Amplitude of Motion - Oscillations and Waves - Past Paper, Exams of Physics

Download Amplitude of Motion - Oscillations and Waves - Past Paper and more Exams Physics in PDF only on Docsity! EXAMINATION PAPER CONTAINS STUDENT’S ANSWERS Please write your 8-digit student number here: The Handbook of Mathematics, Physics and Astronomy Data is provided KEELE UNIVERSITY EXAMINATIONS, 2011/12 Level I Friday 25th May 2012, 09.30-11.30 PHYSICS/ASTROPHYSICS PHY-10020 OSCILLATIONS AND WAVES Candidates should attempt ALL of PARTS A and B, and ONE question from each of PARTS C and D. PARTS A and B should be answered on the exam paper; PARTS C and D 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 30%, PART D yields 30%. Please do not write in the box below A C1 Total B C2 D1 D2 NOT TO BE REMOVED FROM THE EXAMINATION HALL PHY-10020 Page 1 of 12 PART A Tick one box by the answer you judge to be correct (marks are not deducted for incorrect answers) A1 Which one of the following functions is not a general solution to the equation ẍ = −ω2 x? (ω, A, B, and φ0 are all constants.) x(t) = A cos(ωt + φ0) x(t) = A sin(ωt) + B cos(ωt) x(t) = A sin(ωt) + A sin φ0 x(t) = A sin(ωt − φ0) [1] A2 A block of mass m attached to the end of a spring undergoes simple harmonic motion with an angular frequency ω = 12 s−1. A block of mass 4m, attached to the same spring, would have an angular fre- quency of 3 s−1 6 s−1 12 s−1 24 s−1 [1] A3 A string of length L and a bob of mass m form a simple pendulum with period T = 2 s. The period of a pendulum with the same length but a bob of mass 4m is 0.5 s 1 s 2 s 4 s [1] A4 The total mechanical energy of a particle in simple harmonic motion depends on the amplitude of the motion as Etot ∝ A 4 Etot ∝ A 2 Etot ∝ A Etot ∝ A 1/2 [1] A5 The motion of an object in the field of a conservative force is approx- imately simple harmonic near any position where the total energy is a local maximum. total energy is a local minimum. potential energy is a local maximum. potential energy is a local minimum. [1] /Cont’d PHY-10020 Page 2 of 12 PART B Answer all EIGHT questions B1 A particle is in simple harmonic motion with an amplitude of 5 cm. At time t = 0 it passes through its equilibrium position with a velocity of −12 cm s−1. Calculate the period of the oscillation. [3] B2 An object of mass m = 0.080 kg is in simple harmonic motion about x = 0 with angular frequency ω = 3.0 s−1 and a total mechanical energy Etot = 0.0081 J. Find the positions x at which the object has speed |ẋ| = 0.36 m s−1. [3] /Cont’d PHY-10020 Page 5 of 12 B3 A particular oscillator of mass m [kg] has the equation of motion ẍ + 0.40 m ẋ + 0.64 x = 0 , where x is measured in metres and time in seconds. For what values of m is the oscillator overdamped? [3] B4 A damped harmonic oscillator is driven by an external force, F (t) = F0 cos(ωet). Briefly explain what is meant by the transient in the motion of such an oscillator, and write down the general form for the steady-state displacement as a function of time. [3] /Cont’d PHY-10020 Page 6 of 12 B5 Illustrated are the two normal modes of oscillation for a pair of identical blocks on identical springs, coupled by a third spring: Write down a formula for the angular frequency of the symmetric mode. Is this frequency greater than or less than the frequency of the anti-symmetric mode? Justify your answers. [3] B6 Ultrasound travels through human tissue as a harmonic wave with speed 1600 m s−1, wavelength 4.5× 10−4 m, and amplitude 1.8× 10−9 m. Write the displacement s(x, t) of molecules at depth x in the tissue at time t, assuming that s = 0 at x = t = 0. [3] /Cont’d PHY-10020 Page 7 of 12 C2 The displacement x of a harmonic oscillator of mass m, which has a natural angular frequency ω0 and is subjected to a damping force characterized by a constant b, may be given by one of the following three functions of time: (I) x(t) = A0 e −bt/2m sin (ωt + φ0) with ω ≡ √ ω20 − b 2/4m2 (II) x(t) = e−bt/2m (C1 t + C2) (III) x(t) = e−bt/2m ( B1 e qt + B2 e −qt ) with q ≡ √ b2/4m2 − ω20 (a) Write down the equation of motion that is solved by any of the functions (I)–(III). State the physical meaning of each term in the equation of motion. [6] (b) A block of mass m = 0.250 kg is attached to a spring with force constant k = 1.44 N m−1 and damping constant b = 1.50 kg s−1. The block is in equilibrium at time t = 0, when it receives an impulse that gives it an initial velocity of −1.80 m s−1. i. Verify that this system is overdamped, and therefore state which one of equations (I), (II), or (III) above describes the motion of the block at t > 0. [4] ii. Use the initial conditions given to determine both the posi- tion and the velocity of the block at any t ≥ 0. [12] iii. Sketch x(t) for this system. Show on the same sketch and explain (without calculation) how x(t) would change if the value of b were doubled while keeping all other parameters and initial conditions the same. [8] /Cont’d PHY-10020 Page 10 of 12 PART D Answer ONE out of TWO questions D1 (a) Explain why a wave travelling with speed v in one dimension must depend on position x and time t either in the combination (x − vt) or in the combination (x + vt) . [8] (b) A string with uniform linear mass density µ is stretched along the x-axis and kept under a constant tension F . A transverse harmonic wave travels along the string, causing a displacement y(x, t) at position x at time t. The kinetic energy per unit length in the string is given by dK dx = 1 2 µ ( ∂y ∂t )2 and the potential energy per unit length is given by dU dx = 1 2 F ( ∂y ∂x )2 . i. Write a general form for the wave function y(x, t) in terms of the wavenumber and angular frequency. State what each of the partial derivatives ∂y/∂t, ∂2y/∂t2, ∂y/∂x, and ∂2y/∂x2 represents physically. [6] ii. Show that any particle on the string undergoes simple har- monic motion. [4] iii. Use the one-dimensional wave equation to derive the speed of the wave in terms of its angular frequency and wavenumber. [4] iv. Given that dK/dx = dU/dx, infer a formula for the wave speed in terms of F and µ. [8] /Cont’d PHY-10020 Page 11 of 12 D2 (a) Two particular harmonic waves travelling along the x-axis com- bine to produce the standing wave y(x, t) = 2A sin(kx) cos(ωt) . i. Write down the two travelling wave functions involved and verify explicitly that their superposition yields y(x, t) as given. [6] ii. Derive the allowed wavelengths λn if this is a resonant stand- ing wave confined to 0 ≤ x ≤ L, with x = 0 and x = L both nodes. Find the positions of all nodes for the 4th harmonic, and sketch this wave function at t = 0. [10] (b) Two coherent sources, S1 and S2, emit harmonic waves with the same amplitude A, angular frequency ω, wavenumber k, and phase constant φ0. These waves interfere at a point P , which is a distance x1 from source S1 and a distance x2 from source S2. i. Show that the amplitude of the total wave at P is Atot(P ) = 2A cos [k(x1 − x2)/2] . [8] ii. Hence, derive a general relation between the wavelength λ and the path difference (x1 − x2) at points P where there is total destructive interference. [6] PHY-10020 Page 12 of 12