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Consider the integral latex0 By using the substitution latex1 , show that latex2 can also be written latex3 In part (a), you have shown that latex4 Use this relationship to find latex2 . | 1 |
(Based on P4.20, White) A two-dimensional incompressible velocity field has latex0 -component latex1 , for latex2 and latex3 . | 0 |
(Answer to 3 sf) | 0 |
The speed of a D.C. motor is governed by a proportional control system with unity feedback. | 0 |
An equalizer is to be used to process a sound signal with a range of latex0 to boost spoken language while suppressing background noise i.e the equalizer should amplify those frequencies within the normal range of speech ( latex1 ), while suppressing lower and higher frequencies which are outside this range. | 0 |
Which of the following statements correctly describes a fluid? | 1 |
Gas enters the turbine of a gas turbine engine at latex0 , with a mass flow rate of latex1 , and expands adiabatically through a pressure ratio of latex2 . | 0 |
What is the blade length (or height)? | 1 |
Simplify the Euler Equations in the following scenarios,Hydrostatic ( latex0 ). | 1 |
A mild steel beam of rectangular cross-section, latex0 wide, latex1 deep, carries loads which cause a bending moment latex2 and a shear force latex3 at a section latex4 along its length. | 0 |
If latex4 and latex2 is pure imaginary, find latex6 . | 1 |
A real silicon diode with the polarity shown in the diagram above A real C6V3 Zener diode, with the polarity shown in the diagram above An internally short-circuited (i.e blown) diode. | 0 |
Time averages were calculated using all the data recorded. | 0 |
In order for the car to remain on the track, a set performance level must be achieved. | 0 |
Give an expression for: The work done by a fluid between two points with total head, latex0 and latex1 . | 1 |
In ordinary derivatives a function latex0 implicitly defines latex1 and their derivatives obey: latex2 (check e.g for latex3 ). | 0 |
Assuming that there is only molecular diffusion: latex2 . | 0 |
How many parameters are in the equations? | 1 |
The angular momentum of the particle around latex2 is given by latex6 , where latex7 is the velocity of the particle. | 0 |
The velocity in a Poiseuille flow between two parallel plates is characterised by a parabolic profile. | 0 |
We will say that a good approximation is one where the inner product of the difference between the two is small, so the best approximation with latex3 terms is the one that leaves: latex5 the smallest. | 0 |
It is made of titanium alloy ( latex2 , failure stress latex3 ). | 0 |
By what factor is the oscillation amplitude reduced by this slight change in latex14 ? | 1 |
What is the most appropriate "framework" for thermodynamic analysis, i.e closed system (or system or fixed mass) or open system (or control volume), in each of the following situations: An air-standard Otto cycle, modelling the processes in the cylinders of a spark ignition engine? | 1 |
Consider the following two initial values (note that they differ only by the latex22 -th bit): latex94 (i) Apply (6) over latex22 iterations to find latex96 and latex97 (ii) Find latex98 , latex99 , latex100 and latex101 using (5), and comment on their differences No matter how small the difference between two initial conditions is, the respective time histories obtained by the tent map will be remarkably different. | 1 |
The filter should have an input resistance of latex6 . | 0 |
In order to maximise excitement, passengers should feel three times their normal weight at point C. How far below point B should point C be placed? | 1 |
The amount of oxygen and blood pumped by the mammalian heart can be modelled by the following: latex0 latex1 * latex2 Volume of oxygen consumed latex3 * latex4 Cardiac output ( latex5 ) * latex6 Oxygen content in arterial blood ( latex7 ) * latex8 Oxygen content in venous blood ( latex7 ) * latex10 Heartbeat frequency ( latex11 ) * latex12 Stroke volume ( latex13 ) By rearranging these equations, we can work out a value for the stroke volume, latex12 , of the heart, which is more difficult to measure directly than the other variables. | 0 |
If yes, why? | 1 |
What is the Mach number of the train in the tunnel? | 1 |
Compare your solution with the one in the worked solutions. | 1 |
The specific enthalpy of steam at latex10 and latex11 bar. | 0 |
Given that those who have CF have twice as many copies of the *cf* allele as CF carriers, what is the frequency of the *cf* allele (*i.e * what is latex1 ?) | 1 |
* For **(a)** latex6 , while for **(b)** latex7 . | 0 |
In search of an Arts Council grant, the artist Damien Worst wishes to paint the **inside** surface of the musical funnel obtained by rotating the curve latex0 for latex1 about the latex2 -axis. | 0 |
You will also need to use the fact that the boundary conditions are as follows: * At latex1 , latex15 , latex16 , latex17 , * At latex2 , latex19 , latex20 , latex17 . | 0 |
We shall call this toy problem the *logistic map*. | 0 |
Two identical pendulums, each of length 1 m with a mass of 0.5 kg, are coupled with a spring of spring constant 15 Nm latex0 . | 0 |
Show that for a charge moving in the latex4 - latex5 plane under the influence of a magnetic field latex6 , the equations of motion are: latex7 By introducing the complex variable latex8 , show that the equation pair (2) may be written as a single equation: latex9 where latex10 . | 1 |
The car's velocity is in the direction labelled 2, direction 1 is perpendicular to the car's velocity and directed away from the car's position, whilst 3 is the upward normal to the road surface. | 0 |
Advection and diffusion; both in latex0 , latex1 , and latex12 directions. | 0 |
Use latex8 . | 1 |
Note that latex8 and latex10 rotate as the mass rotates. | 0 |
