KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM

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1 KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered on the exam paper; PART C should be answered in the examination booklet which should be attached to the exam paper at the end of the exam with a treasury tag. PART A yields 16% of the marks, PART B yields 24%, PART C yields 60%. You are advised to divide your time in roughly these proportions. Figures in brackets [ ] denote the marks allocated to the various parts of each question. Tables of physical and mathematical data may be obtained from the invigilator. Student Number A C1 Total B C2 C3 C4

2 Page 2 PART A Tick the box by the answer you judge to be correct (marks are not deducted for incorrect answers) A1 Period and angular frequency are related by T = 2πω T = 2π/ω T = ω/2π T = 1/ω [1] A2 The acceleration and displacement of a particle in simple harmonic motion are 90 out of phase 180 out of phase one full cycle out of phase in phase [1] A3 An object is in simple harmonic motion with angular frequency 2 s 1. When it passes through equilibrium, its speed is 0.4 m s 1. Its maximum displacement is 0.1 m 0.2 m 0.8 m 5 m [1] A4 The total mechanical energy of a particle in simple harmonic motion is (in standard notation) 1 2 mω2 x 2 mω 2 x mω2 A 2 mω 2 A 2 [1] A5 An oscillator of mass m = 100 g and natural angular frequency ω 0 = 3 s 1 is subject to a damping force γẋ, with γ = 0.6 kg s 1. This system is critically damped resonantly damped overdamped underdamped [1] A6 The period of a lightly damped oscillator is shorter than longer than equal to unrelated to [1] its natural period. A7 A damped oscillator with natural angular frequency ω 0 is driven by an external force with angular frequency ω e. In the low-frequency limit ω e ω 0, the driving force cancels the damping force replaces the natural restoring force vanishes cancels the natural restoring force [1] A8 At resonance, the steady-state amplitude of a forced oscillator is a maximum a minimum infinite zero [1]

3 Page 3 A9 The speed of a wave on a string under tension F and with linear density µ is v = F/µ v = µ/f v = F/µ v = µ/f [1] A10 Which two of the following are equivalent general forms for a travelling harmonic wave? Tick both. y(x, t) = A sin (kx ± ωt + φ 0 ) y(x, t) = A sin (x ± vt + φ 0 ) y(x, t) = A sin (2πx/λ ± 2πft) y(x, t) = A sin ( 2π [x ± vt] + φ ) λ 0 A11 The maximum particle speed in the wave y(x, t) = 0.4 sin(5x 2t) [m] is 2.5 m s m s m s m s 1 [1] A12 A standing wave has the equation y(x, t) = 0.4 sin(2x) cos(4t), for x and y measured in metres. The speed of the component travelling waves is 1.6 m s 1 2 m s m s m s 1 [1] A13 A one-metre long string is fixed at both ends. Which one of the following is not the wavelength of a normal mode of this string? 1/4 m 1/3 m 2/3 m 3/4 m [1] A14 Interference patterns of the type seen in Young s experiment arise when waves from two sources arrive at the same point from opposite directions having travelled different distances at different times with slightly different frequencies [1] A15 Waves of wavelength λ pass through a single slit of width a. Minima in the diffraction pattern on a distant screen occur at angles satisfying a sin θ = Mλ, M = ±1, ±2,... a sin θ = Mλ, M = 0, ±1, ±2,... a sin θ = (M + 1/2)λ, M = ±1, ±2,... a sin θ = (M + 1/2)λ, M = 0, ±1, ±2,... [1] A16 In quantum mechanics, the energy levels E n of a particle in an infinite potential well are proportional to 1/n 2 1/n n n 2 [1] for n an integer. [1]

4 Page 4 PART B Answer all EIGHT questions B1 Write down the differential equation that characterizes any simple harmonic oscillation, and its general solution. Determine the displacement as a function of time for a block of mass m = 100 g on the end of a spring with k = 0.4 N m 1, if the block is at rest a distance x = +20 cm from equilibrium at time t = 0. [3] B2 A simple pendulum, 30 cm long, oscillates such that the maximum angle it makes with the vertical is θ max = 10. What is the maximum of the angular speed θ, in units of rad s 1? At what point in the swing is this speed achieved? [The acceleration due to gravity is g = 9.81 m s 2.] [3] B3 A particle of mass 100 g executes simple harmonic motion about x = 0 with frequency 0.5 Hz. At a certain instant, its kinetic energy is K = 0.25 J and its potential energy is U = 0.2 J. What is the displacement from equilibrium at this instant? What is the amplitude of the oscillation? [3] B4 A damped oscillator with natural angular frequency ω 0 is driven by an external force F (t) = F 0 cos(ω e t). Write down the general form of the steady-state displacement x(t), and sketch the dependence of its amplitude on ω e for a very lightly damped system. [3]

