedding frequency depends on the deck thickness, th... | Considering the shedding frequency as the main quantity of interested, suggest a list of dimensionless groups to describe the problem.
\nIt is believed that on 07-11-1940, the vortex shedding frequency produced by the Tacoma Narrows bridge (USA) matched one the natural frequency of the bridge and led to its dramatic co... | 3 | 0.5 | 2 | \n\n | \n\n | Wind with speed $U$ blows perpendicularly to a suspension bridge of length $l$. The bridge deck of thickness $h$ disturbs the flow, leading to the periodic release of vortices, the frequency of which is called the vortex shedding frequency, and is denoted by $f$. The shedding frequency depends on the deck thickness, th... | 278 | 9 | 0 | 0 | 1 | 0 | 207 | 3 | 2 | Wind with speed $U$ blows perpendicularly to a suspension bridge of length $l$. Considering the shedding frequency as the main quantity of interested, suggest a list of dimensionless groups to describe the problem. If a $1/100$ scale model were to be tested in a wind tunnel and full dynamic similarity was required: Wha... | 6 |
50c4692a-590e-40c2-a344-c1754d13a64d | 3 | 0 | 0 | 13 | 4 | 2 | 2 | 1 | 
A ship of mass $2.5~\mathrm{Gg}$ is towing another of mass $1.25~\mathrm{Gg}$ using a steel wire rope whose effective elasticity is such that a tension o... | Find the speed of the ships, $v_f$, as they move together.
\nFind the strain energy, $SE$, stored in the rope.
\nFind the minimum length of the rope, $L_{min}$, if the tension in it is not to exceed $500~\mathrm{kN}$.
| 3 | 0.666667 | 4 | Think of this as your usual impact but in reverse, where the impacting body is somehow ahead of the receiving body. Same rules apply.
\nExamine the kinetic energies of the ships before and after the rope became taut. Is there a deficit?
***
Be careful with the units in this question. The masses are given in $\mathrm... | Applying conservation of momentum:
$$
m_sv_s + m_bv_b = (m_s+m_b)v_f
$$
***
Rearranging and simplifying:
$$
v_f = \cfrac{m_sv_s + m_bv_b}{m_s + m_b}
$$
***
Substituting the values of the parameters gives:
$$
v_f = \cfrac{2.5 \times 3 + 1.25 \times 1.5}{2.5 + 1.25}
$$
$$
v_f = 2.5~\mathrm{m/s}
$$
\nThere is no i... | 
A ship of mass $2.5~\mathrm{Gg}$ is towing another of mass $1.25~\mathrm{Gg}$ using a steel wire rope whose effective elasticity is such that a tension o... | 97 | 10 | 35 | 35 | 234 | 5 | 42 | 4 | 1 | A ship of mass $2.5~\mathrm{Gg}$ is towing another of mass $1.25~\mathrm{Gg}$ using a steel wire rope whose effective elasticity is such that a tension of $5~\mathrm{MN}$ produced a $10~\mathrm{\%}$ increase in length. Find the speed of the ships, $v_f$, as they move together. Find the strain energy, $SE$, stored in t... | 4 |
50ea3efd-48ab-4e3a-8b84-24e3650c9f37 | 0 | 1 | 0 | 2 | 1 | 2 | 0 | 7 | Which of the following properties are scalars?
| Which of the following properties are scalars?
| 1 | 0.333333 | 0 | null | null | Which of the following properties are scalars?
| 7 | 0 | 0 | 0 | 0 | 0 | 7 | 0 | 0 | Which of the following properties are scalars? | 1 |
514ab4e7-9f2e-41fc-aee7-89caf911312e | 0 | 1 | 0 | 2 | 1 | 2 | 0 | 3 | If the continuum assumption holds for a particular fluid which of the following statements are true?
| If the continuum assumption holds for a particular fluid which of the following statements are true?
| 1 | 0.333333 | 0 | null | null | If the continuum assumption holds for a particular fluid which of the following statements are true?
| 16 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | If the continuum assumption holds for a particular fluid which of the following statements are true? | 1 |
5189ceb5-42a2-4adc-8722-db1cbd820349 | 6 | 1 | 0 | 11 | 4 | 2 | 5 | 7 | The stator exit flow angle of an axial-flow turbine is $70^{\circ}$. The rotor inlet and exit relative flow angles are symmetrical, i.e. $\beta_3 = -\beta_2$ . The mean radius and axial velocity are constant. If the stator exit gas velocity is $400 ~\mathrm{m/s}$ and the blade speed is $200 ~\mathrm{m/s}$, calculate: ... | The rotor inlet and exit flow angles (relative to the blade).
\nThe rotor exit gas relative velocity.
\nThe rotor exit tangential gas velocity component (magnitude and sign).
\nThe rotor exit absolute gas velocity and flow angle.
| 4 | 0.666667 | 3 | \n\n\n | Draw an initial velocity triangle for state $2$:
*  $ C_{\theta2} $ must be in the same direction as $U$ because $\alpha_2$ is positive;
* Since $ \beta_2 $ is unknown, we choose an arbitrary direction for $ W_{\theta2} $ as our initial guess. 
  
.
\nThe rotor exit gas relative velocity.
\nThe rotor exit tangential gas velocity component (magnitude and sign).
\nThe rotor exit absolute gas velocity and flow angle.
| 0 |
52282ac7-5b8c-4d36-9581-c81512c36a62 | 0 | 8 | 0 | 4 | 3 | 3 | 0 | 3 | In Cartesian coordinates ($x$,$y$,z) and for a mean flow with velocity components ($u$,$v$,$w$) select the equations governing pollutant transport under the conditions that follow.
| Diffusion only and in $x$-direction.
\nDiffusion only and in both $x$ and z-directions.
\nAdvection only and in $x$-direction.
\nAdvection only and in both $x$ and $y$-directions.
\nAdvection and diffusion; both in $x$, $y$, and $z$ directions.
\nSteady flow, with advection and diffusion; both in $x$, $y$, and $z$ dire... | 8 | 0.666667 | 2 | \n\n\n\n\n\n\n | We assume that diffusion is isotropic and homogeneous.
Here, temporal change is balanced by diffusion in one direction, and so
$$
\underbrace{\dfrac{\partial C}{\partial t}}_{\text{unsteady term}}=
\underbrace{D\dfrac{\partial^2 C}{\partial x^2}}_{\text{diffusion in $x$}}.
$$
\nWe assume that diffusion is isotropic a... | In Cartesian coordinates ($x$,$y$,z) and for a mean flow with velocity components ($u$,$v$,$w$) select the equations governing pollutant transport under the conditions that follow.
Diffusion only and in $x$-direction.
\nDiffusion only and in both $x$ and z-directions.
\nAdvection only and in $x$-direction.
\nAdvection ... | 101 | 16 | 11 | 11 | 194 | 0 | 67 | 11 | 0 | In Cartesian coordinates $x$,$y$,z and for a mean flow with velocity components $u$,$v$,$w$ select the equations governing pollutant transport under the conditions that follow. When diffusion is negligible. | 2 |
52f515ac-7136-49bb-938e-1af478bac08c | 0 | 0 | 1 | 9 | 4 | 2 | 6 | 7 | Show that a series LR circuit shown below is a Low-Pass filter if the output is the voltage across the resistor. Set up the differential equation relating $v_\mathrm{o}$ to $v_\mathrm{i}$ and hence derive the corresponding transfer function.
| 
| 1 | 0.666667 | 2 | null |
, Apply KVL round the closed loop, recalling that the voltage across an inductor is given by $L\frac{\mathrm{d}i}{\mathrm{d}t}$:
  
$v_\mathrm{i} = L\frac{\mathrm{d}i}{\mathrm{d}t}+v_\mathrm{o}$
***
***
Apply Ohm's law to eliminate $i$:
  
$v_\mathrm{i} = \frac{L}{R}\frac{\mathrm{d}v_\mathrm... | Show that a series LR circuit shown below is a Low-Pass filter if the output is the voltage across the resistor. Set up the differential equation relating $v_\mathrm{o}$ to $v_\mathrm{i}$ and hence derive the corresponding transfer function.
:
   
The hoop stress can be written as below (from Q6.1b):
   
$$
\begin{align}
\sigma_\theta=P\frac{K^2+1}{K^2-1}
\end{align}
$$
   
***
The radial stre... | A thick-walled cylinder with an outer to inner diameter ratio $K$ and **closed ends** is subjected to increasing internal pressure until yielding first occurs. For a steel cylinder, $\sigma_Y = 310\text{ MPa}$ and $K = 2.4$.
Show that, if the Von Mises criterion applies, yielding will occur at a pressure:
  ... | 125 | 8 | 45 | 45 | 408 | 0 | 93 | 5 | 0 | Show that, if the Von Mises criterion applies, yielding will occur at a pressure: $ P=\sigma_Y\frac{K^2-1}{\sqrt{3K^2}} $ Where $\sigma_Y$ is the tensile yield stress. Find the values of $P$ for both the Von Mises and Tresca yield criteria when the cylinder has closed ends. Find the values of $P$ for both the Von Mises... | 4 |
536be15b-e639-4855-b0e9-1e840852c65e | 4 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | A symmetrical aerofoil with a rounded leading edge is flying at $M=3$ at zero incidence and at sea level conditions ($T_{1}=288 \ \mathrm{K}$,
 $p_1 = 1.013 \cdot 10^{5} \ \mathrm{N/m^2}$). 
  
Hint: a curved shock forms in front of the aerofoil, however just in front of the nose it is a normal ... | Calculate the temperature and pressure at the nose of the aerofoil. 
  
Hint: the nose of the aerofoil is a stagnation point. 
  
\[Note: to enter numbers in scientific notation, use format 1.5e6, for instance]
\nDetermine the pressure coefficient at the nose of the aerofoil. 
\nW... | 3 | 0.333333 | 2 | The stagnation temperature does not change across the shock. So we can use the isentropic relation to find $ T_{01} $. 
***
The stagnation pressure is not conserved across the shock, so we also need to determine the change across the normal shock wave. 
\nRemember that the pressure coefficient can be exp... | The nose of the airfoil is a stagnation point, therefore we need to find the stagnation pressure and temperature behind a shock at $ M=3 $. Since it is air, we can use the tables (though it can also be done with the normal shock equations).
   
***
The stagnation temperature does not change across the sho... | A symmetrical aerofoil with a rounded leading edge is flying at $M=3$ at zero incidence and at sea level conditions ($T_{1}=288 \ \mathrm{K}$,
 $p_1 = 1.013 \cdot 10^{5} \ \mathrm{N/m^2}$). 
  
Hint: a curved shock forms in front of the aerofoil, however just in front of the nose it is a normal ... | 114 | 3 | 25 | 25 | 309 | 7 | 65 | 0 | 0 | Calculate the temperature and pressure at the nose of the aerofoil. Note: to enter numbers in scientific notation, use format 1.5e6, for instance Determine the pressure coefficient at the nose of the aerofoil. What is the pressure coefficient at a point on the aerofoil where the local Mach number is equal to 1? | 3 |
53e5ea9f-f53c-486d-87b9-6c0930b58f67 | 2 | 0 | 0 | 18 | 6 | 1 | 1 | 6 | This question is about space-like and time-like separations. Either answer the following questions by drawing out a space-time diagram, or solve algebraically using the Lorentz transformations. | Two events are separated by $(c\Delta t, \Delta x, \Delta y, \Delta z) = (3,4,0,0)$ in frame $S$. Is there a reference frame $S^\prime$ in which the events are simultaneous, or at the same position? What is the relative velocity between $S$ and the relevant frame $S^\prime$ (give your answer in terms of $c$)?\nTwo even... | 2 | 0.333333 | 2 | First, calculate the value of the invariant interval in this case.
***
   
You should find that this is negative number - is this a space-like or time-like interval?
***
   
