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1 NTURL SCIENCES TRIPOS Part 1 Saturday 7th June, to PHYSICS ttempt all questions from Section and one question from each of Sections B, C, D, and E. Numbers in the right hand margins indicate the approximate distribution of the marks available. Note that Section carries approximately one third of the total marks. The paper consists of 11 sides including this one and is accompanied by a Mathematical Formulae Handbook giving values of constants and containing mathematical formulae which you may quote without proof. nswers to each Section must be written in a separate script booklet. Should you run out of paper please ask the invigilator for another booklet and tie the two together with a tag. The overall coversheet and cover page of each booklet must be completed before leaving the examination. STTIONERY REQUIREMENTS 5 x 8 page script booklets Overall coversheet Rough workpad SPECIL REQUIREMENTS Mathematical Formulae Handbook pproved calculator allowed You may not start to read the questions printed on the subsequent pages of this question paper until instructed that you may do so by the Invigilator.

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3 SECTION ttempt all questions. nswers should be concise and relevant formulae may be quoted without proof. 3 1 mobile phone sends out a 100 mw signal at 1.8 GHz. How many photons per second are produced by the phone and what wavelength do the photons have? [5] 2 positron has the same mass but opposite charge to an electron. positron and electron are orbiting around each other separated by 1µm, in a stable circular orbit about their centre of mass, as a result of their electrostatic attraction. Calculate the period of the orbit. [ ] You may ignore any radiation effects. [5] 3 For low pressure gradients, the volume flow rate Q (units m 3 s 1 ) of water through a pipe having diameter D is related to the viscosity of water η (units Pa s) and the pressure gradient down the length of the pipe dp, by a formula of the form dx ( ) γ dp Q = kη α D β, dx where k is a dimensionless constant. Find α, β and γ. [5] 4 mechanical beam clamped at one end having length L, width w and thickness t is made from a material having Young s modulus E. The deflection d of the unclamped end of the beam that results from a force F applied at the end of the beam in the direction of its thickness is given by d = 4FL3 Ewt 3. Find d and the error in d, given that F = 10 N ±1%, L = (10 ± 0.2) cm, w = (9 ± 0.1) mm, t = (12 ± 0.1) mm and E = Pa ±0.5%. [5] 5 Using the thin-lens equation, prove Newton s lens formula u v = f 2 for real objects and images, where f is the focal length of the lens and u and v are object and image distances measured from the nearest focal points rather than from the lens itself. [5] 6 space ship in the shape of a circular loop of radius r = 1 km can be modeled as a uniform circular line having a mass per unit length of 2000 kg m kg mass lies a distance 150 m away from the ship s centre along a line perpendicular to the plane of the loop that passes through the ship s central point. Calculate the force exerted by the space ship on the mass. [ For the gravitational constant G, you may use a value of m 3 kg 1 s 2] [5] (TURN OVER

4 4 SECTION B nswer one question only. B7 Two masses M and m are connected via a light string passing over a frictionless light pulley, with the mass M sliding on a frictionless plane inclined at angle θ, as shown in the diagram. T M m T x θ The tension in the string is T, the distance of the centre of the mass m beneath the centre of the pulley is x and the string to the left of the pulley is parallel to the slope. Show that ẍ = g (m M sin θ), ( ) m + M where g is the acceleration due to gravity. [3] lso find an expression for the tension T in terms of the variables (m, M, g and θ). [2] The frictionless inclined plane is now replaced by a perfectly rough one and the mass M replaced by a uniform disc of mass M. The disc is free to rotate about the point of attachment to the string and is constrained to move in the vertical plane passing through the pulley and orthogonal to the inclined plane so that the disc is free to roll up and down the slope without slipping. The disc has the same radius a as the pulley, so that the string to the left of the pulley is parallel to the slope. a M T a m T x θ Draw a labelled diagram of the forces acting on the disc of mass M. Why is there now a frictional force F acting parallel to the slope? [3] In the previous case of a simple block on a frictionless slope, the criterion for the mass when released from rest, to begin accelerating up the slope as opposed to accelerating down the slope, was m > M sin θ from equation (*). Discuss whether you would expect the criterion to be the same or different in the current case. [2] Find expressions for T, F and ẍ, again in terms of m, M, g and θ, for this new configuration. [ You may assume the moment of inertia of the disc is 1 2 Ma2. ] [5]

