NJC 2023 Oscillations Problem Set

NATIONAL JUNIOR COLLEGE 1 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. Oscillations Problem Set Exercises Free Oscillations E1. The displacement x (in metres) of a body in oscillatory motion is given by the equation: x = 3.5 sin 4.0 t (a) Determine the amplitude of the motion. (b) Determine the angular frequency. (c) Determine the displacement 0.20 s after oscillation is begun. [ 2.5 m ] (d) Find the expression of velocity in terms of time and, [v = 14 cos 4.0t ] (e) Hence determine the velocity at time t = 0.50 s. [ -5.8 m s-1 ] E2. [N06/1/13] A simple harmonic oscillator has a time period of 10 seconds. Which equation relates its acceleration a and displacement x? A a = -(20/2π)2x B a = -(20π)2x C a = -(20/2π)x D a = -(2π/10)2x Ans: [D] E3. [N08/1/15] The def ining equation for a particle moving in simple harmonic motion 𝑎 = −𝜔2𝑥 where 𝑎 is the acceleration of the particle, 𝑥 is the displacement and is the angular frequency. The graph on the right shows how 𝑎 varies with 𝑥 for a particle moving in simple harmonic motion. Which is the amplitude and period of the motion? Amplitude/cm Period/s A 5.0 0.44 B 5.0 14 C 10 0.44 D 10 14 Ans: [A]

NATIONAL JUNIOR COLLEGE 2 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. E4. A body performs simple harmonic motion with period of 0.063 s. The maximum speed is 3.0 ms-1. What are the values of the amplitude x0 and the angular frequency ? x0/m /rads-1 A 0.030 100 B 0.19 16 C 5.3 16 D 33 100 Ans: [A] E5. An oscillating system performs s.h.m. with an amplitude of 2.0 cm. If the period of the motion is 5.0 s, calculate (a) the magnitude of the velocity of the particle at 0.50 cm from equilibrium position. [2.43 cm s-1] (b) the maximum acceleration of the particle. [3.16 cm s-2] E6. An object moving with simple harmonic motion has an amplitude of 0.020 m and a frequency of 20 Hz. Calculate (a) the period of oscillation. [0.050 s] (b) the velocity at the equilibrium point and at the position of maximum negative displacement. [2.51 m s-1, 0 m s-1] E7. The motion of a piston in a certain car engine is approximately simple harmonic with amplitude 40 mm. The frequency of oscillation is 120 Hz. Determine (a) the maximum acceleration, [2.27 x 104 ms-2] (b) the maximum speed, of the piston. [30.1 ms-1] E8. When the mass M on the spring is 0.040 kg, the vertical displacement y of the mass varies with time t according to the relation y = a cos t where a = 0.010 m and  = 20 rad s-1. Determine (a) the amplitude of the vibration, (b) the period T of the vibration. [0.314 s]

NATIONAL JUNIOR COLLEGE 3 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. Graphs E9. [N11/1/17] The graph below shows the variation of acceleration a with displacement x for an object undergoing simple harmonic motion. The simple harmonic motion system is altered so that it has a period of oscillation twice that of before. Which line could be produced? Ans: [D] Energy of Free Oscillations E10. [N10/1/17] A mass of 8.0 g oscillate in simple harmonic motion with an amplitude of 5.0 mm at a frequency of 40 Hz. What is the total energy of this simple harmonic oscillator? A 0.16 mJ B 6.3 mJ C 13 mJ D 640 mJ Ans: [B]

NATIONAL JUNIOR COLLEGE 4 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. E11. [N16/1/16] For an object undergoing simple harmonic motion, there is a continuous interchange of potential energy (p.e.) and kinetic energy (k.e.). Which energy-displacement graph shows this interchange of energy correctly? Ans: [A] Damping and Resonance E12. Given the same initial displacement and then released, the variation with time t of the displacements x of two objects P and Q are as shown in the given graphs. The two objects are then driven by sinusoidal driving forces of the same constant amplitude and of variable frequency f. Which of the following graphs correctly shows how the amplitudes A of P and Q varies with f?

NATIONAL JUNIOR COLLEGE 5 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. E13. The diagram shows four displacement -time curves for oscillations with various degrees of damping. Which curve would ideally characterise a good car suspension system?

NATIONAL JUNIOR COLLEGE 6 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. Problems (Various concepts combined) P1. Determine whether or not the following quantities can be in the same direction for a simple harmonic oscillator: (a) displacement and velocity, (b) velocity and acceleration, (c) displacement and acceleration. P2. 2018 P1 Q15

NATIONAL JUNIOR COLLEGE 7 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. P3. 2020 P1 Q16 P4. 2014 P3 Q2

NATIONAL JUNIOR COLLEGE 8 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner.

NATIONAL JUNIOR COLLEGE 9 2022 | Science | Physics © 2022 National Junior College All Rights Reserved. No part of this publication may be produced or transmitted in any form or by any means, electronic or me chanical, including photocopy,recording or any other information storage and retrieval system, without prior permission in writing from the copyright owner. P5. At Liverpool the tide is nearly sinusoidal and has a period