This is a *space-like* interval and so there is a frame where the two events are simultaneous...
... | The invariant interval is $s^2=9-16=-7 < 0$.
***
Thus, the events are space-like separated.  
***
This means a reference frame can be found in which the events are simultaneous. 
***
For this frame: $\Delta t^\prime = \gamma(\Delta t-v\Delta x /c^2)=0$. 
***
Therefore we find $v\Delta x/c^2=\Delt... | This question is about space-like and time-like separations. Either answer the following questions by drawing out a space-time diagram, or solve algebraically using the Lorentz transformations.Two events are separated by $(c\Delta t, \Delta x, \Delta y, \Delta z) = (3,4,0,0)$ in frame $S$. Is there a reference frame $S... | 116 | 12 | 4 | 4 | 72 | 2 | 91 | 12 | 0 | Either answer the following questions by drawing out a space-time diagram, or solve algebraically using the Lorentz transformations.Two events are separated by $(c\Delta t, \Delta x, \Delta y, \Delta z) = (3,4,0,0)$ in frame $S$. Is there a reference frame $S^\prime$ in which the events are simultaneous, or at the same... | 4 |
544ecba6-40bd-43ba-96c7-d9d4725fd5fd | 2 | 0 | 0 | 20 | 6 | 1 | 0 | 2 | Define $z=(5+7i)(5+bi)$:
| If $b$ and $z$ are both real, find $b$. 
\nIf $\rm{Im}(b)=4/5$ and $z$ is pure imaginary, find $\rm{Re}(b)$. 
| 2 | 0.666667 | 1 | Expand the brackets...
***
... collect the imaginary terms
***
If $b$ and $z$ are both purely real, what is the imaginary part equal to?
\nStart by expanding the brackets...
***
It is useful to let $b=x+\frac{4}{5}i$, 
***
our goal is to solve for $x$.
***
If $z$ is pure imaginary, what is the real ... | If $b$ and $z$ are both real then the imaginary part is 0,
***
expanding the brackets,
***
$(5+7i)(5+bi)= 25 + 35i + 5bi - 7b$
***
Collecting the imaginary terms,
***
and setting equal to 0,
***
$35+5b=0$, so:
***
$b=-7$
\nExpanding the brackets,
***
$$
z=(5+7i)(5+bi)=25+35i+5bi+7b i^2 = 25-7b+35i+5bi
$$
... | Define $z=(5+7i)(5+bi)$:
If $b$ and $z$ are both real, find $b$. 
\nIf $\rm{Im}(b)=4/5$ and $z$ is pure imaginary, find $\rm{Re}(b)$. 
| 23 | 7 | 15 | 15 | 88 | 6 | 20 | 6 | 0 | Define $z=(5+7i)(5+bi)$: If $b$ and $z$ are both real, find $b$. If $\rm{Im}(b)=4/5$ and $z$ is pure imaginary, find $\rm{Re}(b)$. | 2 |
545a7370-9d91-4c92-a99c-92b64099ab2f | 5 | 0 | 0 | 4 | 3 | 3 | 2 | 4 | Reconsider the sea-river system below.
For this question, the river is discharging $\dot{M}=1\ \mathrm{kg/s}$ of phosphorus into the sea as a contin... | Give the solution to this problem in symbolic form. You may use the far-field solution.
\nCalculate the phosphorous concentration at $x = 1\ \mathrm{km}$ at $y=0,\ 50,\ 100,\ \mathrm{and\ }200\ \mathrm{m}$. Finally, plot the concentration as a function of $y$.
| 2 | 0.666667 | 2 | \n | We use the far-field solution for a continuous release in 2D as
$$
C = \dfrac{\dot{M}}{\sqrt{2 \pi} h U \sigma} \exp(-y^2/(2 \sigma^2)),
$$
where $\sigma^2 = 2 D x / U$.
***
Applying the boundary condition at $y=0$ results in
$$
C = \dfrac{2\dot{M}}{\sqrt{2 \pi} h U \sigma} \exp(-y^2/(2 \sigma^2)).
$$
\nWe use the... | Reconsider the sea-river system below.
For this question, the river is discharging $\dot{M}=1\ \mathrm{kg/s}$ of phosphorus into the sea as a contin... | 102 | 7 | 14 | 14 | 84 | 0 | 34 | 3 | 1 | Reconsider the sea-river system below. The sea has an average depth of $6\ \mathrm{m}$ and it may be assumed that the mixing at the river mouth is sufficient to distribute it uniformly across the depth. Assume the tidal current $U=0.5\ \mathrm{m/s}$, and that the flow has an eddy diffusivity $D_T=1.3\ \mathrm{m^2/s}$. ... | 7 |
5500f226-1737-4c9c-938a-a0eadd134831 | 2 | 0 | 2 | 16 | 6 | 1 | 6 | 5 |  If $u = u(x, y)$ and $x$ and $y$ are related to two new independent variables $s$ and $t$ by
$$
\displaystyle x = st,\quad y = {{s + t} \over {s - t}}
$$
so that $u = \bar u(s, t)$, 
| Use the chain rule to find $\displaystyle{\partial\bar u\over\partial s}$ and $\displaystyle{\partial \bar u\over \partial t}$ in terms of $\displaystyle{\partial u\over\partial x}$ and $\displaystyle{\partial u\over\partial y}$. 
\nHence show that 
$$
\displaystyle 2x{{\partial u}\over {\partial x}} = s{{... | 3 | 0.666667 | 2 | When working on these equivalences, we need to have pairs of variables confined on each side. That is to say $\mathrm{function\ of\ 'old'} \equiv \mathrm{function\ of\ 'new'}$ so aim to find equations of the form  
$$
\displaystyle
\\[5pt]
{\partial\bar{u}\over\partial s} = ... \enspace {\partial\bar{u}\over\par... | Using the chain rule,
$$
\displaystyle
{\partial\bar{u}\over\partial s} = {\partial u\over\partial x}{\partial x\over\partial s} + {\partial u\over\partial y}{\partial y\over\partial s}
$$
Try expressing $\displaystyle {\partial\bar{u}\over\partial t}$ in a similar form using the chain rule.
***
$$
\displaystyle {... |  If $u = u(x, y)$ and $x$ and $y$ are related to two new independent variables $s$ and $t$ by
$$
\displaystyle x = st,\quad y = {{s + t} \over {s - t}}
$$
so that $u = \bar u(s, t)$, 
Use the chain rule to find $\displaystyle{\partial\bar u\over\partial s}$ and $\displaystyle{\partial \bar u\over \partial... | 46 | 13 | 20 | 20 | 113 | 16 | 24 | 6 | 0 | If $u = u(x, y)$ and $x$ and $y$ are related to two new independent variables $s$ and $t$ by $ \displaystyle x = st,\quad y = {{s + t} \over {s - t}} $ so that $u = \bar u(s, t)$, Use the chain rule to find $\displaystyle{\partial\bar u\over\partial s}$ and $\displaystyle{\partial \bar u\over \partial t}$ in terms of ... | 1 |
5594e094-50b1-4072-8bbf-e6c394fb67dc | 1 | 0 | 0 | 14 | 4 | 2 | 8 | 0 | If we denote $\vec{\mathcal{T}}$ as the `traction force' per unit area on each face of a particle; $\underline{\underline{{\sigma}}}$ as the stress tensor; and $\hat{n}$ as a unit normal; what is the order and physical dimensions ([$M$],[$L$],[$T$]) of each of the following terms? | As an example, the answer for $\vec{u}$ is Order 1, $[M]^0$, $[L]^1$, $[T]^{ -1}$.
| 1 | 0.333333 | 1 | null | null | If we denote $\vec{\mathcal{T}}$ as the `traction force' per unit area on each face of a particle; $\underline{\underline{{\sigma}}}$ as the stress tensor; and $\hat{n}$ as a unit normal; what is the order and physical dimensions ([$M$],[$L$],[$T$]) of each of the following terms?As an example, the answer for $\vec{u}$... | 63 | 10 | 0 | 0 | 0 | 0 | 16 | 4 | 0 | If we denote $\vec{\mathcal{T}}$ as the `traction force' per unit area on each face of a particle; $\underline{\underline{{\sigma}}}$ as the stress tensor; and $\hat{n}$ as a unit normal; what is the order and physical dimensions $M$,$L$,$T$ of each of the following terms?As an example, the answer for $\vec{u}$ is Orde... | 1 |
55a12f5a-681a-4375-a9ac-5546802168e2 | 0 | 0 | 3 | 24 | 6 | 1 | 4 | 4 | Verify the following vector operator identities, e.g., by:
* Expanding out each side in cartesian coordinates:
* It is possible to avoid expanding in cartesians by using the vector and differential properties of $\nabla$. One "trick,'' e.g., to evaluate $\nabla\times(\Omega\vec{V})$ is to write $\nabla \equiv \nab... | $$
\nabla\times(\Omega\vec{V})=\Omega(\nabla\times\vec{V})-\vec{V}\times\nabla\Omega
$$
\n$$
\nabla\cdot(\vec{U}\times\vec{V}) = \vec{V}\cdot(\nabla\times\vec{U})-\vec{U}\cdot(\nabla\times\vec{V})
$$
\n\[BONUS]:
$$
\nabla\times(\nabla\times\vec{E})=\nabla(\nabla\cdot \vec{E})-\nabla^2\vec{E}
$$
| 3 | 1 | 4 | There are two methods to approach this problem:
1. Separating $\nabla$ into two parts (see hint in question). This is a non-brute force approach and does not require you to evaluate in Cartesian coordinates.
2. Evaluate the curl $\nabla\times\nabla\vec{V}$ by brute force (i.e., by expressing the curl in Cartesian co... |
, Letting $\nabla = \nabla_\Omega + \nabla_V$:
***
$$
\nabla\times(\Omega\vec{V}) = (\nabla_\Omega+\nabla_V)\times(\Omega\vec{V})
$$
***
Applying the distributive property of the cross product, 
***
$$
= (\nabla_\Omega \Omega \times \vec{V}) + (\Omega\nabla_V \times\vec{V})
$$
In the first term, $\nabla_\O... | Verify the following vector operator identities, e.g., by:
* Expanding out each side in cartesian coordinates:
* It is possible to avoid expanding in cartesians by using the vector and differential properties of $\nabla$. One "trick,'' e.g., to evaluate $\nabla\times(\Omega\vec{V})$ is to write $\nabla \equiv \nab... | 98 | 12 | 49 | 49 | 273 | 25 | 5 | 3 | 0 | Verify the following vector operator identities, e.g., by: Expanding out each side in cartesian coordinates: It is possible to avoid expanding in cartesians by using the vector and differential properties of $ abla$. One "trick,'' e.g., to evaluate $ abla\times(\Omega\vec{V})$ is to write $ abla \equiv abla_\Omega + ... | 2 |
55a2d487-3921-4465-85fa-8aac1bf0f672 | 0 | 0 | 2 | 16 | 6 | 1 | 2 | 6 | Consider the integral
$$
\displaystyle I = \int_0^1 {{{{ {{x}^4} {{(1 - x)}^4 }}} }\over{ {(1 + {x}^2)}}}\ \mathrm{d}x
$$
| Use partial fractions to expand the integrand in order to show that:
$$
\displaystyle I = {22\over7} - \pi.
$$
\n By noting that $1\leq {1 + {x}^2}\leq 2$ in the denominator of $I$, show that:
$$
\displaystyle {{22}\over7} - {1\over{630}} < \pi < {{22}\over7} - {1\over{1260}},
$$
that is to say:
$$
3.14126... ... | 2 | 1 | 2 | First, use the binomial expansion on the numerator.
***
Now divide using whichever method you prefer (e.g. long division or inspection)
***
Remember that $\displaystyle \int {1 \over 1+x^2} \,\mathrm{d}x = \tan^{-1} x + c$
***
You need to divide out the integrand, which is currently in the form $\displaystyle ... | You need to divide out the integrand, which is currently in the form $\displaystyle \frac{\mathrm{8th \,degree}}{\mathrm{quadratic}}$, so that it is in the form $\displaystyle \mathrm{6th\,degree\,polynomial}+\frac{\mathrm{linear}}{\mathrm{quadratic}}$ which you can then integrate as normal.
***
First, use the binomi... | Consider the integral
$$
\displaystyle I = \int_0^1 {{{{ {{x}^4} {{(1 - x)}^4 }}} }\over{ {(1 + {x}^2)}}}\ \mathrm{d}x
$$
Use partial fractions to expand the integrand in order to show that:
$$
\displaystyle I = {22\over7} - \pi.
$$
\n By noting that $1\leq {1 + {x}^2}\leq 2$ in the denominator of $I$, show tha... | 35 | 6 | 5 | 5 | 76 | 9 | 31 | 5 | 0 | Consider the integral $ \displaystyle I = \int_0^1 {{{{ {{x}^4} {{(1 - x)}^4 }}} }\over{ {(1 + {x}^2)}}}\ \mathrm{d}x $ Use partial fractions to expand the integrand in order to show that: $ \displaystyle I = {22\over7} - \pi. | 1 |
561094ab-75df-481c-a5f2-7a7bf711a46d | 0 | 0 | 3 | 21 | 6 | 1 | 1 | 3 | In this question we are looking at the power spectrum of a Fourier series and why that is a useful quantity.
Let $P_n$ be given by: $P_n = a_n^2+b_n^2$. The distribution of the values of, $\lbrace{P_n\rbrace}$, is called the **power spectrum** of the Fourier series.
| Show that writing down the Fourier series in the form:
$$
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}{\alpha_n \cos(nx - \theta_n)} \
$$
$$
$$
is equivalent to the trigonometric Fourier series if:
$$
\begin{aligned}
a_n &= \alpha_n\cos\theta_n, ~~~~~ b_n &= \alpha_n\sin\theta_n \\
\alpha_n^2 &= a_n^2 ... | 3 | 0.666667 | 3 | This question can be separated into two parts. 
1. Start from the given form and prove equivalence to the trigonometric Fourier series by using the definitions of $a_n$ and $b_n$. 
2. Start from the trigonometric Fourier series and prove equivalence to the given equation using the definitions of $\alpha_n$... | To begin, we expand the $\cos(nx-\theta_n)$ term using the compound angle formula:
***
$$
\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \; .
$$
***
Inputting this expression into the given formula, we see that:
***
$$
\begin{aligned}
\alpha_n\cos(nx-\theta_n)
&= \alpha(\cos nx \cos \theta_n + \s... | In this question we are looking at the power spectrum of a Fourier series and why that is a useful quantity.
Let $P_n$ be given by: $P_n = a_n^2+b_n^2$. The distribution of the values of, $\lbrace{P_n\rbrace}$, is called the **power spectrum** of the Fourier series.
Show that writing down the Fourier series in the for... | 117 | 14 | 43 | 43 | 446 | 13 | 72 | 11 | 0 | Let $P_n$ be given by: $P_n = a_n^2+b_n^2$. Show that writing down the Fourier series in the form: $ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}{\alpha_n \cos(nx - \theta_n)} \ $ $ $ is equivalent to the trigonometric Fourier series if: $ \begin{aligned} a_n &= \alpha_n\cos\theta_n, ~~~~~ b_n &= \alpha_n\sin\theta_n \\ ... | 4 |
563d3f4f-c3cb-45c2-8814-261bb1869b10 | 5 | 1 | 0 | 6 | 4 | 1 | 0 | 0 | 
  
You are building a toy car to race in a miniature version of Formula 1. In this racing series, car weight is set at 8kg. You are required to... | The gearbox you have been supplied only came with the gear part numbers, which are as follows…
  
* G1 - 12
* G1 - 30
* G2 - 8
* G2 - 28
  
What is the step-down ratio of the gearbox?
\nGiven that the regulations for Formula Miniature stipulate that your car is allowed to travel no faste... | 4 | 0.666667 | 4 | Use your knowledge of gear selection and part numbers from DTP to determine which gears mesh with each other and how many teeth they each have.
\nAs with previous parts of this question set, use the wheel's circumference as a measure of the distance travelled in 1 revolution. This can then be used to find the wheel rot... | First, you need to determine that these gear part numbers represent a 12-tooth gear meshing with a 30-tooth gear, and an 8-tooth gear meshing with a 28-tooth gear.
Then, you can calculate the step-down ratio produced by the first stage:
step down $=\frac{30}{12}=2.5$
Next, find the step down for the second stage:
s... | 
  