5 B8 The Lorentz transformations linking events in two frames, S and S in standard configuration, are given by ( t = γ t vx ) ) t = γ (t + vx c 2 c 2 x = γ (x vt) x = γ ( x + vt ) y = y y = y z = z z = z 5 Define the terms in these equations, and the meaning of the phrase standard configuration. [2] Use these results to demonstrate the phenomenon of time dilation. [3] Explain why the survival of muons to ground level when emitted in high-altitude cosmic ray showers is evidence in favour of Special Relativity. [2] Muons have a mass of MeV/c 2 and a mean lifetime in their own rest-frame of 2.2 µs. Estimate the minimum total energy an emitted muon would need to possess, as measured by an observer stationary in the Earth frame, for the muon to survive a 20 km flight to the ground, and discuss how the same phenomenon can be explained in the frame of the muon itself. [ In working out the time of flight, you may assume that the muon is travelling sufficiently close to the speed of light that its velocity may be approximated by c for this purpose. ] [3] The dynamics of the process by which muons are produced in cosmic ray showers can be represented via the following decay process particle particle B + a photon in which particle has mass m, particle B has mass m B, and energy and momentum are conserved. Show for such a decay that, in the frame in which particle is initially at rest, the energy of particle B is given by E B = ( ) m 2 + m 2 B c 2. 2m [5] [ You may wish to use the energy-momentum invariant to obtain the momentum of particle B in terms of its energy E B and mass m B. ] (TURN OVER

6 6 B9 Explain the concept of moment of inertia for a rigid body. [2] Calculate the moment of inertia of a uniform thin square plate of side l and mass M about an axis which passes through its centre and is parallel to a side. [2] Use the perpendicular axis theorem to show that the moment of inertia of a thin square plate about any axis in its plane and passing through its centre is the same. [2] Such a plate is suspended from a horizontal hinge along an edge, and hangs vertically. horizontal linear impulse with magnitude K is applied perpendicular to the plate, at a point a distance a vertically below its center of mass, whereupon the center of mass starts moving with an initial speed v. Explain why for general values of a the hinge must supply a horizontal linear reaction impulse of magnitude N in order for this motion to occur, and find expressions for N and v in terms of K, a, l and M. [7] For what value of a, expressed as a multiple of l, does the initial horizontal reaction at the hinge vanish? [2] SECTION C nswer one question only. C10 Explain what is meant by the complex impedance of an electrical component. [2] Starting with the time-domain relationship between voltage and current, derive an expression for the complex impedance of an inductor L at angular frequency ω. [2] aerial radio aerial receives incoming radio waves and converts them into an oscillating aerial voltage V in (t). The aerial is modeled as a zero-impedance voltage source and is connected to a resonant circuit containing a capacitor C, an inductor L and a resistor R as shown in the diagram above.

7 7 Show that when the aerial voltage is given by V in (t) = Re { V 1 e iωt}, where V 1 is a complex voltage and ω is an angular frequency, then the oscillating voltage at the output of the circuit is given by V out (t) = Re { V 2 e iωt}, where the amplitude is given by V 2 = V 1 ωl R 2 + ( ). ωl 1 2 ωc [2] radio transmitter ( transmitter ) emits a radio signal which is converted by the receiving aerial into a sinusoidal aerial voltage V in at a frequency of 9 MHz with an amplitude of 1 mv. What is the amplitude of the voltage at the output of the circuit if R = 50 Ω, L = 9 µh and C = 35 pf? [1] Explain, in terms of the relative phases of the voltage waveforms across each of the impedances in the circuit, how the amplitude of the output voltage can be larger than the amplitude of the aerial voltage. [2] Transmitter is switched off and a second radio transmitter, transmitter B, is switched on. This also results in a sinusoidal aerial voltage of 1 mv but at a frequency of 10 MHz. Compare the amplitude of V out when transmitter B is on with that when transmitter was on and comment on your result. [2] The value of the capacitor is changed to C = 31.5 pf. It is now found that the amplitude of V out is approximately the same when either transmitter or transmitter B is transmitting individually. Explain this change in terms of the change in the resonant frequency of the circuit. [2] What kind of voltage waveform would be seen at the output of the circuit if both transmitters are transmitting at the same time? [2] (TURN OVER

8 C11 simple harmonic oscillator consisting of a mass m attached by a spring of constant k to a fixed surface has an equation of motion of 8 ẍ = kx/m, where x is the extension of the spring. Show, by differentiating the energy with respect to time or otherwise, that the total energy of the system is constant in time. [3] U tube of uniform cross-sectional area is filled with a length l of a liquid of density ρ. Calculate the potential energy of the fluid if it is displaced from equilibrium by an amount y as shown in the diagram below. [2] y y l Calculate the kinetic energy of the fluid assuming the entire liquid is moving at a speed ẏ. [1] Show, by differentiating the total energy with respect to time or otherwise, that the equation of motion of the system when there are no frictional forces is given by ÿ = 2g l y and write down an expression for the frequency of oscillation of the system. [2] drop falls into the liquid in the left-hand side of the tube and sets the fluid in the tube into motion. Sketch and annotate a graph of the liquid height y as a function of time if l = 0.5 m, = m 2, ρ = 1500 kg m 3 and the speed of the fluid immediately after the drop falls into it is 15 mm s 1. [3] [ You may assume that the collision between the drop and the liquid is instantaneous, that there are no frictional forces and that the volume of the drop is negligible compared to the volume of the liquid. ] If a frictional force now exists between the walls of the tube and the liquid such that when the liquid moves at speed v the force is βv per unit length, calculate the Q value of the system when β = 0.3 kg m 1 s 1. [2] ssuming this level of friction, how long after the drop falls into the liquid does the amplitude of the oscillations about the equilibrium level decay to 0.1 mm? [2]

