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As mentioned in Lecture 6, the quantity latex0 is called the \`\`invariant interval'' because it is unchanged under Lorentz transformations. | 0 |
Consider the boundary condition, latex7 and the second condition is given by latex8 but remains arbitrary. | 0 |
latex3 latex4 | 0 |
Find the second-order polynomial that satisfies the criteria above. | 1 |
Using the chain and quotient rules, differentiate the equation above w\.r.t. | 1 |
The system will reach an equilibrium with an angular separation latex3 between the two strings. | 0 |
(Use the Data & Formulae Booklet to find material constants.) | 1 |
A truck travels through shallow water at some speed latex7 . | 0 |
Model the ship as a flat plate, vertically orientated, with the same area in contact with the water, and the same length. | 1 |
Therefore, if we iterate the tent map (3) through a number of iterations, we know how it translates in terms of the original logistic map (1), thanks to (2). | 0 |
\[2022/10/26 - we are aware that the feedback on the box below may take a long time or time out. | 0 |
Consequently, air is considered to behave as an ideal gas. | 0 |
Identify a single equivalent transformation. | 1 |
For small values of latex2 , derive an approximate expression for latex9 in terms of latex2 , keeping terms up to latex16 . | 1 |
**(L10)**: Use Gaussian elimination to calculate the inverses of the following matrices: latex0 latex1 | 1 |
Solve the approximate equation derived in part (e). | 1 |
Give the governing equation, appropriate boundary condition, and solution for the following situations. | 1 |
The heat flux that is applied to the left face of a plane wall is latex0 . | 0 |
Add you own data from the Wind Tunnel and check if it agrees. | 1 |
At the upstream limit (A), and at a certain instance in time, a constant input of sediment with latex4 enters the domain due to changes in the river flow conditions upstream of A. | 0 |
**(L4,6)**: Show that the area of the parallelogram shown below is given by: latex0 Can you convince yourself that this is true? | 1 |
In this question, assume that the bottom plate has infinite extent and the top plate has area latex3 that is very large compared to latex4 . | 0 |
*** \[**Hint:** Let latex2 ]. | 0 |
(i) Draw a block diagram showing the components of the model closed-cycle plant. | 1 |
and P.E changes, is given by \ latex6 where latex7 is the (constant) specific volume. | 0 |
What is the shear stress of the fluid on the bottom plate? | 1 |
Write down expressions for the total force latex6 on the jumper when (i) latex7 and (ii) latex8 and (ii) latex9 , setting the potential energy to zero when latex10 (that is, when the jumper is latex1 metres below the platform). | 1 |
Consider the latex0 matrix, latex1 Calculate latex2 , the determinant of the matrix latex3 . | 1 |
Calculate the gain and the natural frequency of the following second order systems and determine whether they are underdamped or overdamped: latex0 latex1 latex2 latex3 | 1 |
Let latex2 be the angle formed by the car's direction and the pedestrian's line of sight (defined in the solution to part (a)). | 0 |
Calculate the stagnation pressure latex9 behind the oblique shock wave. | 1 |
No body forces. | 0 |
In this model, the valence electron of the atom is connected to the nucleus by a spring and is driven by the oscillating electric field of the light. | 0 |
Airflow in the trachea (density of air is latex1 kg/m latex2 and viscosity of air is latex3 Pa s). | 0 |
The bottom wall (located at latex1 in the Cartesian frame of reference shown is stationary whilst the top wall (located at latex2 ) moves with constant and uniform velocity latex3 . | 0 |
The bridge deck of thickness latex2 disturbs the flow, leading to the periodic release of vortices, the frequency of which is called the vortex shedding frequency, and is denoted by latex3 . | 0 |
What are the relative errors? | 1 |
Calcuate latex8 , the area at the throat. | 1 |
(ii)Then extend the diagram to show that if the Diesel cycle has a higher compression ratio, it can have a higher efficiency than the Otto cycle. | 1 |
Write down expressions for the speed latex4 of the ball when it first hits the table and the time latex5 it takes to drop, both in terms of latex0 . | 1 |
* The centre of mass is located at latex4 from the tip of the cue, where latex5 is the cue's length. | 0 |
(Give 3 decimal places) A fixed area second throat is to be designed for the wind tunnel. | 1 |
The measurement is made using a Pitot-static system - two Pitot tubes are pictured above and they are used in a system as illustrated below: \ \ In this question assume that air at latex1 altitude has the following properties: latex3 , latex4 , latex5 , latex6 . | 0 |
Calculate the period of the oscillations and the decay time (the time for the amplitude to fall by a factor of latex1 ). | 1 |
What is latex2 ? | 1 |
You may use the far-field solution. | 0 |
A **closed** cone has base radius latex0 and vertical height latex1 . | 0 |
A fluid is static if it: | 0 |
As an example, this means that the intensity of light passing through a double slit experiment (representing latex6 ) is equal to the sum of intensities of the scattered light (representing latex7 ). | 0 |
The rate of a chemical reaction often depends on the concentration of its reactants raised to particular powers. | 0 |
A natural-draught cooling tower operates under the conditions given below: (*the notation is the same as in Section 4.4.4 of the lecture notes*). | 0 |
