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Using the approximate solution in part (b), show that if the pendulum starts from rest with displacement latex21 (i.e real) then: latex22 (hint: again remember latex23 ) By taking the real and imaginary parts of the complex solution show that: latex24 Use the above solution to make a sketch of the pendulum's motion in the latex3 - latex4 plane.

are negligible).

Write a simple ordinary differential equation to express this relationship.

Estimate values for the relevant parameters.

The cycle uses R134a and operates between pressures of latex12 bar and latex13 bar.

What is the steady state value of the following system when triggered from rest by a step function with amplitude latex0 ?

Find a suitable thickness for the cylinder wall if the hoop stress is not to exceed latex2 .

The gravitational acceleration is latex12 , the specific heat at constant pressure is latex13 and the heat capacity ratio is latex14 .

Consider two-dimensional flow along a wall located at latex0 , with component latex1 parallel to the wall (in the latex2 -direction), and latex3 perpendicular to the wall (in the latex4 -direction).

The figure below shows the component layout of a microturbine, a small gas turbine engine with external combustion so that the working fluid is air throughout.

We assume that latex0 and latex1 are *bounded*, meaning that there exists a strictly positive real constant latex8 such that latex9 for any point in space and time.

In the lecture we have found that the steady and laminar flow over a flat plate in the absence of a pressure gradient could be described by a third-order non-linear ODE, following a clever change of variable.

What will be the magnitude and direction of the maximum current in the feedback resistor?

Plot the concentration at these times a function of latex8 in a single figure.

**(L10)**: Consider the latex0 matrix, latex1 Show that latex2 has a non-zero determinant.

Hint: a curved shock forms in front of the aerofoil, however just in front of the nose it is a normal shock.

Hence find the vector, latex13 in terms of both the original basis set latex14 and the orthonormal set { latex15 }.

A jumper of mass latex16 kg steps off this bridge attached to a bungee cord of natural length latex17 m and reaches a lowest point latex18 m above the ground.

The transfer function of the process is latex2 .

latex28 has a half-life of latex29 .

[embedded] What step-down ratio is required from the transmission system to get exactly the right speed?

Now treat the gas turbine engine as a reversible open-cycle plant, consisting only of the compressor, the heat exchanger which models the combustor, and the turbine.

Using *natural* logs, convert this equation into a linear form latex15 thinking clearly what the transformed latex16 and latex17 variables are in terms of latex1 and latex0 , and what the gradient and intercept on the transformed axes represent.

Assuming there is no damping, calculate the period of simple harmonic oscillations.

The angle latex4 between the string and the vertical axis remains fixed.

(Use the Data & Formulae Booklet to find material constants.)

You may use the fact that from geometry we have latex11 .

If the particle velocity is not just directed along the latex1 axis, then the velocity transformation formula needs to be written as latex5 Use the same method as in the lecture to work out the transformation rule for one of the perpendicular velocity components, e.g latex6 .

A river section crossing an important city has a section latex0 , a constant flow discharge latex1 and a water depth latex2 .

Sketch the time evolution of the output signal.

How many equations are there?

latex8 where latex5 is defined in the same way as in the previous parts of the question.

** Find the values of latex5 for both the Von Mises and Tresca yield criteria when the cylinder has **closed zero end load.

Describe the resulting motion.

Plot the latex5 component of latex31 , against latex29 (be careful!).

Find the net specific work of the cycle and then use this to find the power supplied by the engine if it is running at 6000 rev/min and the cylinder capacity is 1.7 litres.

A pressure cooker is essentially an enclosed vessel in which the pressure is maintained (by a safety valve in the lid) at a level above atmospheric.

Then, express it as a trigonometric function.

A viscous oil flows steadily down an inclined plate under the action of gravity, forming a film of constant thickness latex0 .

Find the principal values of the following logarithms: latex0 latex1 latex2

You may wish to use .

*** Select your derived answer to this in the answer boxes below.

Now show that latex19 Explain how this result is consistent with the formulae derived in part (a).

Evaluate the integrals: latex0 latex1 latex2

What is the **total** surface area of the cone?

Consider now a river stream with the specific flow rate latex0 , and a water depth of latex1 .

Hence, instead of latex1 , we must have a third event latex27 which occurs at the same time as latex0 in the moving frame, i.e , latex29 .

Compute the rate of temperature change shown by a recorder on board the aircraft.