9 9 SECTION D nswer one question only. D12 Explain briefly the difference between standing and travelling waves. Provide an example of a wave function describing each case. [2] Consider the Schrödinger equation for a quantum mechanical particle of mass m in 1D written in the form d 2 ψ dx + 2m [ ] for x < 0 E V(x) ψ(x) = 0, with potential V(x) = 0 for 0 < x < a 2 h 2 V 0 for x > a as illustrated in the diagram. For the case E > V 0, write the general form of the solution to the Schrödinger equation for the situation in the diagram in each of the three regions x < 0, 0 < x < a and x > a. Include the mathematical relations that the solution has to satisfy at the boundaries x = 0 and x = a. Express the wave vectors in terms of m, E, and V 0, as appropriate. [3] For the case where 0 < E < V 0, show that the solution to the Schrödinger equation can be written in the form ψ(x) = sin(k 1 x) for 0 < x < a, and in the form ψ(x) = Ce k 2x for x > a. Using the boundary condition at x = a, show that bound states (i.e. solutions for 0 < E < V 0 ) exist only if tan (k 1 a) = k 1 /k 2, where k 1 and k 2 are taken to be positive. Express k 1 and k 2 in terms of m, E, and V 0, as appropriate. [4] Explain why bound states exist only if λ 1 < 4a, where λ 1 = 2π/k 1 is the wavelength of the solution in the region 0 < x < a. [ Hint: consider the boundary conditions at x = a. ] [2] Show that the condition for the existence of bound states can be written as V 0 h 2 π 2 /(8ma 2 ). [2] Helium atom is trapped by weak attractive forces above a flat metal surface forming a bound state. ssuming that the surface potential is of the form V(x) given above, where x is the perpendicular distance of the Helium atom from the surface, estimate the minimum value of the parameter a when V 0 = 1 mev. [ Use mhe = kg ]. [2] (TURN OVER

10 10 D13 Explain how phasors can be used in calculating the interference between two waves. [3] Consider a diffraction grating having a large number N of narrow slits of spacing d. Use phasors to derive the angles at which principal maxima in intensity can be seen on a distant screen when the diffraction grating is illuminated at normal incidence with light of wavelength λ. [2] Using phasors, show that if a principal maximum occurs at an angle θ, then the angle between this maximum and the next minimum of intensity is θ = λ/(w cos θ) (see diagram below) where W = Nd is the total width of the grating. [3] For this grating, small subsidiary maxima are seen between the principal maxima. Show that the ratio in the intensity between a principal maximum and the next subsidiary one is approximately 9π 2 /4. [ You may assume that the first subsidiary maximum occurs when the phasor diagram wraps times.] [3] State the Rayleigh criterion for resolving the lines in a spectrum. [1] Naturally occurring potassium is comprised mainly of the 39 K (93%) and 41 K (6%) isotopes, for which the D1 lines in their emission spectra have wavelengths of nm and nm, respectively. Calculate the minimum width of the diffraction grating needed to separate the D1 lines emitted by naturally occurring potassium and comment on the feasibility of making and using such a grating. [3]

11 11 SECTION E nswer one question only. E14 Give a qualitative justification for Gauss s Law for inverse-square-law fields. [2] Gauss s Law for the gravitational flux states that the integral of the gravitational flux, g, over a closed surface that encloses an object of total mass M is of the form g ds = CM, S where C is a constant. State Newton s Law of Gravity and use it to show that the value of the constant C is 4πG. [3] In an attempt to verify the value of the gravitational constant G, a small mass is hung from a 5 m long light string in a service duct located next to a long, straight tunnel having a circular cross section of diameter 7 m that carries water to a hydroelectric power station. The small mass is located 1.5 m horizontally from the edge of the main tunnel. The tunnel is initially full of water. When the water is drained from the main tunnel, the small mass is observed to move a horizontal distance d away from the main tunnel wall. Use Gauss s Law for gravitational flux to estimate the value of d. [ For the gravitational constant G, you may use a value of m 3 kg 1 s 2. ] [6] 5m d 7m 1.5m In a separate attempt to verify the value of G, a mass is suspended from a bridge by a spring having a spring constant such that at equilibrium the spring is extended by 1 m by the weight of the mass. The bridge crosses a wide estuary. Estimate the change in the extension of the spring between low and high tides if the water level in the estuary varies by 8 m. [4] E15 Write short notes on two of the following: (a) fields and potential energy; (b) Faraday and Lenz laws and inductance; (c) mpère and Biot-Savart laws and their uses; (d) orbits in a gravitational field. END OF PPER [7½] [7½] [7½] [7½]

You may not start to read the questions printed on the subsequent pages of this question paper until instructed that you may do so by the Invigilator.

You may not start to read the questions printed on the subsequent pages of this question paper until instructed that you may do so by the Invigilator. NTURL SCIENCES TRIPOS Part I Saturday 9 June 2007 1.30 pm to 4.30 pm PHYSICS nswer the whole of Section and four questions from Sections B, C and D, with at least one question from each of these Sections.