You are building a toy car to race in a miniature version of Formula 1. In this racing series, car weight is set at 8kg. You are required to... | 257 | 1 | 22 | 22 | 487 | 0 | 212 | 1 | 1 | You are required to use a small motor with a speed vs torque curve shown below. embedded The gearbox you have been supplied only came with the gear part numbers, which are as follows… G1 - 12 G1 - 30 G2 - 8 G2 - 28 What is the step-down ratio of the gearbox? Given that the regulations for Formula Miniature stipulat... | 8 |
565ee457-579a-4492-bff9-38cefa3ecdff | 0 | 2 | 1 | 4 | 3 | 3 | 0 | 1 | Consider the mass flux of a contaminant.
| Choose Fick's Law, stating the meaning of the variables on paper.
\nChoose the correct dimensions below of $\boldsymbol{q}$, $D$, and $\nabla C$ in Fick's Law.
\nState the significance of the negative sign in Fick's Law, using a diagram (student's are expected to draw the diagram on paper, and compare with final answer... | 3 | 0.333333 | 2 | \n\nRefer to the theory of Fick's law, and think visually about plotting concentration with respect to spacial distance from the source, and how it changes with repsect to time.
| The mass flux of a contaminant arising from molecular transport can be described by Fick's Law
$$
\bm{q}=-D\nabla C.
$$
Here,
$$
\begin{array}{ll}
\bm{q} & \text{is the mass flux per unit area} \\
D & \text{is the diffusion coefficient} \\
\nabla C & \text{is the concentration gradient}
\end{array}
$$
\n$$
[\bm{q}] ... | Consider the mass flux of a contaminant.
Choose Fick's Law, stating the meaning of the variables on paper.
\nChoose the correct dimensions below of $\boldsymbol{q}$, $D$, and $\nabla C$ in Fick's Law.
\nState the significance of the negative sign in Fick's Law, using a diagram (student's are expected to draw the diagra... | 60 | 3 | 14 | 14 | 115 | 0 | 53 | 3 | 0 | Consider the mass flux of a contaminant. Choose Fick's Law, stating the meaning of the variables on paper. Choose the correct dimensions below of $\boldsymbol{q}$, $D$, and $ abla C$ in Fick's Law. State the significance of the negative sign in Fick's Law, using a diagram student's are expected to draw the diagram on p... | 4 |
57140949-6e46-4482-9225-df71ca7ee0e7 | 2 | 0 | 0 | 23 | 6 | 1 | 1 | 0 | Consider a point 1 m away from a point charge of +1 $\mu$C.
How are far along the radial direction is the potential due to the point charge ...
| 100 V higher?
| 1 | 0 | 1 | Firstly draw a diagram of what is being asked:
  
***
Equation for Electrostatic Potential:
$$
V = \frac{1}{4\pi\epsilon_0}\frac{q}{r}
$$
&... | Firstly draw a diagram of what is being asked:
  
***
Next, solve for the potential $V$ at a point 1m from the charge using the equation for el... | Consider a point 1 m away from a point charge of +1 $\mu$C.
How are far along the radial direction is the potential due to the point charge ...
100 V higher?
| 33 | 1 | 12 | 12 | 109 | 2 | 3 | 0 | 0 | Consider a point 1 m away from a point charge of +1 $\mu$C. How are far along the radial direction is the potential due to the point charge ... 100 V higher? | 2 |
57425408-8cdf-43ee-9d17-04abee93bb6a | 6 | 0 | 1 | 18 | 6 | 1 | 0 | 0 | It is common practice to scale velocities by the speed of light, so we define $\beta=v/c$ and $\gamma = 1/ \sqrt{1-\beta^2}$. This means $\beta$ is a dimensionless number between $-1$ and $1$. | For what positive values of $\beta$ is i) $\gamma=1.1$? ii) $\gamma = 2$? iii) $\gamma = 20$?\nWhat values of $\gamma$ are given by i) $\beta=0.09$? ii) $\beta=0.90$? iii) $\beta=0.99$?\nFor small values of $\beta$, derive an approximate expression for $\gamma$ in terms of $\beta$, keeping terms up to $\beta^2$. (Hint:... | 3 | 0.333333 | 2 | Can you rearrange the expression for $\gamma$ to make $\beta$ the subject?
***
   
You should find that $\beta = \pm \sqrt{1-1/\gamma^2}$\n\nThe binomial expansion formula is:
$$
(1+x)^n \approx 1 + nx + \frac{n(n-1)}{2!}x^2 + ...
$$
***
  ... | $\gamma = 1.1$ implies $1/\sqrt{1-\beta^2}=1.1$ , or $1-\beta^2 = 1/1.21$ . 
***
Thus $\beta = 0.4166$.
 Similarly, for $\gamma = 2$, $\beta = 0.8660$...
***
...and for $\gamma = 20$, $\beta = 0.9987$.\n$\gamma = 1/\sqrt{1-\beta^2}$ 
***
So, for $\beta=0.09$ then $\gamma=1/\sqrt{0.9919} = 1.004$... | It is common practice to scale velocities by the speed of light, so we define $\beta=v/c$ and $\gamma = 1/ \sqrt{1-\beta^2}$. This means $\beta$ is a dimensionless number between $-1$ and $1$.For what positive values of $\beta$ is i) $\gamma=1.1$? ii) $\gamma = 2$? iii) $\gamma = 20$?\nWhat values of $\gamma$ are given... | 102 | 19 | 21 | 21 | 83 | 11 | 72 | 14 | 0 | This means $\beta$ is a dimensionless number between $-1$ and $1$.For what positive values of $\beta$ is i $\gamma=1.1$? ii $\gamma = 2$? iii $\gamma = 20$? What values of $\gamma$ are given by i $\gamma = 1/ \sqrt{1-\beta^2}$0? ii $\gamma = 1/ \sqrt{1-\beta^2}$1? iii $\gamma = 1/ \sqrt{1-\beta^2}$2? For small values o... | 9 |
577b2436-099e-43e0-8081-574f038c8b0e | 0 | 0 | 1 | 18 | 6 | 1 | 1 | 0 | It seems obvious that we can find the inverse Lorentz transformations by swapping the primes and reversing the sign of $v$ (or equivalently $\beta$),
that is
$$
ct^\prime = \gamma (ct-\beta x),\qquad
x^\prime = \gamma(x-\beta ct)
$$
becomes
$$
ct = \gamma (ct^\prime+ \beta x^\prime),\qquad
x = \gamma(x^\prime+ \bet... | It seems obvious that we can find the inverse Lorentz transformations by swapping the primes and reversing the sign of $v$ (or equivalently $\beta$),
that is
$$
ct^\prime = \gamma (ct-\beta x),\qquad
x^\prime = \gamma(x-\beta ct)
$$
becomes
$$
ct = \gamma (ct^\prime+ \beta x^\prime),\qquad
x = \gamma(x^\prime+ \bet... | 1 | 0.333333 | 2 | This question requires no knowledge of relativity.
***
   
There are two equations and two 'unknown' variables, $x$ and $t$ - how do you solve such a system?
***
   
What happens if you multiply the equation for $t'$ by $\beta$ and add it to the equation for $x... | Dividing by $\gamma$ gives
$$
\frac{ct^\prime}{\gamma} = ct-\beta x,\qquad
\frac{x^\prime}{\gamma} = x-\beta ct
$$
***
The second of these gives $\beta x^\prime/\gamma = \beta x-\beta^2 ct$. 
***
Adding this to the first equation leads to...
***
   
$$
\frac... | It seems obvious that we can find the inverse Lorentz transformations by swapping the primes and reversing the sign of $v$ (or equivalently $\beta$),
that is
$$
ct^\prime = \gamma (ct-\beta x),\qquad
x^\prime = \gamma(x-\beta ct)
$$
becomes
$$
ct = \gamma (ct^\prime+ \beta x^\prime),\qquad
x = \gamma(x^\prime+ \bet... | 49 | 6 | 6 | 6 | 39 | 9 | 49 | 6 | 0 | It seems obvious that we can find the inverse Lorentz transformations by swapping the primes and reversing the sign of $v$ or equivalently $\beta$, that is $ ct^\prime = \gamma (ct-\beta x),\qquad x^\prime = \gamma(x-\beta ct) $ becomes $ ct = \gamma (ct^\prime+ \beta x^\prime),\qquad x = \gamma(x^\prime+ \beta ct^\pri... | 1 |
577cc9c4-acd9-4e75-a759-7500c1713a43 | 1 | 0 | 0 | 22 | 6 | 1 | 2 | 2 | Calculate the speed of a longitudinal wave travelling along an aluminium rod. You will need to look up the density and Young’s modulus of the material.
| Calculate the speed of a longitudinal wave travelling along an aluminium rod. You will need to look up the density and Young’s modulus of the material.
| 1 | 0.333333 | 1 | Look up the density and Young's modulus of aluminium. The main purpose here is to look things up correctly and work out how to convert the units if they are not in the units you want. This is an important skill - don't jump to the next hint until you've done this!!
***
The values I found are $Y=69\times 10^9$ Pa and... | Approximate values are $Y=69\times 10^9$ Pa and $\rho = 2710$ kg m$^{-3}$. 
***
The speed is $\sqrt{Y/\rho} = 5046$ m s$^{-1}$.
| Calculate the speed of a longitudinal wave travelling along an aluminium rod. You will need to look up the density and Young’s modulus of the material.
| 26 | 0 | 5 | 5 | 18 | 4 | 26 | 0 | 0 | Calculate the speed of a longitudinal wave travelling along an aluminium rod. | 1 |
5788abb2-f4a6-4bc9-abcb-c2794afe447f | 1 | 0 | 1 | 16 | 6 | 1 | 0 | 4 | If $r(1 + \cos\theta) = 2$, where $r$ and $\theta$ are plane polar coordinates,
| express this equation in terms of Cartesian coordinates $(x, y)$, with $y$ as the subject, and show it is a parabola.
\nSketch the parabola.
| 2 | 0.333333 | 0 | Transformations between Cartesian and Polar coordinates:
\
$r=\sqrt{x^2+y^2}$ $\displaystyle \theta = \arctan{y\over x}$ $x=r\cos\theta$ $y=r\sin\theta$
Now can you express it in Cartesian coordinates?
***
Rearrange to get it into the form of a parabola.
***
Remember that the parabola could be in any... | $$
r(1+\cos\theta)=2
$$
Transformations between Cartesian and Polar coordinates:
\
$r=\sqrt{x^2+y^2}$ $\displaystyle \theta = \arctan{y\over x}$ $x=r\cos\theta$ $y=r\sin\theta$
Now can you express it in Cartesian coordinates? Rearrange to get it into the form of a parabola.
***
Remember that the parabol... | If $r(1 + \cos\theta) = 2$, where $r$ and $\theta$ are plane polar coordinates,
express this equation in terms of Cartesian coordinates $(x, y)$, with $y$ as the subject, and show it is a parabola.
\nSketch the parabola.
| 35 | 5 | 17 | 17 | 120 | 6 | 24 | 2 | 0 | Sketch the parabola. | 1 |
58763411-9d04-476f-8e8e-7f96f3dc99df | 1 | 0 | 0 | 2 | 1 | 2 | 1 | 4 | The velocity distribution in a pipe with diameter $10\,$cm is given to be $u_z(r)=30(1-\frac{r^2}{r_0^2})\,$m/s, where $r_0\,$is the radius of the pipe. What is the shear stress at the wall if the fluid has a viscosity, $\mu$?
| The velocity distribution in a pipe with diameter $10\,$cm is given to be $u_z(r)=30(1-\frac{r^2}{r_0^2})\,$m/s, where $r_0\,$is the radius of the pipe. What is the shear stress at the wall if the fluid has a viscosity, $\mu$?
| 1 | 0.333333 | 1 | Find the *expression* for the shear stress, which is the viscosity multiplied by the normal gradient of the tangential velocity (i.e. the derivative of $u(r)$ with respect to $r$)
***
Take the derivative to find a general expression for the shear stress.
***
Substitute in $r=r_o$ to find the shear stress at the w... | At $r=r_0$, the normal gradient of the tangential velocity is
$\frac{\delta u}{\delta r}=\frac{\delta (30(1-r^2/{r_0}^2))}{\delta r}=-60r/{r_0}^2=-60/r_0=-60/0.05=-1200$
so the shear stress there equals $-1200\mu~kg/m/s^2$ (in the $-z$ direction).
| The velocity distribution in a pipe with diameter $10\,$cm is given to be $u_z(r)=30(1-\frac{r^2}{r_0^2})\,$m/s, where $r_0\,$is the radius of the pipe. What is the shear stress at the wall if the fluid has a viscosity, $\mu$?
| 40 | 4 | 4 | 4 | 23 | 4 | 40 | 4 | 0 | The velocity distribution in a pipe with diameter $10\,$cm is given to be $u_z(r)=30(1-\frac{r^2}{r_0^2})\,$m/s, where $r_0\,$is the radius of the pipe. What is the shear stress at the wall if the fluid has a viscosity, $\mu$? | 2 |
58a0c02b-33e9-4db6-9dea-28a2cf85cb34 | 0 | 0 | 2 | 18 | 6 | 1 | 1 | 7 | The four-velocity is given by $(\gamma_u,\gamma_u\vec{\beta}_u)$, where $\vec{u}$ is the usual three-velocity of a particle, $\vec{\beta}_u = \vec{u}/c$ and $\gamma_u = 1/\sqrt{1-|\vec{\beta}_u|^2}$. | What is the squared-length of the four-velocity?\nApply a Lorentz transformation to the four-velocity formed from the three-velocity $\vec{u}=(u,0,0)$. From this, retrieve the velocity addition formula: $u^\prime = \frac{u-v}{1-uv/c^2}.$ | 2 | 0.333333 | 2 | What is the formula for the length squared of a four vector? How is it different to finding the length of a normal three-vector?
***
   
