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Use the dot product to find the projections of latex3 both parallel ( latex4 and latex5 . | 1 |
A shaft, latex0 diameter, latex1 long, is fitted with a strain gauge, the axis of which is inclined at latex2 to the axis of the shaft. | 0 |
If so, by what factor would it change? | 1 |
Show that latex6 is an eigenvector of the matrix latex3 with eigenvalue latex8 . | 1 |
By inserting the definition of latex1 into the expression for latex10 and expanding the expression fully using the properties of the inner product, show that: latex11 Show that: latex12 To find the coefficients latex13 that minimise latex10 , we can view latex10 as a function of all the coefficients, latex16 , and then differentiate to find the minimum. | 1 |
Like nuclei, particles such as muons decay on average with an exponential function latex1 , where latex2 is the initial number of particles and latex3 is the number of particles that have not yet decayed after time latex4 . | 0 |
In Lecture 4 we derived the velocity addition formula for a particle moving with speed latex0 along the latex1 axis, in a frame moving at velocity latex2 in the latex1 direction. | 1 |
Estimate the aircraft speed. | 1 |
What is the velocity of the centre of mass? | 1 |
The aircraft speed is deduced from measurement of the difference between the stagnation and static pressures. | 0 |
Use these to show that latex16 only depends on latex7 and also that latex18 Solve the equation you found in Part 8.14d) and apply the boundary conditions on latex19 to find latex19 as a function of latex14 , latex22 , latex23 and latex1 . | 1 |
Check your results using the usual small-angle expansion of the cosine function for a single variable. | 1 |
An emitter-follower is to be used to control a resistive heating element with a power rating of latex0 at latex1 . | 0 |
This is the inverse problem to the one in the same lecture, where we checked time dilation by assuming length contraction. | 0 |
Recall that the gradient of a vector latex1 in a cylindrical coordinate system is: latex2 whereas the curl of a vector in cylindrical coordinates is: latex3 Write the velocity gradient tensor in cylindrical coordinates Write the deformation tensor in cylindrical coordinates Write the spin tensor in cylindrical coordinates Write the vorticity vector in cylindrical coordinates | 1 |
Show that the width in latex2 of latex1 is proportional to latex14 when latex15 . | 1 |
**\[Boas 4.1.6]** For latex0 , Verify that latex1 . | 1 |
The density of the water is latex3 . | 0 |
Find an expression for the work latex24 done along path 2. | 1 |
What is the speed of the particle at time latex3 s? | 1 |
The bed material is rounded gravel whose latex2 is latex3 . | 0 |
What is latex2 for latex5 , latex6 and latex7 ? | 1 |
Find the centre of mass of a uniform thin wire in the form of a circular arc of radius latex0 , subtending an angle latex1 at the centre. | 1 |
BZX 79 Zener diodes, rated at latex2 , are available for C2V4 and C3V6 but not for C6V0. | 0 |
Body forces can be neglected. | 0 |
Neglect atmospheric resistance and the variation of gravity with altitude. | 0 |
**(L7)**: Use Cramer's rule to find the intersection of the lines latex0 | 1 |
For the chlorophyll-a dimer in the reaction centre of photosystem I (PS-I), the difference in energy between the ground and excited states is about 171 latex0 . | 0 |
Identify the order of magnitude of each term in the equation and, given the limit latex15 , suggest an approximation to the equation appropriate in the given limit (i.e neglect a term) and solve the approximate equation.If it's not clear what the question is asking, you can reveal a step in the worked solutions to see where we are heading. | 1 |
Calculate the collector current. | 1 |
From now on, assume that you know latex14 , latex0 , latex9 , latex12 and latex13 only. | 0 |
The diffusivity latex0 and the time latex1 are the only quantities that determine the spread latex2 of a contaminant released instantaneously at a single point at latex3 . | 0 |
The following figure shows an oblique shock forming when air ( latex0 ) flowing at latex1 over a wall finds a corner with deflection of latex2 . | 0 |
Write an equation relating the heat transfer to the change in properties of the gas. | 1 |
Calculate the values of latex9 and latex24 from this data set using linear regression (use Excel if you want). | 1 |
We are going to test a latex8 scale model of a latex9 long ship. | 0 |
The energy decay time, latex6 , is the time taken for the energy of the oscillations to reduce by a factor of latex5 . | 0 |
The figure above illustrates the behaviour of the concentration at the point of release and at the solid boundary. | 0 |
What constraint must the syphon satisfy on its height, so that it can still function? | 1 |
Neglecting any dissipation, calculate the maximum speed attained by the pendulum. | 1 |
Justify your answer. | 1 |
Hence determine the sensitivity of the detector in latex3 . | 1 |
This question is a bonus question for strong students. | 0 |
The force resisting the motion of the moving part is given by latex2 , where latex3 is the velocity and latex4 is latex5 . | 0 |
What effect will the pressure gradient, derived in part (f), have on the location of boundary layer separation on the surface of the cylinder? | 1 |
Express the following functions of latex0 in the form of latex1 , where latex2 and latex3 are the real and imaginary parts of these functions. | 1 |
Using this assumption the sediment concentration profile is simplified to latex4 You may wish to use . | 1 |