5 Page 5 B5 Show that a wave travelling in the +x direction is described by a function of the form y(x, t) = f(x vt). [3] B6 Write down the one-dimensional wave equation, and show that the function y(x, t) = x 3 6x 2 t + 12xt 2 8t 3 is a solution. What is the speed of this wave, if x and y are measured in metres and t in seconds? [3] B7 A violin string must be tuned to vibrate at a frequency of 660 Hz in its fundamental mode. The vibrating part of the string is 33 cm long, and the linear density is 3 g m 1. What is the tension when the string is in tune? [3] B8 Electrons from a slow beam (speed 365 km s 1 ) pass through two narrow slits separated by 1 µm and strike a flat screen 1 m away. Find the de Broglie wavelength of the electrons and calculate the distance on the screen from the central peak of electron intensity to the first minimum. [3]

6 Page 6 PART C Answer TWO out of FOUR questions C1 A spherical star cluster of total mass M and radius R has a uniform density. (a) Find the force on a single star of mass m moving purely in the radial (r) direction inside the cluster, and thus show that the motion of such a star is simple harmonic with a period T = 2π R 3 /GM. Where is the stable equilibrium about which the star oscillates? [10] (b) Suppose the star in (a) is instantaneously at rest at a radius r 0 at time t = 0. Give equations for the position r, the velocity ṙ, and the acceleration r of the star as functions of time, all in terms of r 0, R, M, and Newton s G. [6] (c) At what time does the star first reach the centre of the cluster, and what is its speed at that point? Again, give these in terms of r 0, R, M, and G. [4] (d) Obtain equations for the kinetic energy K and the potential energy U of the star as functions of time (assuming U = 0 at equilibrium). Use these to show that the total mechanical energy is a constant, E tot = GMmr 2 0/2R 3. [6] (e) Sketch the dependences of K and U on position inside the cluster. [4] C2 The equation of motion of a damped oscillator is ẍ + γ m ẋ + ω2 0 x = 0. (a) Briefly state what each term in this equation represents physically. [3] (b) Consider the special case in which γ = 2mω 0. i. What is this type of damping called? [1] ii. Re-write the equation of motion with ω 0 as the only constant. [1] iii. Show, by direct substitution into the differential equation in (ii), that the function x(t) = e ω 0t is a solution. [6] iv. Show that the function x(t) = te ω 0t is also a solution. [9] v. Justify the conclusion that the general solution to the equation of motion in this case is x(t) = C 1 t e ω0t + C 2 e ω0t, with C 1 and C 2 arbitrary constants. [2]

7 Page 7 (c) Name the type of damping that results for larger γ > 2mω 0. Sketch the typical dependence of x on t in this case, clearly showing all of the main physical features. What is the primary distinction between this type of damping and that in (b)? [4] (d) Name the type of damping that results for smaller γ < 2mω 0. Sketch the typical dependence of x on t in this case, clearly showing all of the main physical features. [4] C3 (a) Write the wavefunctions for two harmonic waves travelling on the x-axis in opposite directions but with identical amplitudes A, wavenumbers k, and angular frequencies ω. (Take the phase constant to be φ 0 = 0 for both waves.) [4] (b) Show that the standing wave formed by the two waves in (a) is y(x, t) = 2A sin(kx) cos(ωt). [4] (c) Suppose this is a transverse wave on a string with ends at x = 0 and x = L, both of which are fixed with y = 0 at all times. Derive the allowed wavelengths λ n and frequencies f n of the normal modes. [6] (d) Sketch the wavefunction y as a function of x at t = 0 for the first three harmonics. Indicate the positions of all nodes and antinodes in each case. [6] (e) Discuss qualitatively the connections between this analysis and the treatment in quantum mechanics of a particle trapped in an infinite potential well [U(x) = 0 for 0 x L, and U(x) = otherwise]. Specifically mention the time-independent Schrödinger equation; the importance of boundary conditions; and the significance of nodes and antinodes. [6] (f) By interpreting the λ n of (c) as de Broglie wavelengths, infer the allowed momenta p n of a particle trapped in an infinite potential well. From this, find the allowed kinetic energies E n. What is the ground-state energy? [4]

8 Page 8 C4 (a) Two sources, S 1 and S 2, emit harmonic waves with the same amplitude A, wavenumber k, and angular frequency ω. A point P in space is located a distance x 1 away from source S 1 and a distance x 2 from source S 2. If the sources are in phase, with φ 0 = 0, show that the total wave function at P is [ ] [ ] k(x1 x 2 ) k(x1 + x 2 ) y = 2A cos sin + ωt. [10] 2 2 (b) Hence, infer relations between the path difference (x 1 x 2 ) and wavelength λ, for which the interference at P is (i) totally destructive (y = 0 at any time); and (ii) fully constructive (y = maximum possible at any time). [5] (c) Show that in a two-slit (Young s) experiment, the conditions for constructive and destructive interference on a distant screen are d sin θ = m λ, m = 0, ±1 ± 2... and d sin θ = (m + 1/2) λ, m = 0, ±1 ± 2... Use the results from (b) in your derivation, and clearly define (with the aid of a diagram) the quantities d and θ. [10] (d) Light of wavelength 500 nm illuminates two narrow slits separated by 0.5 mm. The first-order minimum in the intensity pattern on a distant screen is found 5 mm away from the central intensity peak. What is the distance to the screen? [5]