The squared length of the four vector $(\gamma_u,\gamma_u\vec{\beta}_u)$ is given by $\gamma_u^2 - \gamma_u^2|\vec{\beta}_u|^2$\nHow do you ap... | The squared length is given by $\gamma_u^2 - \gamma_u^2 |\vec{\beta}_u|^2 = \gamma_u^2 (1-|\vec{\beta}_u|^2) = 1$.\nApplying the Lorentz transformation to the given four-velocity gives
***
$$
\gamma_u^\prime = \gamma (\gamma_u - \beta \gamma_u \beta_u),\qquad
\gamma_u^\prime \beta_u^\prime = \gamma (\gamma_u\beta_u -... | The four-velocity is given by $(\gamma_u,\gamma_u\vec{\beta}_u)$, where $\vec{u}$ is the usual three-velocity of a particle, $\vec{\beta}_u = \vec{u}/c$ and $\gamma_u = 1/\sqrt{1-|\vec{\beta}_u|^2}$.What is the squared-length of the four-velocity?\nApply a Lorentz transformation to the four-velocity formed from the thr... | 46 | 6 | 6 | 6 | 45 | 4 | 27 | 2 | 0 | The four-velocity is given by $(\gamma_u,\gamma_u\vec{\beta}_u)$, where $\vec{u}$ is the usual three-velocity of a particle, $\vec{\beta}_u = \vec{u}/c$ and $\gamma_u = 1/\sqrt{1-[table]^2}$.What is the squared-length of the four-velocity? Apply a Lorentz transformation to the four-velocity formed from the three-veloci... | 2 |
58e2a175-e436-40fb-9613-87a62c94fcc9 | 1 | 1 | 0 | 4 | 3 | 3 | 6 | 2 | A company releases a sand mining disposal in the water surface of a wide river of water depth $h =6.91\ \mathrm{m}$ and average river slope of $S = 1.26 \times 10^{-4}$ . The sand disposal is characterised by a medium sediment size $d_{50} = 0.25\ \mathrm{mm}$.
| Due to the potential contamination by the sediment disposal, the company is interested in knowing whether the material will be transported in suspension with the river or whether it will settle on the bed.
\nCompute the maximum river specific flow rate that would lead to sediment deposition.
| 2 | 0.666667 | 2 | \n | The bed shear stress is computed from
$$
\tau_0 = \rho g S h = 1000\cdot 9.81 \cdot 6.91 \cdot 1.26 \times 10^{-4} = 8.541\ \mathrm{N/m^2}.
$$
The shear velocity is given as
$$
u_* = \sqrt{\dfrac{\tau_0}{\rho}} = \sqrt{8.541 \cdot 1000} = 0.09242\ \mathrm{m/s}.
$$
***
We then compute the settling velocity for the ... | A company releases a sand mining disposal in the water surface of a wide river of water depth $h =6.91\ \mathrm{m}$ and average river slope of $S = 1.26 \times 10^{-4}$ . The sand disposal is characterised by a medium sediment size $d_{50} = 0.25\ \mathrm{mm}$.
Due to the potential contamination by the sediment disposa... | 85 | 3 | 19 | 19 | 197 | 0 | 47 | 0 | 0 | Compute the maximum river specific flow rate that would lead to sediment deposition. | 1 |
58eaf562-be8a-41a5-8583-6195c9ac6abf | 2 | 0 | 0 | 9 | 4 | 2 | 6 | 10 | A permanent magnet D.C. motor has the following characteristics: 
  
Moment of inertia: $\hspace{10 mm} J = 2 \times 10^{-4}~\mathrm{kgm^2}$ 
Armature resistance: $\hspace{6 mm} R_\mathrm{a} = 300 ~\Omega$ 
Back e.m.f. const: $\hspace{10 mm} K_\mathrm{e} = 0.4 ~\mathrm{Vs/rad}$ 
Motor... | Calculate the motor time constant and the final speed with $200 ~\mathrm{V}$ D.C. applied.
| 1 | 0.666667 | 2 | null | From the lecture notes Section 7.3.5, the gain and time constant of a D.C. motor can be written 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 full ... | A permanent magnet D.C. motor has the following characteristics: 
  
Moment of inertia: $\hspace{10 mm} J = 2 \times 10^{-4}~\mathrm{kgm^2}$ 
Armature resistance: $\hspace{6 mm} R_\mathrm{a} = 300 ~\Omega$ 
Back e.m.f. const: $\hspace{10 mm} K_\mathrm{e} = 0.4 ~\mathrm{Vs/rad}$ 
Motor... | 45 | 6 | 8 | 8 | 87 | 0 | 13 | 1 | 0 | const: $\hspace{10 mm} K_\mathrm{e} = 0.4 ~\mathrm{Vs/rad}$ Motor constant: $\hspace{13 mm} K_\mathrm{t} = 0.4 ~\mathrm{Nm/A}$ Viscous friction:$\hspace{13 mm} K_\mathrm{f} = 0$ Calculate the motor time constant and the final speed with $200 ~\mathrm{V}$ D.C. applied. | 1 |
5949ab93-b491-4ccb-8e93-b7fb4578d384 | 0 | 0 | 1 | 11 | 4 | 2 | 7 | 5 | How is the enthalpy of formation of $\mathrm{CO}_2$ defined? How is it related to the calorific value of carbon (graphite)?
| How is the enthalpy of formation of $\mathrm{CO}_2$ defined? How is it related to the calorific value of carbon (graphite)?
| 1 | 0.333333 | 0 | null | null | How is the enthalpy of formation of $\mathrm{CO}_2$ defined? How is it related to the calorific value of carbon (graphite)?
| 20 | 1 | 0 | 0 | 0 | 0 | 20 | 1 | 0 | How is the enthalpy of formation of $\mathrm{CO}_2$ defined? How is it related to the calorific value of carbon graphite? | 2 |
598da216-0699-4875-8c08-04fb6362dc7b | 0 | 0 | 1 | 4 | 3 | 3 | 1 | 12 | For the two diffusivities mentioned in the previous question ($D = 0.202\ \mathrm{cm^{2}/s}$ and $D = 10^4\ \mathrm{cm^{2}/s}$), plot the distance in the $x$ direction where the concentration exceeds $C = 4\ \mathrm{mg/m^3}$ ($y$-axis) as a function of time ($x$-axis).
Hint: You may need to use the $\tt erfcinv()$ f... | For the two diffusivities mentioned in the previous question ($D = 0.202\ \mathrm{cm^{2}/s}$ and $D = 10^4\ \mathrm{cm^{2}/s}$), plot the distance in the $x$ direction where the concentration exceeds $C = 4\ \mathrm{mg/m^3}$ ($y$-axis) as a function of time ($x$-axis).
Hint: You may need to use the $\tt erfcinv()$ f... | 1 | 1 | 2 | null | Treating the garage as infinitely long $(L \rightarrow \infty)$, we determine the values of $x = x^*$ for which $C = 4\ \mathrm{mg/m^3}$ as a function of time $t$. Therefore, $C > 4\ \mathrm{mg/m^3}$ for $x < x^{*}$. From $C(x, t) = \frac{C_0}{2}\mathrm{erfc}\left(\frac{x}{\sqrt{2} \sigma}\right)$ we get
$$
C^* - \dfr... | For the two diffusivities mentioned in the previous question ($D = 0.202\ \mathrm{cm^{2}/s}$ and $D = 10^4\ \mathrm{cm^{2}/s}$), plot the distance in the $x$ direction where the concentration exceeds $C = 4\ \mathrm{mg/m^3}$ ($y$-axis) as a function of time ($x$-axis).
Hint: You may need to use the $\tt erfcinv()$ f... | 48 | 7 | 5 | 5 | 82 | 0 | 48 | 7 | 0 | For the two diffusivities mentioned in the previous question $D = 0.202\ \mathrm{cm^{2}/s}$ and $D = 10^4\ \mathrm{cm^{2}/s}$, plot the distance in the $x$ direction where the concentration exceeds $C = 4\ \mathrm{mg/m^3}$ $y$-axis as a function of time $x$-axis. Hint: You may need to use the $\tt erfcinv()$ function i... | 2 |
59ab6d91-f676-4b4f-8b80-ffc9c0302de0 | 6 | 0 | 1 | 22 | 6 | 1 | 1 | 6 | Two equal masses (each of mass $m$) are connected to each other and to two adjacent walls by identical springs (spring constant $\kappa$). The walls are a distance $L$ apart. The equilibrium configuration is shown in the figure. The masses can only move perpendicular to the walls.
 from equilibrium, show that their equations of motion are
	 $m \ddot{x}_1 = -2\kappa x_1 + \kappa x_2$,
	 $m \ddot{x}_2 = \kappa x_1 - 2\kappa x_2$.	
\nWe now look for normal modes of the system. These are solutions where both masses oscillat... | 3 | 0.666667 | 4 | First figure out the the extensions of each of the three springs in terms of $x_1$ and $x_2$, taking care about the signs.
***
Work out the total force on mass 1, bearing in mind that there is a force coming from the left spring and from the middle spring.
Do the same for mass 2.
Apply Newton's second law.
\nWork ... | Let the displacements from equilibrium be $x_1$ and $x_2$. That means that the extensions of the three springs, from left to right are $x_1$, $x_2-x_1$ and $-x_2$. 
***
The total force on mass 1 is 
	 $F_1 = -\kappa x_1 + \kappa(x_2-x_1) = -2\kappa x_1 + \kappa x_2 = m \ddot{x}_1$.
The total force on... | Two equal masses (each of mass $m$) are connected to each other and to two adjacent walls by identical springs (spring constant $\kappa$). The walls are a distance $L$ apart. The equilibrium configuration is shown in the figure. The masses can only move perpendicular to the walls.
 = A_1\cos(\omeg... | 4 |
5a2a2425-d81a-4117-86e5-314af49e923f | 0 | 1 | 0 | 7 | 4 | 2 | 0 | 1 | The heat flux that is applied to the left face of a plane wall is $q^{\prime \prime}=25 \mathrm{~W} / \mathrm{m}^2$. The wall is of thickness $L=12 \mathrm{~mm}$ and of thermal conductivity $k=12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. If the surface temperatures of the wall are measured to be $50^{\circ} \mathrm{C... | The heat flux that is applied to the left face of a plane wall is $q^{\prime \prime}=25 \mathrm{~W} / \mathrm{m}^2$. The wall is of thickness $L=12 \mathrm{~mm}$ and of thermal conductivity $k=12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. If the surface temperatures of the wall are measured to be $50^{\circ} \mathrm{C... | 1 | 0.333333 | 0 | null | If steady-state conditions exist, Fourier's Law can be applied:
  
$q^{\prime\prime} = k\frac{\Delta T}{L}$
***
Calculate $q^{\prime\prime}$ assuming steady-state conditions:
  
$q^{\prime\prime} = 12\times\frac{50-30}{0.012} = 20000~\mathrm{W/m^2}$
  
which is much higher than the... | The heat flux that is applied to the left face of a plane wall is $q^{\prime \prime}=25 \mathrm{~W} / \mathrm{m}^2$. The wall is of thickness $L=12 \mathrm{~mm}$ and of thermal conductivity $k=12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. If the surface temperatures of the wall are measured to be $50^{\circ} \mathrm{C... | 55 | 5 | 4 | 4 | 56 | 0 | 55 | 5 | 0 | The heat flux that is applied to the left face of a plane wall is $q^{\prime \prime}=25 \mathrm{~W} / \mathrm{m}^2$. If the surface temperatures of the wall are measured to be $50^{\circ} \mathrm{C}$ on the left side and $30^{\circ} \mathrm{C}$ on the right side, do steady-state conditions exist? | 2 |
5a42f703-3644-4f9b-b2f7-12f8d5361b29 | 1 | 0 | 0 | 11 | 4 | 2 | 2 | 2 | A sample of a wet steam is passed through a well-insulated throttling calorimeter in order to measure its dryness fraction. The inlet pressure $P_1$ is measured as $8.99$ bar gauge and the sample is throttled to atmospheric pressure $P_2$, measured as $1010$ mbar. The calorimeter exit temperature $T_2$ is measured as $... | Sketch the process on an $h - s$ diagram and evaluate the dryness fraction $x_1$ of the wet steam sample.
| 1 | 0.666667 | 2 | null | In an $h-s$ diagram, the bell is slightly skewed to the left. The question states that state $1$ is in the wet steam region. Moreover, a throttling process is isenthalpic (see below) but entropy increases. Therefore, the diagram looks something like this:
  
The... | Determine the velocity $U$ of the centre of the ball as it starts to roll up the incline in terms of its initial velocity $v = 10~\mathrm{m/s}$, and the angle of the incline $\theta = 30^o$.
*(Take rotation in the clockwise direction to be positive)*
\nFind the percentage of energy lost in the collision.
| 2 | 0.666667 | 4 | The first step to solving this problem is the same as the previous question –write down the angular momentum equations about a suitable point before and after impact, again in terms of the usual variables.
***
Try choosing the point at which the ball will first come in contact with the incline as it rolls towards it... | 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 uniform spherical ball rolls without slipping along a horizontal plane towards an incline. The ball neither slips nor rebounds, as pictured below:
The... | 82 | 5 | 21 | 21 | 137 | 1 | 52 | 3 | 1 | Determine the velocity $U$ of the centre of the ball as it starts to roll up the incline in terms of its initial velocity $v = 10~\mathrm{m/s}$, and the angle of the incline $\theta = 30^o$. Take rotation in the clockwise direction to be positive Find the percentage of energy lost in the collision. | 2 |
5b97641a-6952-44f2-b98a-388f8b0f6d82 | 0 | 0 | 4 | 14 | 4 | 2 | 9 | 1 | Consider a generic velocity vector $\vec{u}=\left[u ~~ v ~~ w\right]^{T}$ in a Cartesian frame of reference. | Write the velocity gradient tensor in Cartesian coordinates\nWrite the deformation tensor in Cartesian coordinates\nWrite the spin tensor in Cartesian coordinates\nWrite the vorticity vector in Cartesian coordinates | 4 | 0.333333 | 2 | \n\n\n | \n\n\n | Consider a generic velocity vector $\vec{u}=\left[u ~~ v ~~ w\right]^{T}$ in a Cartesian frame of reference.Write the velocity gradient tensor in Cartesian coordinates\nWrite the deformation tensor in Cartesian coordinates\nWrite the spin tensor in Cartesian coordinates\nWrite the vorticity vector in Cartesian coordina... | 37 | 1 | 0 | 0 | 1 | 0 | 26 | 0 | 0 | Consider a generic velocity vector $\vec{u}=\left[u ~~ v ~~ w\right]^{T}$ in a Cartesian frame of reference.Write the velocity gradient tensor in Cartesian coordinates Write the deformation tensor in Cartesian coordinates Write the spin tensor in Cartesian coordinates Write the vorticity vector in Cartesian coordinates | 1 |
5c9b67cf-710b-407c-96f6-c44f08c658fc | 3 | 3 | 0 | 2 | 1 | 2 | 6 | 4 | Let us revisit the problem on the sudden expansion in a previous problem set.
| Write down an expression for the total head at surface 1, $p_{T1}$, if gravitational effects can be neglected.
\nWrite down an expression for the total head at surface 2, $p_{T2}$, if gravitational effects can be neglected.
\nRecall from the previous problem that we have $u_2=\frac{A_1}{A_2}u_1$ and $p_2=p_1+\rho u_1^2... | 5 | 0.333333 | 2 | Your gravitational term can be left out.
\n\nSubstitute the values given in the question into the equation for $p_{T2}$ and simplify.
\n\nThe volumetric flow is $u_1A_1$, and the energy lost per unit volume is $p_{T1}-p_{T2}$.
So if we multiply the volumetric flow by the energy lost per unit volume, we will get the to... | \n\n$p_{T2}=p_2+\frac12\rho u_2^2$
$=p_1+\rho u_1^2\frac{A_1}{A_2}(1-\frac{A_1}{A_2})+\frac12\rho \frac{A_1^2}{A_2^2}u_1^2$
$=p_1+\frac12\rho u_1^2\frac{A_1}{A_2}(2-\frac{A_1}{A_2}).$
\nA sketch of the quadratic function appearing at the end of the expression for $p_{T2}$ is shown below:
Write down an expression for the total head at surface 1, $p_{T1}$, if ... | 156 | 14 | 15 | 15 | 105 | 4 | 141 | 14 | 1 | Let us revisit the problem on the sudden expansion in a previous problem set. Write down an expression for the total head at surface 1, $p_{T1}$, if gravitational effects can be neglected. Write down an expression for the total head at surface 2, $p_{T2}$, if gravitational effects can be neglected. Use this to give an ... | 7 |
5cecb4ff-f8df-4f8a-a4cd-6dad1b117cfc | 4 | 0 | 0 | 11 | 4 | 2 | 0 | 8 | Nitrogen enters a turbine at a pressure of $4$ bar (absolute) and a temperature of $30^{\circ}\mathrm{C}$, and leaves at a pressure of $1$ bar (absolute) and a temperature of $–50^{\circ}\mathrm{C}$. Kinetic energies at entry and exit are negligible and nitrogen can be modelled as a perfect gas under these conditions.
| If the process is reversible, and its path on the $T–s$ diagram is a straight line (not necessarily isentropic), evaluate the heat transfer and the shaft work per unit mass of nitrogen. (*Hint: recall the significance of the area under a reversible process path on a $T–s$ diagram.*)
\nA different process between the sa... | 2 | 0.666667 | 2 | \n | Draw out the process on a $T-S$ diagram, bearing in mind that the entropy change is yet unknown:
  