List all the necessary assumptions. | 1 |
Consider the one-dimensional diffusion of a contaminant which is released instantaneously at latex0 , at a time of latex1 . | 0 |
Surely, this cannot be right. | 0 |
Sketch the concentration profile latex2 for the release on an unbounded domain. | 1 |
The contaminant diffuses at a rate of latex3 . | 0 |
Let latex0 (to read 'psi'), the stream function, be a function of the streamwise ( latex1 ) and wall-normal ( latex2 ) coordinates such that: latex3 where latex4 is the streamwise component of the velocity. | 0 |
This problem aims at deriving an approximate solution. | 0 |
For latex18 , *first find* latex19 ). | 1 |
Find the Taylor series expansions of the following functions in increasing powers of latex0 , as far as terms in latex1 . | 1 |
Calculate stagnation temperature latex10 behind the oblique shock wave. | 1 |
Determine: The mass fraction of each component. | 1 |
)* (iii)Why can real diesel engines take advantage of high compression ratios, unlike real spark-ignition engines? | 1 |
Inside a micro-pump a fin extends into a uniform, steady, two-dimensional flow in a region where body forces and pressure gradients are negligible. | 0 |
Remember latex3 . | 0 |
If not, look at the '*Final Answer*' to check if your equation of intersection is correct. | 1 |
To answer this question, look at your order of magnitude answer to part (b). | 1 |
In this question consider a **steady flow** of people through the airport. | 0 |
A thin-walled, open-ended, cylinder of radius latex0 and wall thickness latex1 is pressurised internally until it yields at pressure latex2 . | 0 |
The gravimetric analysis (meaning analysis by mass) of air is given in the Data & Formulae book. | 0 |
A gas turbine (GT) with an efficiency of latex0 is used in a conventional gas-steam combined cycle. | 0 |
The motor has a normal running speed of latex13 . | 0 |
Find the two eigenvalues latex4 of the matrix latex3 . | 1 |
The complete cardioid ('heart' shape) latex13 . | 0 |
Is the behaviour any different? | 1 |
Assuming that latex2 , calculate the effective stress intensity factor, latex3 , that is corrected for plasticity. | 1 |
Find the speeds of pulleys latex0 and latex5 when the belts have stopped slipping. | 1 |
What are the heads latex6 and latex7 before and after the pump, respectively? | 1 |
**(L13)**: Given that, latex0 Find latex1 and comment on its significance. | 1 |
The following functions are orthogonal on the interval (**True** / **False**): latex0 and latex1 over latex2 latex0 and latex1 over latex5 latex0 and latex1 over latex8 | 0 |
What do you learn by equating the torque and the rate of change of angular momentum about latex1 ? | 1 |
*Hint: you may need to revisit ME1 Maths to remind yourself how to solve a linear 2nd order ODE. | 0 |
For each of the following fields and closed paths oriented anti-clockwise as seen from latex0 , evaluate: latex1 where latex2 stands for counter-clockwise. | 1 |
We proceed again, counting from the symbol list, until latex31 is reached. | 0 |
The attached sketch illustrates a bed form called antidune, occurring when the Froude number is close or above 1 (supercritical). | 0 |
A gas mixture consists of latex0 of latex1 , latex2 of latex3 and latex4 of latex5 . | 0 |
*Hint: Shear stress on neutral axis of I-beam latex3 where latex4 . | 0 |
What is the component of the force exerted by the fluid on the block parallel to the surface of the block? | 1 |
What is the Reynolds number, does this mean the flow is stable or unstable? | 1 |
If necessary, write your answer as a surd. | 1 |
You can assume that the fluid in the manometer is water with density latex4 kg/m latex5 and latex6 cm. | 0 |
Given that latex5 and latex6 are distinct solutions to latex7 and thus are also solutions to latex8 , show that and latex9 must be real. | 1 |
Compute the roughness scale, latex3 \[m]. | 1 |
You are building a toy car to race in a miniature version of Formula 1. | 0 |
Find the trigonometric Fourier series for latex1 defined on the interval latex5 . | 1 |
(Based on P2.58 in White) | 0 |
For the case where one end of the tube is open and the other is closed, find the values of latex5 and latex6 and show that the wavelength of the sound wave is latex11 , where latex8 is an integer. | 1 |
(Based on P4.6, White) Assume for simplicity that flow in the converging nozzle shown below has the form latex0 . | 0 |
Given that you must travel the 10m straight in 4 seconds (accelerating from rest), and that the engine can only operate in one operating point and at max throttle per run, you need to determine the rpm required for each torque value to achieve exactly this time. | 1 |
We can assume the water is Newtonian with dynamic viscosity, latex7 . | 0 |
You have measured the conditions at the start of the compression process (state 1) and have found the temperature and pressure to be 25°C and 1 atm respectively. | 0 |
From the following sketches, pick the most accurate one for the graph you drew in the previous part. | 1 |
latex4 The stress intensity factor *K*, for this geometry is given by the formula latex5 where latex6 is a function of crack depth to sample width latex7 to overall depth tabulated below and latex8 is the nominal bending stress at the tensile face. | 0 |
Consider three planes in latex0 specified by the equations, latex1 Let latex2 denote the matrix of coefficients for this system of three linear equations in three unknowns. | 0 |
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