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For the function latex0 Find latex1 . | 1 |
Secondly, we will do the calculation in terms of the velocity components and the pressure. | 1 |
Indicate the minimum kinetic energy required for an alpha particle (two protons and two neutrons) to fission into two deuterons. | 1 |
*You did most of this on Problem Sheet 2,* *Question 4*. | 0 |
Ignore the effects of gravity. | 0 |
A Diesel engine uses latex0 of the oil fuel specified in Table 8.4 of the lecture notes to produce a work output of latex1 . | 0 |
Using the R134a tables, calculate: The coefficient of performance. | 1 |
In other words, the distance as well as the time between two events changes when transforming to a different frame, but latex1 does not. | 0 |
Which function do you suggest would be a good starting point to modelling this variation in insolation by latitude? | 1 |
Take the kinematic viscosity to be latex0 , an approximate value for water at 20 latex1 C. What proportion of the boundary layer is laminar? | 1 |
Find the angular velocity latex3 that the gyroscope rotor should spin at if the car is travelling with velocity latex4 . | 1 |
Define the displacement thickness, latex3 , of a boundary layer both in words and with an equation of the form latex4 , where latex5 is the perpendicular distance from the wall and latex6 is the component of velocity parallel to the wall. | 1 |
Note: this is probably the most challenging question in the module! | 0 |
What would be the normal vector to planes of constant pressure for this fluid? | 1 |
The gas can be treated as perfect, with latex5 and latex6 . | 0 |
What impact would this filter have on a signal with a frequency of latex4 , latex5 and at latex6 ? | 1 |
They have each been charged to the same charge latex2 . | 0 |
Which way did you draw your negative normal direction, latex2 ? | 1 |
Use constants latex14 and latex15 . | 0 |
Estimate the Rouse number. | 1 |
Compute the bed shear stress, latex5 latex6 . | 1 |
If its volume latex1 is fixed... Find the side lengths that minimise the surface area latex2 of the box. | 1 |
Imperial College eccentric Dr. Archie Meades dreams that he has to design the main attraction for the next (!) | 0 |
What assumptions have you made in doing this calculation? | 1 |
Show that: latex4 This is called Parseval’s theorem for Fourier transforms. | 1 |
Consider the one-dimensional diffusion of a contaminant for which a quantity latex0 is released instantaneously at latex1 , latex2 . | 0 |
Find the volume of the resulting solid. | 1 |
First, show that: latex5 Show that: latex6 By inserting latex7 and using integration by parts show that: latex8 Finally pull it all together to show that we indeed have a delta function. | 1 |
A buffer stop at the end of a railway track (purple in the image below) has a moving part, which has a mass of latex0 and can move latex1 parallel to the track. | 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 latex0 , where latex1 is the drag force, latex2 is the ship speed, latex3 is its length, latex4 is its width, latex5 and latex6 are the sea water density and viscosity, and latex7 is the acceleration due to gravity.Find the non-dimensional groups that describe the problem. | 1 |
Think of two or more potential functions, at least one of which should be smooth and differentiable at its minimum, for which small oscillations about the minimum are **not** simple harmonic oscillations. | 1 |
Derive a general solution to the equation given the boundary condition. | 1 |
What happens to the entropy when heat is transferred from the system to the environment? | 1 |
Which of the following equations relates wavelength in metres to standard reduction potential in volts? | 1 |
Bernoulli's equation | 0 |
Consider a turbulent and incompressible flow.Apply Reynolds decomposition to the velocity field and time-average the continuity equation to show that: latex0 Show that the velocity fluctuations also satisfy the continuity equation, that is: latex1 | 1 |
Hence evaluate latex6 . | 1 |
Find the work done by the force latex0 in moving an object \[in the latex1 plane] from latex2 to \ latex3 \phi$] From latex2 to $(1,1) latex5 (0,1) latex7 \vec{F} latex8 \Omega(x,y) latex9 \vec{F} = \frac{\partial\Omega}{\partial x}\mathbf{\hat{i}} + \frac{\partial\Omega}{\partial y}\mathbf{\hat{j}} $ | 1 |
The natural oscillation frequency of the electron, latex0 , is the transition (angular) frequency of the atom. | 0 |
Steam enters a turbine at a pressure of latex0 bar and at a temperature of latex1 with a velocity of latex2 . | 0 |
How many `cubes' or cells do you need to cover the domain and resolve the smallest motions? | 1 |
The manufacturer claims that the train, which has a maximum cross-section area latex1 , will negotiate tunnels at a speed latex2 . | 0 |
Show how latex5 can be expressed in terms of the latex0 th roots of unity. | 1 |
Calculate the power dissipation in the transistor. | 1 |
Write an equation relating the turbine work output to the change in properties of the working fluid (Assuming the K.E. | 1 |
Different situations require different thermodynamic analyses. | 0 |
Assume that there are no interactions with boundaries. | 0 |
The position of a D.C. motor is regulated by a PD control system, with latex0 . | 0 |
The Navier--Stokes equations for steady flow in two dimensions are latex8 latex9 Hence sketch the streamlines. | 1 |