***
The area under a $T-s$ diagram (for a... | Nitrogen enters a turbine at a pressure of $4$ bar (absolute) and a temperature of $30^{\circ}\mathrm{C}$, and leaves at a pressure of $1$ bar (absolute) and a temperature of $–50^{\circ}\mathrm{C}$. Kinetic energies at entry and exit are negligible and nitrogen can be modelled as a perfect gas under these conditions.
... | 139 | 6 | 17 | 17 | 280 | 0 | 87 | 2 | 0 | A different process between the same end states is adiabatic; evaluate the shaft work per unit mass of nitrogen and the change in specific entropy of the gas in this process, and determine whether it is reversible or irreversible. | 1 |
5d175063-e928-4c16-972d-c11878a6fb66 | 0 | 0 | 1 | 4 | 3 | 3 | 4 | 5 | Explain in your own words the physical meaning of shear velocity.
| Explain in your own words the physical meaning of shear velocity.
| 1 | 0.666667 | 1 | null | null | Explain in your own words the physical meaning of shear velocity.
| 11 | 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 | Explain in your own words the physical meaning of shear velocity. | 1 |
5d926bad-dc4a-4cbf-9a72-df393075ff0a | 1 | 0 | 0 | 24 | 6 | 1 | 0 | 15 | *\[Riley 6.18]* Sketch the domain of integration for the integral:
$$
\cal{I} = \int_0^1 \int_{x=y}^{1/y} \frac{y^3}{x} \exp \left[ y^2 (x^2 + x^{-2})\right]\, dx\, dy
$$
and characterise its boundaries in terms of new variables $u=xy$ and $v=y/x$. Show that the Jacobian for the change from $(x,y)$ to $(u,v)$ is equ... | *\[Riley 6.18]* Sketch the domain of integration for the integral:
$$
\cal{I} = \int_0^1 \int_{x=y}^{1/y} \frac{y^3}{x} \exp \left[ y^2 (x^2 + x^{-2})\right]\, dx\, dy
$$
and characterise its boundaries in terms of new variables $u=xy$ and $v=y/x$. Show that the Jacobian for the change from $(x,y)$ to $(u,v)$ is equ... | 1 | 1 | 3 | Changing variables in integration involves the following:
1. Changing limits
2. Changing the integrand
3. Changing the differential using the Jacobian. 
***
Sketch the region in the $xy$ plane and identify the boundaries.
***
Convert the $(x,y)$ limits to $(u,v)$. You should have found one limit to be $y... | Region of integration in $xy$ plane:
Starting by changing the limits $y=1/x$ and $y=x$ to the new coordinate system $(u,v)$ (we will deal with $y=0$ later)... | *\[Riley 6.18]* Sketch the domain of integration for the integral:
$$
\cal{I} = \int_0^1 \int_{x=y}^{1/y} \frac{y^3}{x} \exp \left[ y^2 (x^2 + x^{-2})\right]\, dx\, dy
$$
and characterise its boundaries in terms of new variables $u=xy$ and $v=y/x$. Show that the Jacobian for the change from $(x,y)$ to $(u,v)$ is equ... | 102 | 13 | 29 | 29 | 172 | 16 | 102 | 13 | 0 | Show that the Jacobian for the change from $(x,y)$ to $(u,v)$ is equal to $(2v)^{-1}$. To understand what is going on, look at the transformation of the line $ y= $ constant to the $u=xy$0 plane, for constant $u=xy$1, and this should explain what happens to the line $u=xy$2. | 2 |
5e72706e-d114-47ad-a37d-d01f74ec6e6b | 0 | 0 | 4 | 14 | 4 | 2 | 9 | 4 | Solve Problem 10.4 by considering a cylindrical frame of reference attached to the ground, with origin in the centre of the tank and with axial direction $\vec{e}_z$. | Determine an expression for the velocity field\nDetermine the deformation tensor.\nDetermine the spin tensor\nDetermine the vorticity | 4 | 0.333333 | 3 | \n\n\n | \n\n\n | Solve Problem 10.4 by considering a cylindrical frame of reference attached to the ground, with origin in the centre of the tank and with axial direction $\vec{e}_z$.Determine an expression for the velocity field\nDetermine the deformation tensor.\nDetermine the spin tensor\nDetermine the vorticity | 42 | 1 | 0 | 0 | 1 | 0 | 15 | 0 | 0 | Solve Problem 10.4 by considering a cylindrical frame of reference attached to the ground, with origin in the centre of the tank and with axial direction $\vec{e}_z$.Determine an expression for the velocity field Determine the deformation tensor. Determine the spin tensor Determine the vorticity | 2 |
5e9131cd-ed1c-40a2-9bb7-bf14211657a1 | 4 | 0 | 0 | 11 | 4 | 2 | 3 | 9 | A mixture of $1 ~\mathrm{kg} ~ \mathrm{CO}$ and $1~ \mathrm{kg} ~ \mathrm{H}_2$ in a sealed, rigid container is at a pressure of $2$ bar and a temperature of $20^{\circ}\mathrm{C}$. The container is heated until the pressure is $4$ bar. Find:
| The final temperature of the mixture.
\nThe change in specific enthalpy of the mixture.
\nThe change in specific entropy of the mixture.
\nThe magnitude of the heat transfer.
| 4 | 0.666667 | 2 | \n\n\n | Since the mixture can be treated as a perfect gas, the ideal gas equation can be used:
  
$PV = n\bar{R}T$
***
where $n$ is the total number of moles in the mixture and can be calculated as follows:
  
$n =\Sigma (\frac{m_\mathrm{i}}{M_\mathrm{i}})$
***
Substituting in numbers:
  ... | A mixture of $1 ~\mathrm{kg} ~ \mathrm{CO}$ and $1~ \mathrm{kg} ~ \mathrm{H}_2$ in a sealed, rigid container is at a pressure of $2$ bar and a temperature of $20^{\circ}\mathrm{C}$. The container is heated until the pressure is $4$ bar. Find:
The final temperature of the mixture.
\nThe change in specific enthalpy of th... | 63 | 5 | 25 | 25 | 213 | 0 | 28 | 0 | 0 | Find: The final temperature of the mixture. | 1 |
5eabf4ab-0d86-49d4-a486-8e28845a586a | 1 | 0 | 0 | 13 | 4 | 2 | 0 | 1 | The Imperial Racing Green car below is initially horizontal, has a mass of $2.1~\mathrm{tonnes}$ and its centre of mass is $1.5~\mathrm{m}$ above and $2.0~\mathrm{m}$ forward of the point where the rear wheel touches the ground. The moment of inertia of the car about its centre of mass is $3000~\mathrm{kgm^2}$.
*
| 1 | 0.666667 | 3 | Think also about the acceleration of the centre of gravity of the car in all directions. Again, pick a suitable point around which to take moments (which point does the car rotate about whilst doing a wheelie?).
***
You will need to refer to the vector equation in your notes to relate the accelerations of different ... | Free body diagram and kinematic diagram:
***
Resolving forces in the x-axis:
$$
\sum F_x = ma_{G_x}
$$
$$
F_B = ma_{G_x}~~\mathrm{(Equation 1)}
$... | The Imperial Racing Green car below is initially horizontal, has a mass of $2.1~\mathrm{tonnes}$ and its centre of mass is $1.5~\mathrm{m}$ above and $2.0~\mathrm{m}$ forward of the point where the rear wheel touches the ground. The moment of inertia of the car about its centre of mass is $3000~\mathrm{kgm^2}$.
 ...
***
... The problem is that the question only calculates how the work done changes the KE of the walker, ignoring the fact that the train also speeds up or slows down. What law is this a consequence ... | No, the walker does not use $41$ times more energy.
***
When viewed from the inertial frame of reference with velocity equal to the initial velocity of the train carriage, the person’s journey looks exactly the same in both cases. Since both frames are inertial, Newton’s laws apply equally well in both. The amount of... | [In a previous year, students were challenged to work with their tutorial groups to invent new questions to appear on the problem sheets. Here is one of the most interesting questions they came up with.]
  
A person of mass $m = 60\,$kg, standing in a stationary train carriage, starts walking towards the ... | 192 | 17 | 35 | 35 | 460 | 14 | 192 | 17 | 0 | According to the work-energy theorem which states that the work done equals the change in KE: $ \text{work done on person} = \text{KE}_{\text{final}} - \text{KE}_{\text{initial}} = \frac{1}{2}m v_w^2 - 0 = 30\,\text{J}. When the same person accelerates from rest in the carriage to a final velocity of $u = 0\,$1m$\cdot$... | 5 |
5eef3bdd-0639-4b55-aef4-81fbb780fb52 | 3 | 0 | 2 | 4 | 3 | 3 | 3 | 2 | A wide river is dredged to make it more navigable. This releases a significant mass $M$ of contaminated soil into the river. The dredged river-section is sufficiently large to model this problem as 1-dimensional (in $z$). Assume that the suspended soil behaves as a passive scalar that is released instantaneously at the... | Starting from the three-dimensional case, show that the above assumptions allow us to reduce the governing equations for the contaminant to one dimension. Give the corresponding boundary conditions.
\nThe eddy viscosity in the channel can be described by (see sediment transport lectures):
$$
\nu_T = u_* \kappa z\left(... | 4 | 1 | 2 | \n\n\n | \nSince $Sc_T = 1$, $ D_T = \nu_T = u_* \kappa z\left(1-\dfrac{z}{h}\right) $. 
We then take the integral of the expression of $D_T$ and averaging it over $h$ with
$$
\displaystyle \langle D_T \rangle = \dfrac{1}{h}\int_{0}^{h} {u_* \kappa z(1-\dfrac{z}{h})}\mathrm{d}z.
$$
***
This can be evaluated further un... | A wide river is dredged to make it more navigable. This releases a significant mass $M$ of contaminated soil into the river. The dredged river-section is sufficiently large to model this problem as 1-dimensional (in $z$). Assume that the suspended soil behaves as a passive scalar that is released instantaneously at the... | 170 | 6 | 14 | 14 | 101 | 0 | 100 | 4 | 1 | Assume that the suspended soil behaves as a passive scalar that is released instantaneously at the bed and that the flow conditions are those usually encountered for steady uniform open channel flows. Starting from the three-dimensional case, show that the above assumptions allow us to reduce the governing equations fo... | 6 |
5f1a9fd0-9e55-422f-be15-10c651210951 | 0 | 10 | 0 | 2 | 1 | 2 | 0 | 9 | Are the following flows steady? Explain your answer.
| Blood flow in the heart is
\nFlow in a river is
\nWater flow from a tap
\nFlow in a vein
| 4 | 0.333333 | 0 | \n\n\n | \n\n\n | Are the following flows steady? Explain your answer.
Blood flow in the heart is
\nFlow in a river is
\nWater flow from a tap
\nFlow in a vein
| 28 | 0 | 0 | 0 | 1 | 0 | 20 | 0 | 0 | Are the following flows steady? Explain your answer. | 2 |
5f89db98-8d5c-4e61-9c36-a7726b260a1f | 4 | 1 | 0 | 4 | 3 | 3 | 6 | 3 | A river stream has a flow rate of $q = 20.98\ \mathrm{m^2 /s}$ and a river depth of $h = 8.23\ \mathrm{m}$. The sediment of the bed has a typical sediment size of $d_{50} = 0.28\ \mathrm{mm}$.
For relatively large particles in large water depths, sediment suspension occurs in a small layer close to the bed. In such ca... | Using this simplified expression, calculate the sediment concentration at the following vertical elevations with respect to the bed: $z_1 = 0.0020\ \mathrm{m}$, $z_2 = 2.0\ \mathrm{m}$, and $z_3 = 4.0\ \mathrm{m}$.
\nCompare your values with the original Rouse profile. Is the assumption of suspended sediment transport ... | 3 | 1 | 2 | \n\n | Assuming a rough turbulent flow $z_0 = \dfrac{k_s}{30} =\dfrac{2.5 d_{50}}{30} = 2.333 \times 10^{-5}\ \mathrm{m}$, the friction coefficient is computed as
$$
C_f = \left[ \dfrac{1}{\kappa} \left( \ln \left( \dfrac{h}{z_0} \right) -1 \right) \right]^{-2}
= \left[ \dfrac{1}{0.41} \left( \ln \left( \dfrac{8.23}{2.333 \t... | A river stream has a flow rate of $q = 20.98\ \mathrm{m^2 /s}$ and a river depth of $h = 8.23\ \mathrm{m}$. The sediment of the bed has a typical sediment size of $d_{50} = 0.28\ \mathrm{mm}$.
For relatively large particles in large water depths, sediment suspension occurs in a small layer close to the bed. In such ca... | 130 | 8 | 30 | 30 | 266 | 0 | 56 | 3 | 0 | In such cases, we can assume that $z<<h$. Using this assumption the sediment concentration profile is simplified to $ \overline{C} = C_a \left( \dfrac{z}{z_a} \right)^{-b}. $ You may wish to use . Using this simplified expression, calculate the sediment concentration at the following vertical elevations with respect to... | 7 |
5fca4d66-12e3-4564-8582-c6cb93a7fa9c | 2 | 0 | 0 | 16 | 6 | 1 | 6 | 3 | Are the following exact differentials? If so - of what functions?
Yes - Input the function
No - Type 'none'
| $$
e^y \mathrm{d}x+ x(e^y + 1) \mathrm{d}y
$$
\n$$
(e^y + ye^x) \mathrm{d}x + (e^x + xe^y + 1) \, \mathrm{d}y
$$
| 2 | 0.333333 | 2 | The equation $P(x,y)\mathrm{d}x + Q(x,y)\mathrm{d}y$ is an exact differential when
$$
\begin{aligned} {\partial P\over\partial y} = {\partial Q\over\partial x} \end{aligned}
$$
is satisfied.
Now you can try identifying $P(x,y)$ and $Q(x,y)$, and then can you see if this is satisfied?
\nThe equation $P(x,y)\mathrm{d}... | The equation $P(x,y)\mathrm{d}x + Q(x,y)\mathrm{d}y$ is an exact differential when
$$
\begin{aligned} {\partial P\over\partial y} = {\partial Q\over\partial x} \end{aligned}
$$
is satisfied.
Now you can try identifying $P(x,y)$ and $Q(x,y)$, and then can you see if this is satisfied?
***
$$
\begin{aligned}
P(x,y) ... | Are the following exact differentials? If so - of what functions?
Yes - Input the function
No - Type 'none'
$$
e^y \mathrm{d}x+ x(e^y + 1) \mathrm{d}y
$$
\n$$
(e^y + ye^x) \mathrm{d}x + (e^x + xe^y + 1) \, \mathrm{d}y
$$
| 23 | 2 | 45 | 45 | 306 | 23 | 3 | 2 | 0 | Are the following exact differentials? If so - of what functions? | 2 |
5ff4fab4-25d8-4f7b-97b7-e9e89b8969e8 | 4 | 0 | 1 | 8 | 4 | 2 | 2 | 1 | A cylindrical pressure vessel is $0.6\text{ m}$ diameter, with hemispherical ends, and is made of steel with a yield strength of $250 \text{ MPa}$, $13 \text{ mm}$ thick. It is pressurised to $100 \text{ bar}.$
| Calculate the safety factor for the hemispherical ends, using both Tresca and Von Mises criteria.
\nCalculate the safety factor for the cylindrical part, using both criteria.
\nExplain, by drawing a yield locus, why the answers are the same in (a), but different in (b).
| 3 | 0.333333 | 2 | \n\n | For this pressure vessel with hemispherical ends, we know that:
* $\text{Diameter } (d)=0.6\text{ m}$
* $\text{Thickness } (t)=13\text{ mm}$
* $\text{Pressure } (P)=100\text{ bar}$
* $\text{Yield Strength } (\sigma_y)=250\text{ MPa}$
   
