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Show that: latex9 Find latex1 for any positive value of latex3 .

Figures 1 and 2 are two views of a Formula 1 race.

Air, supplied by a reservoir at latex0 flows through a converging-diverging nozzle whose throat area is latex1 .

Figure 1 Figure 2 Figure 1 is consistent with: Figure 2 is consistent with:

Using experimental data given in the table, find the thrust generated by a geometrically similar propeller of diameter latex4 , spinning at latex5 at a forward velocity of latex6 , while operating at an altitude where the density is half that at sea level.

The following statements are true at *both* plates (1) and (2): o Plate area is latex0 .

Note that latex24 is the relative speed experienced by the cups, assumed constant for each half rotation of the anemometer.

Challenge yourself to answer this question with no further information.

Draw a circuit diagram of the system and select a suitable resistor value.

A square plate fixed along all four edges has a set of normal modes characterized by two integers latex0 and latex1 .

the latex5 component of latex23 .

By how much must the fracture toughness of the material be improved in order to tolerate a longer crack latex11 at the same fracture load?

In the lectures we looked at the interference pattern due to a pair of point sources with separation latex0 .

In this part of the question you will *begin* to find the latex22 component, latex11 , of the flow Given that the flow does not depend on latex12 or latex13 , simplify the continuity equation and solve it to find the dependency of latex11 upon latex12 , latex13 , latex29 and latex30 .

Use the cross product to find the angle between latex5 and latex6 .

The card can sample at latex3 .

Use a frame of reference fixed to the tank and assume no relative motion between the fluid particles in that frame of reference.

Find the values of latex2 and latex3 at latex4 .

Make use of Green's Theorem.

Consider the function latex0 , where latex1 , is analytic.

Using this figure, estimate the maximum operating temperature of a gas turbine blade made out of this material which is to withstand a stress of latex0 for a duration of latex1 hours.

The mass flow rate latex4 is the same on each side, so that the temperature difference latex5 between the hot and cold sides is constant throughout the heat exchanger.

The ratio of latex7 with latex1 is known as the Froude number (noted latex10 ).

It is often easier to understand where they come from and rederive them when needed.

Is the discrepancy in air-speed computations significant in this case?

There is damping in the system because the accelerating electron dissipates energy by radiation.

Show that the maximum value of latex2 is 	 latex3 , and that 	 latex4 The half-power frequencies are defined as the values of latex5 at which latex2 is half the maximum value.

Draw an element of material at the point of interest, and indicate the stresses acting on its faces.

In the channel flow pictured below, with water entering from the left, in which region (labelled A to F) is the flow field approximately the same as the 'straining flow' field described above?

An 'integrated' risk assessment, for simple risks such as everyday activities.

This problem leads you through an alternate derivation based on the invariance of the speed of light.

Force fields without potentials are called non-conservative.

The geometry is illustrated in the figure below, where we’ve assumed latex1 is odd (though the results will not depend on that).

** Calculate the safety factor against **buckling in the plane of the diagram**.

The general solution of the driven oscillator with light damping, where the driving force is latex0 , is the sum of the steady solution 	 latex1 and the transient solution 	 latex2 .

The combustion process in the real engine is to be modelled by a heat transfer latex6 to the air of latex7 .

Calculate the latex15 in a heat pump application.

A wide freshwater stream has a stream flow of latex0 at a uniform depth of latex1 .

Define a control volume in the reference frame moving with the compression front.

In the quantum picture, the power is dissipated by emitting photons, each one having energy latex11 .

The Reynolds number describes the ratio of inertial to viscous effects.

(You may assume throughout this question that all physical quantities are given in dimensionless units, and there are no external forces acting on the particle.)

The pressure is latex7 Pa, where latex8 is the distance parallel to the flow (measured in metres).

What is the maximum deflection angle for which a flow at latex2 will remain attached?

*** The centroid is the Cartesian point latex1 such that latex2 and latex3 where latex4 is the region of integration and latex5 is the area of that region.

By recognising: latex8 as a Fourier transform of a function from *Question 3*, evaluate the integral: latex9 through the use of Parseval’s theorem.

It branches in two repeatedly, and at each level, the volume of each daughter branch is one quarter that of its parent.