Atmospheric pressure at sea level is latex0 Pa, and air density can be taken to have the constant value 1.2 kg/m latex1 Pipe 1 has cross-sectional area latex2 m latex3 and contains water flowing with speed latex4 m/s, whilst Pipe 2 has cross-sectional area latex5 m latex3 with water at speed latex7 m/s. | 0 |
*** The equation we will need is: latex0 where latex1 is called the *moment of inertia*, latex2 is the angle between the vertical and the cue, and latex3 is the distance from the pivot point to the cue's centre of mass. | 0 |
What might the problem be with the solution to the last question? | 1 |
A random sample of animals is captured using traps over a fixed period, and then marked, and released. | 0 |
The tank is cooled until the temperature of the contents is latex4 . | 0 |
If the collar has a mass latex3 , find the velocity of the collar when latex4 , if it is released from rest. | 1 |
Photosynthesis relies on electrons in chlorophyll molecules absorbing photons of light energy. | 0 |
Try this yourself or use the separate parts for suggestions on how to break this problem down. | 1 |
Two charged spheres of mass latex0 are attached by light (i.e massless) strings of length latex1 to the same point on the ceiling. | 0 |
The shaft material has a shear modulus of latex7 . | 0 |
Consider the instantaneous release of latex0 of contaminant at latex1 . | 0 |
Compare the solution for latex9 (with latex15 at latex16 ) to the one with no coupling latex22 (maybe form the ratio between the two) and discuss why it can be very difficult to measure if a system has a small coupling. | 1 |
The walls are a distance latex2 apart. | 0 |
\[Online]. | 0 |
Compute the maximum flow rate latex5 to obtain a stable channel bed with no sediment motion. | 1 |
To avoid such occurrence a rate-feedback system can be used instead, whilst retaining full capability in limiting the overshooting. | 0 |
**(L10)**: Consider the matrices: latex0 where latex1 are known as *elementary matrices*. | 0 |
The container is heated until the pressure is latex4 bar. | 0 |
In lectures we showed that the change in momentum of an object is equal to the impulse applied: latex0 Derive the three-dimensional vector equivalent: latex1 | 1 |
The initial position of the system is shown in the diagram below. | 0 |
When an electron in chlorophyll absorbs light, it is promoted to higher energy orbital. | 1 |
Does the system of linear equations formed by the planes have a unique solution? | 1 |
The open-loop system shown below requires an operator to set a motorised valve setting, latex0 , where latex1 , so that two fluids are mixed together. | 0 |
This releases a significant mass latex0 of contaminated soil into the river. | 0 |
Determine the spin tensor Determine the vorticity | 1 |
A ping-pong (table tennis) ball makes a head-on elastic collision with a much heavier basketball, which is initially stationary. | 0 |
Draw Mohr's strain circle from this data and determine the principal strains and the inclination of gauge latex0 to the direction of the greater principal strain. | 1 |
Plot the concentration as a function of time at that station. | 1 |
Find the differential latex2 of latex3 . | 1 |
A latex0 flow of steam at latex1 bar, latex2 is throttled to latex3 bar, with negligible heat loss and negligible change in kinetic energy. | 0 |
On an initial trapping session, latex0 rabbits were caught and individually marked. | 0 |
Consider the polar form of a complex number latex0 . | 0 |
Let the proportionality constant be latex2 and for simplicity take the outside temperature to be zero. | 0 |
Show that the function latex0 has three stationary points - two local minima and one saddle point. | 1 |
Use the oblique shock tables to find the wave angle of the oblique shock shown in the figure (at least two decimal spaces). | 1 |
The relationship between latex2 and latex24 is shown below: latex25 * latex2 Number of moles of strong base ( latex27 ) [embedded] * latex24 Concentration of protons ( latex29 ) [embedded] * latex30 Ionic product of water (a constant, latex31 ) [embedded] * latex32 Acid dissociation constant of the buffering agent ( latex29 ) [embedded] * latex34 Concentration of the buffering agent ( latex29 ) Note that the equation is expressed in terms of latex24 rather than latex1 . | 0 |
Hence, find latex2 . | 1 |
Consider latex0 such that latex1 Calculate latex2 Calculate latex3 | 1 |
You should now be able to retrieve the length contraction formula. | 0 |
The electric drive of a record turntable develops a constant torque of latex0 and takes latex1 seconds to accelerate to the turntable to its operating speed of latex2 . | 0 |
It has a sanding disc which rotates at latex0 in the direction shown. | 0 |
Separate the variables (i.e rearrange to get all the latex0 and latex5 on one side and all the latex1 and latex7 on the other) and then integrate both sides to get an *indefinite* integral. | 1 |
At a point in a material, the resultant stress on a plane A is latex0 inclined at latex1 to the normal and on a plane B is latex2 inclined at latex3 to the normal. | 0 |
Find an expression for latex1 . | 1 |
The equation for the profile is latex10 where latex11 and latex12 is a constant. | 0 |
This *is* the Henderson-Hasselbalch equation. | 0 |
The distance from point source latex8 to P is latex9 . | 0 |
A wavefront is defined as a surface over which the phase of the wave is a constant at any given moment in time. | 0 |
Show that when these waves are added together the total wave can be written as latex2 and write down expressions for latex3 , latex4 , latex5 and latex6 . | 1 |
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