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EXAMINATION PAPER CONTAINS STUDENT’S ANSWERS
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The Handbook of Mathematics, Physics and
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KEELE UNIVERSITY
EXAMINATIONS, 2010/
Level I
Tuesday 24th^ May 2011, 09.30-11.
PHYSICS/ASTROPHYSICS
PHY-
OSCILLATIONS AND WAVES
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 exami- nation 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%.
Please do not write in the box below A C1 Total B C C C
NOT TO BE REMOVED FROM THE EXAMINATION HALL
PART A Tick one box by the answer you judge to be correct (marks are not deducted for incorrect answers)
A1 A block of mass m, hanging from a spring with force constant k and natural length L, oscillates purely in the vertical direction. The an- gular frequency of the oscillation is ω =
√ k/m ω =
√ g/L ω =
√ mg/k ω =
√ k/mg [1] where g is the acceleration due to gravity.
A2 A simple pendulum on the surface of the Earth has a period of 1 second. On the moon, where the acceleration due to gravity is 6 times lower, the period of the same pendulum would be (^6) √ seconds 16 seconds 6 seconds √^16 seconds [1]
A3 An object is in simple harmonic motion about an equilibrium position, with an angular frequency of 3 s−^1 and an amplitude of 0.4 m. The speed of the object at the equilibrium position is 3 .6 m s−^1 2 .4 m s−^1 1 .2 m s−^1 0 [1]
A4 A particle is in simple harmonic motion. At how many times dur- ing one oscillation cycle are the kinetic and potential energies of the particle equal to each other? eight times four times two times one time [1]
A5 An oscillator with mass 300 g and natural angular frequency 6.00 s−^1 is damped by a force Fdamp = −γ x˙. The critical damping constant is γ = 1.80 kg s−^1 γ = 2.55 kg s−^1 γ = 3.60 kg s−^1 γ = 10.8 kg s−^1 [1]
/Cont’d
A11 The wavenumber k of a harmonic wave is, in standard notation,
k = mω^2 k = ωv k = 2π/λ k = 2πλ [1]
A12 In the wave y(x, t) = 0.03 sin(4x + 8t) (where x and y are in metres and t is in seconds), the particle velocity at x = 0 at t = 0 is 2 m s−^1 0 .24 m s−^1 0 .12 m s−^1 0 [1]
A13 Two travelling harmonic waves combine to produce the standing wave y(x, t) = 0.01 sin(40x) cos(60t) (for x and y in metres, and t in sec- onds). The amplitude of each of the travelling waves is A = 0.005 m A = 0.01 m A = 0.02 m A = 0.1 m [1]
A14 A string of length L with both ends fixed vibrates in its nth^ harmonic. The distance between adjacent nodes on the string is 2 L/n L/n nL 2 nL [1]
A15 Two waves with the same intensity I 0 interfere at a point P in space. The maximum possible intensity of the total wave at P is I 0 / 2 I 0 2 I 0 4 I 0 [1]
A16 Interference patterns of the type seen in Young’s double-slit experi- ment arise when waves emitted in phase by two sources arrive at a point in space from opposite directions having travelled different distances at different times with slightly different frequencies [1]
/Cont’d
PART B Answer all EIGHT questions
B1 A block of mass m = 100 g attached to a horizontal spring with k = 40 N m−^1 has a displacement given by x(t) = 0.05 sin(ωt) m. Calculate the velocity at time t = T /2, where T is the period of oscillation. [3]
B2 An object of mass m = 0.4 kg is in simple harmonic motion about x = 0 with angular frequency ω = 3 s−^1. Its total mechanical energy is Etot = 4. 5 × 10 −^3 J. Find the speed of the object when its displacement is x = 0.03 m. [3]
/Cont’d
B5 Give a sketch illustrating the two normal modes of oscillation for a coupled pair of identical blocks on identical springs. [3]
B6 A long string carries a transverse harmonic wave travelling in the negative-x direction with amplitude 2 cm, wavelength 60 cm, and frequency 440 Hz. The displacement of the string at x = 0 at t = 0 is y = 0. Write the wave function, y(x, t). [3]
/Cont’d
B7 A travelling wave has the function y(x, t) = 8 x^2 + 6x + 8xt + 3t + 2t^2 for x in centimetres and t in seconds. Use the one-dimensional wave equation to find the phase speed of the wave. [3]
B8 A transverse wave travels at speed 330 m s−^1 on a piano wire that has a total mass of 10 grams and a length of 64 cm. What is the tension in the wire? [3]
/Cont’d
C2 (a) The displacement of an undriven, underdamped harmonic oscil- lator is given by x(t) = A 0 e−γ t/(2m)^ sin (ωt + φ 0 ) , in which ω ≡
√ ω^20 − γ^2 /(4m^2 ). i. Sketch a representative x(t) curve, indicating clearly all main physical features of the motion. [6] ii. A block with m = 0.4 kg is attached to a damped spring having k = 2.5 N m−^1 and γ = 0.56 kg s−^1. The block is in equilibrium at t = 0, when it receives an impulse giving it an initial velocity of +0.6 m s−^1. A. Verify that this system is underdamped. [4] B. Determine the displacement and the velocity of the block as functions of time for t > 0. [14] (b) Explain what is meant by the transient and the steady state for the motion of an underdamped oscillator that is driven by an external force of the form F (t) = F 0 cos(ωet). Write down the general form of the displacement x(t) in the steady state. [6]
/Cont’d
C3 (a) Give an argument as to why a wave travelling with speed v in one dimension must depend on position x and time t only in one of the combinations (x − vt) or (x + vt). [6] (b) The function y(x, t) = A sin [k(x − vt) + φ 0 ] describes a travelling harmonic wave. For such a wave: i. The wavelength λ is defined as the smallest length such that y(x + λ, t 0 ) = y(x, t 0 ) for any x at a fixed t 0. Use this to derive the standard relation between k and λ. [6] ii. Show that y undergoes simple harmonic oscillation at any fixed position x in the wave. Thus, express the angular frequency ω of the wave in terms of k and v. [6] (c) Consider the function y(x, t) = 4 ex−^2 t^ − e^3 x−^6 t^. i. Verify that this function is a solution to the one-dimensional wave equation. [6] ii. Show that ∂^2 y/∂t^2 + 8 ∂y/∂t + 12 y = 0. Thus, what kind of oscillation drives this wave? [6]
/Cont’d