***
The stress in the hemispherical ends is the same i... | A cylindrical pressure vessel is $0.6\text{ m}$ diameter, with hemispherical ends, and is made of steel with a yield strength of $250 \text{ MPa}$, $13 \text{ mm}$ thick. It is pressurised to $100 \text{ bar}.$
Calculate the safety factor for the hemispherical ends, using both Tresca and Von Mises criteria.
\nCalculate... | 73 | 4 | 10 | 10 | 112 | 0 | 44 | 0 | 0 | It is pressurised to $100 \text{ bar}.$ Calculate the safety factor for the hemispherical ends, using both Tresca and Von Mises criteria. Calculate the safety factor for the cylindrical part, using both criteria. Explain, by drawing a yield locus, why the answers are the same in a, but different in b. | 3 |
6124b15c-8d77-49b9-8d46-129ffdf45e53 | 4 | 0 | 0 | 10 | 4 | 2 | 7 | 0 | A crack of length $10\text{ mm}$ has been found at the center of a large thin plate, with an end load of $150\text{ kN}$. The plate is $100\text{ mm}$ wide and $5\text{ mm}$ thick. Assuming that the material is **perfectly linear elastic.** 
 stress on the plate.
\nAssuming that $Y=\sqrt\pi$, calculate the stress intensity factor, $K$
\nGiven that the materials fracture toughness is $80\text{ MPa}\sqrt{\text{m}}$, calculate the critical crack length in the material for fast (unstable brittle) fracture to occur.
\nCalculate the... | 4 | 0.666667 | 2 | \n\n\n | $$
\text{Remote Stress}=\frac{F}{A}
= \frac{150\times10^3}{(0.1)(0.005)}=\boxed{300\text{ MPa}}
$$
\nSince the crack is in the center of the plate, the crack length $(a)$ is equal to half of the crack length.
   
$$
a=10\text{ mm}
$$
   
***
Therefore $K$ can be calculated as below:
  &#... | A crack of length $10\text{ mm}$ has been found at the center of a large thin plate, with an end load of $150\text{ kN}$. The plate is $100\text{ mm}$ wide and $5\text{ mm}$ thick. Assuming that the material is **perfectly linear elastic.** 
$, where $F_D$ is the drag force, $U$ is the ship speed, $l$ is its length, $B$ is its width, $\rho$ and $\mu$ are the sea water density and viscosity, and $g$ is the accele... | Find the non-dimensional groups that describe the problem.
\nIf we are to test a model of the ship to determine the drag, what are the requirements for dynamic similarity?\nWe are going to test a $1/25$ scale model of a $100\,\mathrm{m}$ long ship. If the maximum velocity of the full-scale ship is $10\,\mathrm{m/s}$, w... | 4 | 0.5 | 2 | \n\n\n | \n\n\n | The resistance of a sea-going ship is due to wave-making and viscous drag, and it may be expressed in functional form as $F_D = f(U,l,B,\rho,\mu,g)$, where $F_D$ is the drag force, $U$ is the ship speed, $l$ is its length, $B$ is its width, $\rho$ and $\mu$ are the sea water density and viscosity, and $g$ is the accele... | 157 | 14 | 0 | 0 | 1 | 0 | 97 | 6 | 0 | The resistance of a sea-going ship is due to wave-making and viscous drag, and it may be expressed in functional form as $F_D = f(U,l,B,\rho,\mu,g)$, where $F_D$ is the drag force, $U$ is the ship speed, $l$ is its length, $B$ is its width, $\rho$ and $\mu$ are the sea water density and viscosity, and $g$ is the accele... | 4 |
613e706e-06a3-4f0b-a712-d1ae8794752b | 1 | 0 | 0 | 15 | 5 | 0 | 0 | 4 | The Q-cycle is a part of the electron transport chain in mitochondria, plastids, and many prokaryotes. The specific version found in mitochondria oxidise ubiquinol ($\mathrm{UQH_{2}}$) to ubiquinone ($\mathrm{UQ}$) by removing two electrons (and two protons) and passing them to the next component of the electron transp... | How many molecules of $\mathrm{cyt\text{-}c}$ are reduced by the complete oxidation of a single $\mathrm{UQH_{2}}$ molecule to $\mathrm{UQ}$? There are two ways to get the answer, and one of them requires an infinite sum. See if you can see both paths to the solution.
| 1 | 1 | 1 | null | The answer can be worked out from the chemistry without recourse to the maths: if we cancel the terms that appear on both sides of the chemical formula:
$$
\mathrm{2\,UQH_2 + 1\,UQ + 2\,cyt\text{-}c_{ox}\rightarrow 2\,UQ + 1\,UQH_2 + 2\,cyt\text{-}c_{red}}
$$
we get:
$$
\mathrm{UQH_2 + 2\,cyt\text{-}c_{ox} \rightarr... | The Q-cycle is a part of the electron transport chain in mitochondria, plastids, and many prokaryotes. The specific version found in mitochondria oxidise ubiquinol ($\mathrm{UQH_{2}}$) to ubiquinone ($\mathrm{UQ}$) by removing two electrons (and two protons) and passing them to the next component of the electron transp... | 279 | 26 | 47 | 47 | 355 | 0 | 45 | 3 | 1 | The specific version found in mitochondria oxidise ubiquinol $\mathrm{UQH_{2}}$ to ubiquinone $\mathrm{UQ}$ by removing two electrons and two protons and passing them to the next component of the electron transport chain: cytochrome-c. $\mathrm{cyt\text{-}c}$ $\mathrm{UQH_{2}}$ has two electrons to pass on, and $\mathr... | 3 |
614376cf-25d5-4712-b4d1-2260204101f7 | 5 | 0 | 0 | 17 | 6 | 1 | 3 | 7 | A ball is dropped onto a table from an initial height $h_0$. If it lands with speed $v$, it bounces up with speed $ev$, where the coefficient of restitution $e$ is a positive constant less than 1.
| Write down expressions for the speed $v_0$ of the ball when it first hits the table and the time $t_0$ it takes to drop, both in terms of $h_0$. 
\nFind the height $h_1$ reached by the ball after its first bounce and the corresponding drop time $t_1$. Hence write down expressions for the height $h_n$ and drop time... | 3 | 0.666667 | 2 | Use conservation of energy or SUVAT equations.
\nRecall from the question that if the ball hits the ground with speed $v$, it bounces up with speed $ev$.
***
As in part (a), use conservation of energy/SUVAT equations to find $h_1$ and $t_1$.
***
Write $h_1$ and $t_1$ in terms of $h_0$ and $t_0$ respectively.
**... | Using conservation of energy,
***
The speed $v_0$ of the ball when it first hits the ground is related to the initial height $h_0$ by $mgh_0 = \frac{1}{2}mv_0^2$. This gives:
***
$$
v_0 = \sqrt{2gh_0}.
$$
(Could also have used the SUVAT equation $v^2 = u^2 + 2as$.) 
***
Since $v_0 = g t_0$,
***
$$
t_0 = \... | A ball is dropped onto a table from an initial height $h_0$. If it lands with speed $v$, it bounces up with speed $ev$, where the coefficient of restitution $e$ is a positive constant less than 1.
Write down expressions for the speed $v_0$ of the ball when it first hits the table and the time $t_0$ it takes to drop, b... | 154 | 18 | 30 | 30 | 215 | 18 | 114 | 14 | 0 | Write down expressions for the speed $v_0$ of the ball when it first hits the table and the time $t_0$ it takes to drop, both in terms of $h_0$. Find the height $h_1$ reached by the ball after its first bounce and the corresponding drop time $t_1$. Hence write down expressions for the height $h_n$ and drop time $v$0 of... | 6 |
61444717-1285-46e9-8574-5e6ee1d6cdc8 | 0 | 2 | 2 | 21 | 6 | 1 | 2 | 4 | In this problem, we will consider some practical applications of the Fourier transform in optics. You may use the result from lectures that the amplitude $U$ of a diffraction pattern is given by:
$$
U(l) = \frac{\lambda}{\sqrt{2\pi}}A(2\pi l/\lambda) \; ,
$$
where $l=\sin \theta$ with $\theta$ the scattering angle, $... | Consider two infinitely narrow slits that are separated by a distance $d$. Calculate the amplitude of the diffraction pattern as a function of $l$. Then, calculate the corresponding intensity that would be recorded by a detector.
***
In the answer box below, input the form of the intensity diffraction pattern that yo... | 4 | 1 | 4 | First, what is $a(x)$ (the *aperture function*) for two infinitely narrow slits...?
***
... Consider that an infinitely narrow slit can be represented by a delta function.
***
Now, your goal is to find $A(\omega)$, the Fourier transform of $a(x)$ (**Note:** you will let $\omega = 2\pi l/\lambda$ later on), in ord... | In the lectures, we derived that the Fourier transform of the aperture function, $a(x)$, gives the form of an interference pattern. 
***
If $x=0$ is the centre of aperture, the slits are at $x=+d/2$ and $x=-d/2$:
***
 = \frac{\lambda}{\sqrt{2\pi}}A(2\pi l/\lambda) \; ,
$$
where $l=\sin \theta$ with $\theta$ the scattering angle, $... | 298 | 17 | 8 | 8 | 250 | 24 | 226 | 9 | 0 | You may use the result from lectures that the amplitude $U$ of a diffraction pattern is given by: $ U(l) = \frac{\lambda}{\sqrt{2\pi}}A(2\pi l/\lambda) \; , $ where $l=\sin \theta$ with $\theta$ the scattering angle, $\lambda$ the wavelength of the light, and $A$ is the Fourier Transform of the aperture function $a(x)$... | 15 |
61591033-37c0-4c98-bf3c-29b4c44617ec | 2 | 0 | 0 | 0 | 0 | 2 | 5 | 3 | For a specific design Mach number $M$, the geometry of an inlet can make the reflected shock wave cancel. Consider the geometry shown in the figure below (not to scale) for a supersonic diffuser to reduce the Mach number from $M=3.5$. The deflection angle is $\theta = 10^{\circ}$. 
 for a supersonic diffuser to reduce the Mach number from $M=3.5$. The deflection angle is $\theta = 10^{\circ}$. 
. In the antidune system the water surface is in phase with the bed configuration with a larger water depth at the antidune crest than at the antidune trough.
 of an antidune system if the total sediment transport is downstream? Discuss your answer.
| 2 | 0.666667 | 2 | \n | \n | The attached sketch illustrates a bed form called antidune, occurring when the Froude number is close or above 1 (supercritical). In the antidune system the water surface is in phase with the bed configuration with a larger water depth at the antidune crest than at the antidune trough.
 = \frac{cx}{x^2 + a^2}
$$
where $a$ and $c$ are constants.
| Find the position of **stable** equilibrium and the value of $U(x)$ at that point. Sketch $U(x)$.
\nIf the particle starts from the position of stable equilibrium with velocity $v$, find the range of velocities for which it:
* (i) escapes to $+\infty$
* (ii) escapes to $-\infty$
* (iii) oscillates
(**Note:** ... | 3 | 1 | 4 | What is the relationship between $F(x)$ and $U(x)$? 
***
Equilibrium occurs when $F(x)=0$. Therefore, solve for $x$ and plug this back into $U(x)$.
***
Therefore, solve for $x$ and plug this back into $U(x)$. You should obtain **two** equilibrium points...
***
... Which equilibrium point is stable? There ... | The force $F(x)$ corresponding to the given potential function $U(x)$ is:
***
$$
F(x) = -\frac{dU(x)}{dx}
$$
Apply the quotient rule:
***
$$
F(x) = -\frac{dU(x)}{dx} = \frac{2cx^2 - c(x^2 + a^2)}{(x^2 +
a^2)^2} = \frac{c(x^2 - a^2)}{(x^2 + a^2)^2}.
$$
***
At the position of equilibrium the force mus... | A particle of mass $m$ moves in the $x$ direction under the action of a conservative force. Its potential energy is given by
$$
U(x) = \frac{cx}{x^2 + a^2}
$$
where $a$ and $c$ are constants.
Find the position of **stable** equilibrium and the value of $U(x)$ at that point. Sketch $U(x)$.
\nIf the particle starts ... | 118 | 10 | 51 | 51 | 413 | 28 | 88 | 5 | 0 | Its potential energy is given by $ U(x) = \frac{cx}{x^2 + a^2} $ where $a$ and $c$ are constants. Find the position of stable equilibrium and the value of $U(x)$ at that point. Sketch $U(x)$. If the particle starts from the position of stable equilibrium with velocity $v$, find the range of velocities for which it: i ... | 5 |
619b472c-839f-4fd4-9e7a-c581134f37b6 | 2 | 0 | 0 | 2 | 1 | 2 | 2 | 3 | The pressure force in a hydrostatic fluid is a vector given by $F_{pressure}=-\nabla p$.
| In cartesian co-ordinates, what is the $x$-component of the pressure?
\nWhat would we expect the x-component of the pressure to equal? Why?
| 2 | 0.333333 | 0 | The vector describing the pressure force is $ \nabla p=\frac{\delta p}{\delta x}+\frac{\delta p}{\delta y}+\frac{\delta p}{\delta z} $. What is the x-component of this vector?
***
The x-component is $\frac{\delta p}{\delta x}$. Think about the forces acting on the fluid and which direction they act in. How will thi... | The vector describing the pressure force is $ -\nabla p=-(\frac{\delta p}{\delta x}+\frac{\delta p}{\delta y}+\frac{\delta p}{\delta z}) $. The x-component of this vector is $-\frac{\delta p}{\delta x}$
\n
In a hydrostatic fluid the only forces acting are due to pressure and gravity. The convention for gravity is th... | The pressure force in a hydrostatic fluid is a vector given by $F_{pressure}=-\nabla p$.
In cartesian co-ordinates, what is the $x$-component of the pressure?
\nWhat would we expect the x-component of the pressure to equal? Why?
| 37 | 2 | 4 | 4 | 78 | 2 | 23 | 1 | 0 | The pressure force in a hydrostatic fluid is a vector given by $F_{pressure}=- abla p$. In cartesian co-ordinates, what is the $x$-component of the pressure? What would we expect the x-component of the pressure to equal? Why? | 4 |
627119a8-7b2f-406c-a366-45b4cc230be6 | 1 | 1 | 1 | 17 | 6 | 1 | 6 | 1 | The figure below shows two equal masses attached to either end of a light stiff rod, which is mounted in the middle on a vertical axle.
The rod and... | Does the axle exert a net force on the rotating rod? 
\nWhen the rod rotates fast enough, the bearing breaks. Why?
\nFind vector expressions for the angular momentum of each mass and add the two vectors to get the total angular momentum $\vec{\boldsymbol{L}}_{\text{tot}}$ of the contraption. Show that the torque ... | 3 | 0.666667 | 3 | Remember that we are ignoring gravity in this question.
***
The rod provides a force to keep the masses in orbit.
***
Consider N3. 
\nWhat happens to the required torque as $\omega$ increases?
***
Consider that the bearing has a limit to the torques it can exert.
\nBefore starting this question, use the r... | The axle supplies the centripetal forces required to keep the two masses orbiting in circles, but these are always equal and opposite (N3). 
***
The nett force is zero.
\nThe axle generates the centripetal forces required to keep the masses in orbit by applying a torque to the rod via the bearing. 
***
Si... | The figure below shows two equal masses attached to either end of a light stiff rod, which is mounted in the middle on a vertical axle.
The rod and... | 247 | 13 | 21 | 21 | 297 | 15 | 146 | 11 | 1 | Ignore the effects of gravity. Does the axle exert a net force on the rotating rod? When the rod rotates fast enough, the bearing breaks. Why? Find vector expressions for the angular momentum of each mass and add the two vectors to get the total angular momentum $\vec{\boldsymbol{L}}_{\text{tot}}$ of the contraption. S... | 7 |
6281e7a1-0b80-40fc-9b0f-718dc0d5bbb5 | 0 | 3 | 0 | 21 | 6 | 1 | 0 | 1 | The following functions are orthogonal on the interval (**True** / **False**):
|  $\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $-L\leq x\leq L$
\n$\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $0\leq x\leq 2L$
\n$\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $0\leq x\leq L$
| 3 | 0.666667 | 2 | If the two functions are orthogonal, the *inner product* of the functions must be 0 (see section **2.3** of the notes). This is like the dot product of two orthogonal vectors. 
**Note:** we are not testing for Kronecker-delta here because there is no $n,m$ dependence in the functions.
***
After setting-up th... | First, performing the substitution $k=\pi x/L$ for simplification:
$$
\begin{aligned}
&= \frac{L}{\pi}\int_{-\pi}^{\pi}{\sin(k) \cos(2k) dk} \\
\end{aligned}
$$
***
Then, applying the formulae for $\sin$ and $\cos$ in imaginary terms:
$$
\begin{aligned}
\sin{\theta} = \frac{e^{i\theta}-e^{-i\theta}}{2}\\
\cos... | The following functions are orthogonal on the interval (**True** / **False**):
 $\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $-L\leq x\leq L$
\n$\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $0\leq x\leq 2L$
\n$\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $0\leq x\leq L$
| 29 | 9 | 11 | 11 | 388 | 6 | 18 | 9 | 0 | The following functions are orthogonal on the interval True / False: $\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $-L\leq x\leq L$ $\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $0\leq x\leq 2L$ $\sin(\pi x/L)$ and $\cos(2\pi x/L)$ over $0\leq x\leq L$ | 1 |
62a5b748-6653-44bb-9eea-3eac5a486698 | 0 | 1 | 1 | 11 | 4 | 2 | 4 | 0 | Refrigerators and heat pumps are both examples of reversed heat engines.
| What is the difference between a refrigerator and a heat pump? 
\nIf a refrigerator and a heat pump operate on the same cycle, which has the higher $COP$?
| 2 | 0.333333 | 0 | \n | \nFor a refrigerator:
  