Assume that the fluid velocity is uniform in latex0 and latex1 . | 0 |
Determine the minimum value of latex2 necessary to contain the steady state error within latex3 , when a step input of amplitude latex4 is applied. | 1 |
Give a quantitative answer by examining how much original latex3 mass would remain at this point in terms of latex27 . | 1 |
This corresponds to a local maximum in the latex0 vs. latex1 plot. | 0 |
Show that for these initial conditions latex9 and latex10 . | 1 |
Find the angular velocity latex5 of the wheel at the instant when it leaves the path and starts to mount the step. | 1 |
Find the directional derivative latex2 in the direction of the (un-normalised) vector latex3 at the point latex4 . | 1 |
Neglecting friction, what is the minimum height, latex8 , of latex4 above latex5 at which the car can start from rest without falling off the track at latex11 ? | 1 |
Since then, bridge designers test their designs in wind tunnels. | 0 |
The fluids are Newtonian with constant and uniform densities ( latex7 ) and dynamic viscosities ( latex8 ). | 0 |
The carrier, which is freely pivoted about latex4 , has a total mass of latex5 and may be considered as two uniform rods, each of length latex6 . | 0 |
After leaving Waterloo station the driver requires a steady speed of latex1 . | 0 |
What is the critical bed shear stress, latex4 , for this channel? | 1 |
Ignore all forces other than gravity. | 0 |
Don’t forget that $1/x latex6 x^{-1} latex7 \frac{A}{B}=0 latex8 A latex9 A latex10 B$ is not zero too, but it isn’t in this case because all the terms in the denominator are positive). | 0 |
For what values of latex4 does latex7 ? | 1 |
If a latex6 scale model were to be tested in a wind tunnel and full dynamic similarity was required: What air velocity would be necessary in the wind tunnel compared to the wind velocity experienced in real life (use latex7 for velocity)? | 1 |
As it falls, it undergoes both linear and rotational acceleration. | 0 |
What is: The change in energy flow rate. | 1 |
The situation is modelled as follows: In the diagram, the coordinate system is at the centre of the car's circular motion. | 1 |
The figure shows the train in the tunnel, in a frame of reference attached to the train. | 0 |
An electric resistance heater with latex0 efficiency. | 0 |
Sketch the processes on a latex9 diagram, numbering the state points to correspond with the figure above. | 1 |
Having now seen the Lorentz transformations, we will check that we can reproduce these results explicitly.We will start with time dilation. | 0 |
What would have happened if, instead of throwing the rock sideways, Biff had thrown it up at an angle? | 1 |
Find the maximum deflection of the loaded beam. | 1 |
The sanding disc, which may be considered to be a uniform disc of mass latex1 , has a diameter latex2 . | 0 |
That is, the power spectrum is the same if latex8 is replaced by latex9 , where latex10 is an arbitrary phase. | 0 |
The latex0 roots of unity are the solutions to the equation latex1 for any **positive integer latex2 . | 0 |
Does the length of the rod matter? | 1 |
Can you work out the boat’s velocity latex2 and displacement latex3 in this case? | 1 |
* Gravity can be neglected. | 0 |
There are two ways to get the answer, and one of them requires an infinite sum. | 0 |
As with the petrol engine, you need to evaluate the conditions at each stage in the engine. | 1 |
At latex3 , Matt emits a light pulse to Rick and another to Lucy. | 0 |
How many conditions on latex17 do you need to find to determine the coefficients? | 1 |
Calculate the wall shear-stress latex24 . | 1 |
Sketch a diagram of the plant layout. | 1 |
There is a dam at one end of width latex1 m as shown. | 0 |
In practice, the repeated sequence will be truncated to fit in memory. | 0 |
If a population of cells is exposed to radioactive UTP labelled with latex0 (a radioactive isotope of phosphorus), the mRNA being formed at that moment will be labelled. | 0 |
State whether this cycle is suitable for maintaining a temperature of latex4 in the cold space. | 0 |
Find the inverse of each of these functions and the domain of each inverse; latex0 , for all real latex1 latex2 , where latex3 | 1 |
(Based on P4.10, White) The flows in this question are all two-dimensional, incompressible and steady. | 0 |
Use an analytical method to confirm the answers found in (a). | 1 |
What is the distance latex2 between Earth and the centre of the Milky Way as seen by an observer in the spaceship's inertial reference frame? | 1 |
In lectures, we worked out a formula for the distance travelled by a boat initially moving at speed latex0 , gradually brought to rest by a drag force of the form latex1 . | 0 |
In doing so we derived the values of the constants needed to match the general solution; however more generally we need a method to do this matching when the general solution is not known. | 0 |
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 latex13 . | 1 |
Using the Cartesian frame of reference shown: If latex5 , show that latex6 . | 1 |
A mass of 1.5 kg is on a spring of spring constant 80 N/m. | 0 |
What does it mean if the filter function is real but negative, i.e $G(\omega) [embedded] * If aircraft humming (below 200 Hz) should be seriously reduced, what is the longest the delay can be before the signal is (negatively) replayed? | 1 |
The mass of the wheels may be ignored. | 0 |
A damped harmonic oscillator has the equation of motion latex0 When latex1 , the damping is light. | 0 |
To answer this question, note carefully the definitions of latex17 and latex18 in Figure 16.4. latex22 where latex23 is the frontal projected area and latex24 is the relative flow speed. | 0 |
Three-quarters of its initial mass is fuel. | 0 |
Contrast these results with a heat loss of approximately latex7 under normal room conditions. | 1 |
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