Find the acceleration of a disc of mass latex0 and radius latex1 , which has been released from rest, as it rolls (without slipping) down an inclined plane which is at an angle latex2 to the horizontal.

The sediment size at the river bed is described by latex2 .

If the planes intersect at a point, input the point of intersection.

Lubricant viscosity is VG32 (32 mPa.s).

Determine the dryness fraction latex6 at the final state of the steam

A crack of length latex0 has been found at the center of a large thin plate, with an end load of latex1 .

A passive contaminant or tracer.

The Navier latex0 Stokes equations describing the flow of an incompressible Newtonian fluid are as follows: latex36 latex37 latex38 In this part you will find the latex12 - and latex13 -components of the flow, latex9 and latex10 , in terms of the pressure field, latex43 .

Assume that advection is the only process of importance.

(Based on P3.164, White) The nozzle shown below ejects a jet of water into air at atmospheric pressure (101 kPa).

Why do these not hold?

Is the channel bed stable in terms of initiation of motion for this flow?

What is the highest frequency the card can sample without the danger of aliasing?

(*Hint: find* latex12 *and* latex13 *from the fact that process* latex14 *is isentropic, then* latex10 *from the 1st Law, then* latex11 *from the equation of state*.)

* You may wish to abbreviate that to latex41 for clarity.

(iii) Calculate the net specific work and hence the thermal efficiency of the cycle.

A constant force latex6 N acts for 5 s. Find the acceleration of the block and hence its final speed and final kinetic energy.

However, there is a simple classical model that works surprisingly well.

What is the probability of a randomly synthesised 6 nucleotide-long piece of DNA having the sequence AACCGT (assume all bases have equal probabilities of occurrence)?

Calculate the total resistance as measured between points **a** and **b** in the network below:

*\[Riley 10.11]* The hyperboloid latex0 is parameterised by latex1 , latex2 , latex3 .

Use Cramer's rule to find a solution to the following system of linear equations: latex0

Hint: Are there any symmetries that allow reduction of the spatial dimensions?

What is the deflection angle that will make the flow behind a shock at latex1 sonic?

You should assume that the conditions in the reservoir are steady.

Consider two infinitely narrow slits that are separated by a distance latex8 .

What is the total resultant force of the water on the dam?

Give its numerical value at ground level: Aircraft designers assume air to be a continuum medium.

A force of latex1 N acts on the mass for 4 seconds.

Assume acceleration is uniform up until the point of reaching maximum speed.

Assume the channel to be steady and uniform.

What is the pressure at the bottom of a (hypothetical) latex2 mm high column of mercury, latex3 kg/ latex4 ?

Find the pressures and temperatures at the end of the expansion process.

Hint: You may need to use the latex6 function in Matlab.

What is the critical bed shear stress, latex5 , for this channel?

Consider a river stream given by the specific flow rate latex0 , where the water depth is latex1 .

The water is present in both liquid and vapour phases, and has a total mass of latex3 .

Let’s write the frequencies of these normal modes in the form latex2 .

Answer the following questions.

If so - of what functions?

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 5000 rev/min and the cylinder capacity is 2 litres ( latex1 ) Using the answer to part c, determine the thermal efficiency of the cycle, the Carnot efficiency, and the effective efficiency of the engine as a percentage of the Carnot.

Is the assumption of suspended sediment transport occurring in a small layer close to the bed valid?

Repeat this calculation for other time values.

Call this number of nuclei latex27 , and use it to obtain a particular solution to the ODE.

The phone has an internal resistance of latex2 and requires a latex3 input.

A snow plough starts at noon latex1 and moves along a road removing snow at a constant rate.

Now consider the burning of the log and describe the temperature latex5 as a convolution of the constant heat provided by the log with the temperature evolution latex6 from part (a).

The minus sign in the answer arises because the radial direction points out of the sphere while the flux is into the sphere.

**(L13)**: A parallelogram latex0 (in latex1 ) anchored at the origin is defined by the sides with vectors with components latex2 and latex3 .

It can be shown that the mean-free path in the hard-sphere model is latex7 .

latex0 Which of the following is correct:- Select all that apply Does the function latex1 satisfy Laplace's equation?

You must choose your own unit normals --- do this with care regarding the signs.

the station.