$COP_\mathrm{R} = \frac{\dot{Q}_\mathrm{in}}{|\dot{W}_\mathrm{net}|}$
  
where $\dot{Q}_\mathrm{in}$ is the heat transferred from the cold space into the refrigerant.
***
For a heat pump:
  
$COP_\mathrm{HP} = \frac{|\dot{Q}_\mathrm{out}|}{|\dot{W}_\mathrm{... | Refrigerators and heat pumps are both examples of reversed heat engines.
What is the difference between a refrigerator and a heat pump? 
\nIf a refrigerator and a heat pump operate on the same cycle, which has the higher $COP$?
| 40 | 1 | 8 | 8 | 61 | 0 | 29 | 1 | 0 | What is the difference between a refrigerator and a heat pump? If a refrigerator and a heat pump operate on the same cycle, which has the higher $COP$? | 2 |
637204d2-f4c7-494e-844c-6b169616db05 | 3 | 0 | 0 | 24 | 6 | 1 | 0 | 11 | **\[Boas 5.4.1 (a),(b)]** For the disk $\rho\leq a$, find by integration using polar coordinates:
| The area of the disk.
\nThe centroid of one quadrant \[take the positive quadrant] of the disk.
***
The centroid is the Cartesian point $(X,Y)$ such that $X = \frac{1}{A} \iint_R xdxdy$ and $Y = \frac{1}{A} \iint_R ydxdy$ where $R$ is the region of integration and $A$ is the area of that region. 
| 2 | 0.666667 | 2 | $$
A = \iint_R{dA}
$$
Can you express $dA$ in polar coordinates? ...
***
... This should be of the form $|J|d\rho d\phi$, where $|J|$ is the *Jacobian* in polar coordinates.
***
Sketch the disk. What are the limits of $\rho$ and $\phi$?
***
Hence perform the double integral, keeping the variable you are not i... | Sketch of the disk:
***
The area of the disk is the sum of the area elements:
$$
A = \iint_{R}{dA}
$$
This integral could be performed in Cartesian co... | **\[Boas 5.4.1 (a),(b)]** For the disk $\rho\leq a$, find by integration using polar coordinates:
The area of the disk.
\nThe centroid of one quadrant \[take the positive quadrant] of the disk.
***
The centroid is the Cartesian point $(X,Y)$ such that $X = \frac{1}{A} \iint_R xdxdy$ and $Y = \frac{1}{A} \iint_R ydxd... | 59 | 6 | 18 | 18 | 195 | 10 | 45 | 5 | 0 | Boas 5.4.1 a,b For the disk $\rho\leq a$, find by integration using polar coordinates: The area of the disk. | 1 |
638eb85e-0b37-46ad-b4c7-021442fe87f1 | 2 | 0 | 0 | 14 | 4 | 2 | 5 | 4 | Newton had already formulated that the propagation of sound in air followed from the oscillatory motion of the fluid particles (which he viewed as ``the law of the oscillating pendulum''). While he did not state it explicitly, it appears that he considered such a process to be isothermal. | Rederive the speed of sound for an ideal gas, but assume the successive compression/expansion of the fluid particle to be isothermal. Give your answer in terms of the gas temperature. 
\nWhat is the relative error between Newton's sound speed for ideal gases and the one obtained in class? Is it a function of the t... | 2 | 0.5 | 2 | \n | \n | Newton had already formulated that the propagation of sound in air followed from the oscillatory motion of the fluid particles (which he viewed as ``the law of the oscillating pendulum''). While he did not state it explicitly, it appears that he considered such a process to be isothermal.Rederive the speed of sound for... | 118 | 1 | 0 | 0 | 1 | 0 | 71 | 1 | 0 | Newton had already formulated that the propagation of sound in air followed from the oscillatory motion of the fluid particles which he viewed as ``the law of the oscillating pendulum''. While he did not state it explicitly, it appears that he considered such a process to be isothermal.Rederive the speed of sound for a... | 6 |
6396919f-7055-4a64-9fea-9aafa18f0068 | 4 | 0 | 0 | 0 | 0 | 2 | 4 | 0 | Determine the following quantities downstream of the continuous $10^{\circ}$ convex corner shown below, with the flow upstream of the corner being at Mach 2: 
***
... | Determine the following quantities downstream of the continuous $10^{\circ}$ convex corner shown below, with the flow upstream of the corner being at Mach 2: 
 *every other particle*. Express $d\boldsymbol{\vec{p}_i}/dt$...
***
... The internal forces should be expressed as a summation over every particle $j \ne i$.
***
Sum up the forces to find the total force acting on the system.
**... | Label the particles $i = 1, 2, \ldots, N$. N2 applies to all of them:
***
$$
\begin{aligned}
\frac{d\boldsymbol{\vec{p}}_1}{dt}
&= \boldsymbol{\vec{f}}_1^{\text{ ext}} +
\sum_{j\,(\neq 1)}\boldsymbol{\vec{f}}_{j\text{ on }1},
& \text{(particle $1$)}\\
& \ldots & \\
& \ldots & \\
\frac{d... | In lectures, you saw how to use Newton’s second law for two particles,
$$
\begin{aligned}
\frac{d\boldsymbol{\vec{p}}_1}{dt} &= \boldsymbol{\vec{f}}_1^{\text{ ext}} + \boldsymbol{\vec{f}}_{2\text{ on }1}
,
&
\frac{d\boldsymbol{\vec{p}}_2}{dt} &= \boldsymbol{\vec{f}}_2^{\text{ ext}} + ... | 50 | 4 | 13 | 13 | 166 | 3 | 50 | 4 | 0 | In lectures, you saw how to use Newton’s second law for two particles, $ \begin{aligned} \frac{d\boldsymbol{\vec{p}}_1}{dt} &= \boldsymbol{\vec{f}}_1^{\text{ ext}} + \boldsymbol{\vec{f}}_{2\text{ on }1} , & \frac{d\boldsymbol{\vec{p}}_2}{dt} &= \boldsymbol{\vec{f}}_2^{\text{ ext}} + \boldsymbol{\vec{f}}_{1\text{ on }2}... | 1 |
63d7bf5e-3b9e-4583-bf45-a2c0665a35ab | 2 | 5 | 2 | 2 | 1 | 2 | 7 | 13 | In this question you will derive the flow profile of an incompressible Newtonian fluid that is flowing with volume flux $Q$ in a channel of width $h$ created by the gap between two plates, each of length $L$ and width $W$, at $y=0$ and $y=h$.
  ... | What do each of the assumptions imply about the velocity components $u(x,y,z,t)$, $v(x,y,z,t)$ and $w(x,y,z,t)$ and the pressure $p(x,y,z,t)$?
\nShow that the continuity equation implies that $v$ is constant.
\nWrite down the boundary conditions at the top and bottom of the channel. What does this tell you about $v$?
\... | 6 | 1 | 4 | \nWrite out the expanded form of the continuity equation *i.e.* the one used throughout Question 8.2.
***
Since $u$ and $v$ don't depend on $x$ what does the equation $\frac{\delta u}{\delta x}+\frac{\delta v}{\delta y}=0$ simplify to? 
***
What does the solution of the simplified PDE suggest about $v$?
\nPr... | We have
* The assumption of two-dimensional flow means that $u$, $v$ and $p$ do not depend on $z$ and that $w=0$. Thus we have $u(x,y,t)$, $v(x,y,t)$, $p(x,y,t)$.
* The assumption of steady flow means that none of the variables depend on $t$. Thus $u(x,y)$, $v(x,y)$, $p(x,y)$,
* The assumption of fully develop... | In this question you will derive the flow profile of an incompressible Newtonian fluid that is flowing with volume flux $Q$ in a channel of width $h$ created by the gap between two plates, each of length $L$ and width $W$, at $y=0$ and $y=h$.
  ... | 244 | 31 | 57 | 57 | 248 | 36 | 140 | 23 | 1 | In this question you will derive the flow profile of an incompressible Newtonian fluid that is flowing with volume flux $Q$ in a channel of width $h$ created by the gap between two plates, each of length $L$ and width $W$, at $y=0$ and $y=h$. We will make the following assumptions: Two-dimensional flow no dependence o... | 11 |
644c4af4-fec2-4fcd-93d7-b775dd08d41b | 4 | 0 | 0 | 0 | 0 | 2 | 2 | 3 | The figure below shows a convergent-divergent nozzle. It is observed that the flow (air, $\gamma=1.4$) exits the nozzle as a perfectly expanded (isentropic) supersonic flow. 
   