Compute the shear stress on the bottom plate.

Considering a deflection of latex2 , use linar supersonic theory to find the pressure coefficients in regions 1, 2, 3, 4, and 5.

A submerged submarine is towed horizontally at a steady speed latex0 in deep still water.

If acting as expert for the legislator, do you approve the manufacturer's claim?

Why does the net power from a gas turbine engine depend strongly on the isentropic efficiencies of the compressor and turbine?

Show that the same result holds for a system of latex0 particles.

Advection only and in both latex0 and latex1 -directions.

Calculate the lengths latex1 of the vectors latex2 and latex3 .

Show that latex15 is the only component of the viscous stress tensor different from zero and that latex16 .

If the coolant is air and the corresponding convection coefficient is latex3 , what is the maximum allowable chip power?

Now suppose you *don't* know how to solve a quadratic equation (the analogy here is that you don't know how to solve the N-S equations).

A clinometer is a tube attached to a protractor with a weighted string (see the diagram below), which can be used to measure angles between the ground and a distant point.

An aircraft flies at a speed of latex0 , at an altitude where the pressure and temperature are latex1 and latex2 respectively.

Comment on the feasibility of such a wind tunnel.

Re-write that solution using a Taylor series expansion about latex8 , up to terms latex9 .

The contaminant concentration at latex1 (after the release has stopped) is given by the triangular profile below.

The sphere will align with the centre of mass

Saturated vapour leaving the evaporator is compressed adiabatically to latex1 bar, then condensed to the saturated liquid state at entry to the expansion valve.

A particle of mass 2 kg has an initial velocity of latex0 .

Sketch the process on an latex6 diagram and evaluate the dryness fraction latex7 of the wet steam sample.

Mutations can be induced in spores of the fungus *Aspergillus nidulans* by irradiation of its colonies with UV light.

Type latex1 for no eigenvector.

You may wish to use .

If copper and lead balls were dropped together from the top of the Tower of Pisa, which would hit the ground first?

In this question you will find the pressure latex3 and the force latex4 provided by the flange bolts required to hold the nozzle in place.

Draw a graph illustrating the output voltage latex3 , as a function of the input voltage latex8 .

In the Earth's frame, we can explain the same effect by time dilation, as the muon lifetime can be treated as a physical clock.

The distance between the inner faces of the outer discs is latex1 cm The inner cylinder of the spool has diameter between its outer surfaces latex2 cm.

Bernoulli's equation Flow of honey from a spoon.

A rotation of latex4 in the latex0 plane, followed by a rotation latex4 in the latex5 plane.

Relate this finding to your observation of breaking waves on the beach.

Consider again the straining flow studied in Q2.1 latex0 where latex1 is a constant real number.Derive an expression for the position of a particle, latex2 , that is located at latex3 when latex4 .

This question is about space-like and time-like separations.

Find the principal stresses at the point and show the position of the two planes A and B relative to the principal planes.

It is observed that the flow (air, latex0 ) exits the nozzle as a perfectly expanded (isentropic) supersonic flow.

You do not have any information about the engines and must determine their performance parameters from limited practically attained data.

A particle starts at position latex0 with a velocity latex1 .

Would the separation change if you moved the Tower of Pisa to the Moon, where latex5 is approximately six times smaller than on Earth?

Therefore there is a happy medium between killing all the spores, and getting enough mutations: 1% survival is sufficiently small for the colonies to be easily countable, whilst providing enough of them to study.

Use the boundary condition that at time latex9 , the concentration of caffeine is latex10 .

Do you expect the profile of the concentration to be symmetric around the maximum concentration in all directions?

At the top of the plate, latex2 .

A fluid with density latex0 , flows at latex1 over a flat plate of length latex2 and width latex3 .

A square isothermal chip is of width latex0 on a side and is mounted in a substrate such that its side and back surfaces are well insulated; the front surface is exposed to the flow of a coolant at latex1 .

Calculate the stagnation pressure in the reservoir.

Please indicate whether the following statements are true or false.

Suppose that the bottom wall is slightly rising in the wave direction.

Bernoulli's equation Flow of air over an aeroplane wing during cruising.

Unfortunately your mobile phone charger has broken.

(*Hint: Use the fact that* latex17 .

Calculate latex7 , the temperature at section 1.

regions labels 1, 2, 3, 4 and 5. latex1 refers to free stream.