This implies that any light ray directed from one focus is reflected by a mirrored ellipse to the other focus. | 0 |
Hence, find the motion of the point latex19 in the stationary frame of the planet as a function of latex2 . | 1 |
The torque required to turn the shaft is measured, and used to determine the viscosity. | 0 |
Make a sketch of the intensity pattern. | 1 |
Do your answers (b) and (c) agree with your intuition? | 1 |
An accelerometer with an output of latex0 to latex1 is used to monitor the vibrations of a frame of a train. | 0 |
Find expressions for: The velocity of the mass a function of latex1 . | 1 |
The drivetrain you have been tasked to build must rotate a gyroscope at an angular speed of latex0 rad/s. | 0 |
Using an idealised model of the engine and its processes, you need to determine the conditions at each key point in the cycle. | 1 |
Compute the bed shear stress, latex3 latex4 . | 1 |
Is this the same pressure ratio at which cycle efficiency is a maximum, for the same maximum temperature? | 1 |
The African bush elephant is the largest and heaviest land mammal, with recorded weights up to 10.4 tonnes (i.e a mass of 10,400 kg). | 0 |
Consider the integral latex0 Use partial fractions to expand the integrand in order to show that: latex1 By noting that latex2 in the denominator of latex3 , show that: latex4 that is to say: latex5 | 1 |
We want to decelerate air at a Mach number of 3.2 and at sea level conditions. | 0 |
(i) (ii) The figure above shows an elastic collision as seen (i) from the centre-of-mass frame, and (ii) from the lab frame. | 0 |
Remember that resistors can be connected in parallel. | 0 |
The same process can then be used to remove components along latex4 and latex7 from latex10 before normalising that result. | 0 |
In other words, show that the centre-of-mass frame is also the zero-momentum frame. | 1 |
Calculate the amplitude of the wave, the wavelength, the frequency, the speed, the transverse velocity of the string at latex1 and latex2 s. | 1 |
The number of nuclei present at the death of the organism (time zero, latex26 ) can be used to set such a boundary. | 1 |
The space and time axes of the moving frame of the car are also shown in the spacetime diagram. | 0 |
Obtain the total sediment transport rate. | 1 |
Plot the acceleration of Person (1) as it varies in time, i.e plot the latex5 component of latex25 against latex8 . | 1 |
This ripple can be removed using a low pass filter. | 0 |
(iii) By calculating the compressor specific work latex13 and turbine specific work latex14 , show that the net specific work latex15 is the same as that found in part (b)(iii). | 1 |
The pressure in the pipe can fluctuate, such that in a given week it can spend the following times at the following pressures at a constant temperate of latex0 . | 0 |
* The beam was lightened by removing discs of metal from the web (as shown in the figure.) | 0 |
The way the Q-cycle does this is quite complicated: one of the electrons from the oxidation of a molecule of latex0 is passed to latex2 but the other electron is passed *back* to a different latex1 quinone molecule, which it half-reduces to a latex8 free-radical. | 0 |
We know that latex13 is very small so we consider the case of high Reynolds number i.e latex14 In anticipation of approximation, non-dimensionalise the main equation. | 0 |
Compute the mass flow rate across the top surface of the control volume (noted ''ab'' in the figure). | 1 |
So let latex0 be the real function, and latex1 the approximation to latex0 that includes latex3 terms. | 0 |
Use the Taylor series about latex11 , i.e latex12 to simplify the general solution in this limit. | 1 |
Refrigerant R134a enters the expansion valve as saturated liquid at latex0 in cycle (a) and as sub-cooled liquid at latex1 , latex2 bar in cycle (b). | 0 |
It is new and part of an FYP. | 0 |
(see discussion in worked solutions) How many unknown, dependent variables are in the equations? | 1 |
Why are the properties appearing in the two equations different? | 1 |
Calculate the pressure latex7 behind the oblique shock wave. | 1 |
You've experienced convection cooling if you've ever extended your hand out the window of a moving vehicle or into a flowing water stream. | 0 |
We need to approximate. | 0 |
Hence, find latex4 and latex5 in terms of latex12 and latex13 as a function of latex2 . | 1 |
Show that the total (complex) wave at P is latex14 where latex15 . | 1 |
The system is regulated by a proportional control with unity feedback. | 0 |
Find the products latex11 and latex12 and show they do not commute. | 1 |
The pressure of wet steam with dryness fraction latex3 at latex4 . | 0 |
Indicate any position(s) of **un**stable equilibrium. | 1 |
Express the force per unit area (a vector) exerted by the fluid on each plate in terms of latex0 , latex7 , and latex11 . | 1 |
The density of the sediment is latex3 . | 0 |
A block of mass latex0 and with surface area latex1 is pulled at a constant velocity latex2 across the surface of a pool of water of depth latex3 , as shown in the diagram below. | 0 |
What will be the pressure in the test section? | 1 |
Determine analytically the maximum shear stress in the material at the point concerned and the applied torque. | 1 |
What is the maximum mass flow rate possible in the gap between the train and the tunnel? | 1 |
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