  
\nDetermine the pressure at the nozzle exit ($p_{exit}$).
\nThe pressure at the nozzle exit is gradually increased from its original value. For what range of values of $p_{exit}$ would the mass flow rate through... | 4 | 1 | 3 | Mass flow rate is constant throughout the nozzle, so we can analyze it at any point. At what section is there enough information to do this? Hint: the only quantities remaining to be determined are $ \rho^{*} $ and $ v^{*} $. 
***
To determine $ \rho^{*} $, use the ideal gas law. To determine the any quantit... | We will analyse the mass flow rate at the throat, where we know that $ M^{*} = 1 $, and $A^{*} = 0.1 \ \mathrm{m^{2}}$. The expression for mass flow rate here is
   
 $ \dot{m} = \rho^{*} \cdot A^{*} \cdot v^{*} $.
 
 Thus the quantities we need to determine are $ \rho^{*} $ and $ v^{*} ... | The figure below shows a convergent-divergent nozzle. It is observed that the flow (air, $\gamma=1.4$) exits the nozzle as a perfectly expanded (isentropic) supersonic flow. 
   
  
**: Use Cramer's rule to find the intersection of the lines
$$
\begin{aligned}
x+ y&=1\, ,\\
x - y&= -2\, .
\end{aligned}
$$
| **(L7)**: Use Cramer's rule to find the intersection of the lines
$$
\begin{aligned}
x+ y&=1\, ,\\
x - y&= -2\, .
\end{aligned}
$$
| 1 | 0.333333 | 0 | Form the determinants $\Delta,\Delta_1,\Delta_2$ (see **section 2.4**). Hence find $x,y$ using these determinants.
| Start by evaluating the 3 determinants: $\Delta, \Delta_1, \Delta_2$ (see **section 2.4**): 
***
$$
\displaystyle \Delta \ =\ \begin{array}{|c c|}
1 & 1\\
1 & -1
\end{array} = -1-1 = -2
$$
***
$$
\displaystyle \Delta _{1} \ =\ \begin{array}{|c c|}
\hskip6pt 1 & \hskip6pt 1\\
-2 & -1
\end{array} = -1+2 = 1
$$
... | **(L7)**: Use Cramer's rule to find the intersection of the lines
$$
\begin{aligned}
x+ y&=1\, ,\\
x - y&= -2\, .
\end{aligned}
$$
| 12 | 1 | 6 | 6 | 23 | 2 | 12 | 1 | 0 | L7: Use Cramer's rule to find the intersection of the lines $ \begin{aligned} x+ y&=1\, ,\\ x - y&= -2\, . \end{aligned} $ | 2 |
65990727-92f1-40a7-9db9-c12b6e215476 | 4 | 0 | 0 | 19 | 6 | 1 | 8 | 3 | **(L15)**: Consider the matrix ${\mathbf{\text{A}}}=\left(\begin{array}{cr}5&\hskip3pt -7\\1&\hskip3pt -3\end{array}\right).$
| Find the eigenvalues and normalised eigenvectors of ${\mathbf{\text{A}}}$.
\nWrite down ${\mathbf{\text{S}_A}} = (\bf{x}_1, \bf{x}_2)$, the matrix of eigenvectors for ${\mathbf{\text{A}}}$. Use ${\mathbf{\text{S}_A}}$ to diagonalise ${\mathbf{\text{A}}}$.
| 2 | 0.333333 | 2 | To solve for the eigenvalues, solve the characteristic equation $\det{(\text{A}-\lambda\mathbb{I})}=0$ (**section 3.19**).
***
Then, for each eigenvalue, solve $(\text{A}-\lambda\mathbb{I})\mathbf{\underline{x}}$. 
***
Normalise this result. 
\nWrite the matrix ${\mathbf{\text{S}}} = (\bf{x}_1, \bf{x}_2... | The eigenvalues of ${\mathbf{\text{A}}}$ are determined from the characteristic equation $\det{(\text{A}-\lambda\mathbb{I})}=0$:
***
$$
\begin{aligned}
\det({\mathbf{\text{A}}}-\lambda\mathbb{I})&=
\left|
\begin{array}{cc}
5-\lambda&\hskip3pt -7\\
1&\hskip3pt -3-\lambda
\end{array}\right|=-(5-\lambda)(3+\lambda)+7\no... | **(L15)**: Consider the matrix ${\mathbf{\text{A}}}=\left(\begin{array}{cr}5&\hskip3pt -7\\1&\hskip3pt -3\end{array}\right).$
Find the eigenvalues and normalised eigenvectors of ${\mathbf{\text{A}}}$.
\nWrite down ${\mathbf{\text{S}_A}} = (\bf{x}_1, \bf{x}_2)$, the matrix of eigenvectors for ${\mathbf{\text{A}}}$. Use ... | 31 | 6 | 27 | 27 | 159 | 7 | 26 | 5 | 0 | L15: Consider the matrix ${\mathbf{\text{A}}}=\left(\begin{array}{cr}5&\hskip3pt -7\\1&\hskip3pt -3\end{array}\right).$ Find the eigenvalues and normalised eigenvectors of ${\mathbf{\text{A}}}$. Write down ${\mathbf{\text{S}_A}} = (\bf{x}_1, \bf{x}_2)$, the matrix of eigenvectors for ${\mathbf{\text{A}}}$. Use ${\mathb... | 3 |
65ba8a09-46a5-4797-9493-cd5a35137a9f | 2 | 0 | 0 | 17 | 6 | 1 | 1 | 0 | A block of mass $m = 2\,$kg slides on a horizontal frictionless surface. The block is initially moving at velocity $v = -1\,$m$\cdot$ s$^{-1}$. (Notice that $v$ is negative, so the block is initially moving in the $-x$ direction.) A constant force $F = +10\,$N acts for 5 s.
| Find the acceleration of the block and hence its final speed and final kinetic energy. How much kinetic energy has the block gained?
\nHow far from its initial position was the block after 5 s? Calculate the work done by the force and show that this is the same as the kinetic energy gained by the block.
| 2 | 0.333333 | 1 | Use N2 to find the acceleration of the block. 
***
Can you think of a SUVAT equation that finds final velocity from initial velocity and acceleration?
***
Kinetic energy, $K=(1/2)mv^2$. Hence find the *change* in $K$. 
\nUse a SUVAT equation to find the displacement at $t=5$.
***
Since the force is c... | The acceleration $a$ of the block is $F/m = 5\,\text{m}\cdot\text{s}^{-2}$. This is positive, so the block is accelerating to the right.
***
The final velocity is given by the familiar SUVAT equation:
***
$$
v = u + at = -1 + 5 \times 5 = 24\,\text{m}\cdot\text{s}^{-1}. \\
$$
***
The initial and final kinetic en... | A block of mass $m = 2\,$kg slides on a horizontal frictionless surface. The block is initially moving at velocity $v = -1\,$m$\cdot$ s$^{-1}$. (Notice that $v$ is negative, so the block is initially moving in the $-x$ direction.) A constant force $F = +10\,$N acts for 5 s.
Find the acceleration of the block and hence... | 104 | 7 | 17 | 17 | 288 | 7 | 56 | 0 | 0 | Notice that $v$ is negative, so the block is initially moving in the $-x$ direction. A constant force $F = +10\,$N acts for 5 s. Find the acceleration of the block and hence its final speed and final kinetic energy. How much kinetic energy has the block gained? How far from its initial position was the block after 5 s?... | 5 |
65ef6b13-0e9c-4858-b3cf-fa98de6f5bba | 3 | 0 | 2 | 14 | 4 | 2 | 7 | 1 | A scale model of an aeroplane is to be tested in a wind tunnel at $M=2$. The tunnel is supplied from an air reservoir at $21^{\circ}\mathrm{C}$, and the working section is held at atmospheric pressure $100\space\mathrm{kPa}$. Assume air to be a perfect gas with $\gamma=1.4$ and $R=287\mathrm{J/(kg K)}$.
}$.
}$. Assuming isentropic flow, what must the reservoir pressure be? What is the air speed at the test section? If the cross-sectional area at the test section is $0.2 \> \mathrm{m}^{2}$, what is the mass rate of the flow? Sketch the pressure dist... | 6 |
66688299-b090-4830-8280-7b06bf819c0b | 3 | 0 | 1 | 11 | 4 | 2 | 3 | 12 | An air conditioning unit comprises a cooling coil with condensate removal followed by an electrical heater. It is designed to deliver $10~ \mathrm{m^3/min}$ of conditioned air. On test the following temperatures (in $^{\circ}\mathrm{C}$) were recorded:
| | Dry Bulb | Wet Bulb |
| :----- | :------- | :------- |
... | Draw the processes on the psychrometric chart
\nCalculate the mass flow rate of condensate.
\nCalculate the rate of heat removal at the cooling coil.
\nCalculate the heater electrical power input.
| 4 | 1 | 3 | \n\n\n | \nIt can be helpful to draw a diagram:
  
***
The specific humidity ($\omega$) at the inlet and outlet can be found using the psychrometric ... | An air conditioning unit comprises a cooling coil with condensate removal followed by an electrical heater. It is designed to deliver $10~ \mathrm{m^3/min}$ of conditioned air. On test the following temperatures (in $^{\circ}\mathrm{C}$) were recorded:
| | Dry Bulb | Wet Bulb |
| :----- | :------- | :------- |
... | 94 | 2 | 58 | 58 | 518 | 0 | 30 | 0 | 0 | An air conditioning unit comprises a cooling coil with condensate removal followed by an electrical heater. On test the following temperatures in $^{\circ}\mathrm{C}$ were recorded: table Draw the processes on the psychrometric chart Calculate the mass flow rate of condensate. Calculate the rate of heat removal at the ... | 4 |
675f9153-b04f-4db0-a59d-3c8ac43e4430 | 0 | 1 | 0 | 11 | 4 | 2 | 7 | 2 | A fuel is burnt first with the stoichiometric amount of air and then with the stoichiometric amount of pure oxygen. 
| For which case will the adiabatic flame temperature be higher?
| 1 | 0.333333 | 0 | null | null | A fuel is burnt first with the stoichiometric amount of air and then with the stoichiometric amount of pure oxygen. 
For which case will the adiabatic flame temperature be higher?
| 30 | 0 | 0 | 0 | 0 | 0 | 10 | 0 | 0 | For which case will the adiabatic flame temperature be higher? | 1 |
678a7d0c-71db-4140-b767-b5017c34fe00 | 1 | 0 | 3 | 24 | 6 | 1 | 2 | 0 | Consider a surface $S$ defined as the portion of the plane $x + 2y -3z = 0$ with $0<x<1$ and $0<y<1$.
| Show that the surface element $d\vec{S}$ on the plane is given by:
$$
d\vec{S}=\left(\mathbf{\hat{i}}+\frac{\partial z}{\partial x}\mathbf{\hat{k}}\right)\times\left(\mathbf{\hat{j}}+\frac{\partial{z}}{\partial y}\mathbf{\hat{k}}\right)\,dx\,dy
$$
\nHence show that:
$$
d\vec{S} = \left(-\frac{1}{3}\,\mathbf{\hat{i}} ... | 4 | 0.333333 | 1 | The position vector on the plane is $\vec{r}=x\mathbf{\hat{i}}+y\mathbf{\hat{j}}+z(x,y)\mathbf{\hat{k}}$.
***
$$
d\vec{S}=\left(\frac{\partial \vec{r}}{\partial x}\times \frac{\partial \vec{r}}{\partial y}\right)\,dx\,dy
$$
***
Leave this in a general form and do not insert the equation of the plane. 
\nWhat... | The position vector on the plane is given by:
$$
\begin{aligned}
\vec{r} &= x\mathbf{\hat{i}}+y\mathbf{\hat{j}}+z(x,y)\mathbf{\hat{k}}\\
\end{aligned}
$$
***
Then, the surface area element is given by:
$$
d\vec{S}=\left(\frac{\partial \vec{r}}{\partial x}\times \frac{\partial \vec{r}}{\partial y}\right)\,dx\,dy
$$
... | Consider a surface $S$ defined as the portion of the plane $x + 2y -3z = 0$ with $0<x<1$ and $0<y<1$.
Show that the surface element $d\vec{S}$ on the plane is given by:
$$
d\vec{S}=\left(\mathbf{\hat{i}}+\frac{\partial z}{\partial x}\mathbf{\hat{k}}\right)\times\left(\mathbf{\hat{j}}+\frac{\partial{z}}{\partial y}\mat... | 80 | 13 | 24 | 24 | 133 | 8 | 63 | 9 | 0 | Consider a surface $S$ defined as the portion of the plane $x + 2y -3z = 0$ with $0<x<1$ and $0<y<1$. Show that the surface element $d\vec{S}$ on the plane is given by: $ d\vec{S}=\left(\mathbf{\hat{i}}+\frac{\partial z}{\partial x}\mathbf{\hat{k}}\right)\times\left(\mathbf{\hat{j}}+\frac{\partial{z}}{\partial y}\mathb... | 2 |
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