JPJC 2026 Oscillations Tutorial

1 JURONG PIONEER JUNIOR COLLEGE 9478 H2 PHYSICS OSCILLATIONS TUTORIAL Self-Check Questions S1 What is meant by displacement, amplitude, period, frequency and angular frequency of a simple harmonic motion. S2 Express the period T in terms of frequency f and angular frequency  ? S3 Define simple harmonic motion. State the defining equation of simple harmonic motion. S4 Write down a solution for a simple harmonic oscillator which starts its motion from the equilibrium position. Express its velocity and acceleration in terms of time t. S5 Sketch graphs to show the variation with time t of the displacement, velocity and acceleration of the oscillator in S4. S6 Sketch graphs to show how the velocity and acceleration of the oscillator in S4 vary with displacement? S7 Describe the variation in between kinetic and potential energy with time during simple harmonic motion of a mass attached to a horizontal spring ? The mass is moving on a frictionless surface. Describe how the frequency of the energy varies in comparison to that of the oscillation itself. S8 Explain what is meant by damping? Give one practical example of a lightly damped oscillation. Why is critical damping in a car suspension system important? S9 Explain what is meant by forced oscillations and resonance? S10 Sketch a set of graphs, using the same axes, to show how the amplitude of forced oscillation varies with driving frequency for very light, moderate and heavy damping. Explain the features of your graphs. Self-Practice Questions 1 In which list are all three quantities constant when a particle moves in undamped simple harmonic motion? A acceleration angular frequency total energy B amplitude force total energy C acceleration angular frequency force D total energy amplitude angular frequency

2 2 In a fair ground shooting game, a gun fires at a moving target. The gun fires itself at random times. The player has to point the gun in a fixed direction, and the target moves from side to side with simple harmonic motion. At which region should the player take a fixed aim in order to score the greatest number of hits? A 3 B either 1 or 5 C either 2 or 4 D any of 1, 2, 3, 4 and 5 3 The graphs in Fig. 3.1 show how the displacement x, velocity v and the acceleration a of a body vary with time t when it is oscillating with simple harmonic motion. Determine the period T. Fig. 3.1 4 The diagram shows the graph of displacement against time for a body performing simple harmonic motion. At which point are the velocity and acceleration in opposite directions? displacement time A 0 B C D

3 5 An object undergoing a forced oscillation has displacement y, as shown in Fig. 5.1. Fig. 5.1 (a) Use Fig. 5.1 to determine, for this oscillation, (i) its amplitude, (ii) its period, (iii) its frequency, (iv) its angular frequency. (b) Use Fig. 5.1 to state, for each of the following, a time at which the oscillating object has (i) maximum positive velocity, (ii) maximum positive acceleration, (iii) maximum negative acceleration, (iv) maximum kinetic energy, (v) maximum potential energy. 6 A trolley of mass 2.0 kg with free -running wheels is attached to two fixed points P and Q by two identical springs as shown. The trolley is displaced a small distance of 0.050 m towards Q by a force of 10 N and is released. The trolley then oscillates horizontally with simple harmonic motion. What is the angular frequency? A 0.50 rad s−1 B 1.0 rad s−1 C 2.0 rad s−1 D 10 rad s−1

4 7 A mass hanging from a spring suspended from a ceiling is pulled down and released. The mass then oscillates vertically with simple harmonic motion of period T. The graph shows how its distance from the ceiling varies with time. What can be deduced from this graph? A The speed is a maximum at 4 Tt = . B The amplitude of the oscillation is 70 cm. C The kinetic energy is a maximum at 2 Tt = . D The restoring force on the mass increases between 0t = and 4 Tt = . 8 The displacement of a particle P which moves with simple harmonic motion can be described by the expression ( )0.050sin 8xt = where x is in metres and t in seconds. Determine (a) the amplitude of the motion, (b) the frequency of the motion, (c) the length of time it takes for P to complete one oscillation, (d) the velocity of P as it passes through its equilibrium position, (e) the maximum acceleration of P during its motion. Answer Key to Self-Practice Questions 1 E 2 B 3 2.1 s 4 C 5(a) (i) 0.11 m, (ii) 1.7 s, (iii) 0.59 s, (iv) 3.7 rad s−1 5(b) (i) 0 s, 1.7 s, 3.4 s, 5.1 s, (ii) 1.3 s, 3.0 s, 4.7 s, (iii) 0.4 s, 2.1 s, 3.8 s, 5.5 s, (iv) 0 s, 0,85 s, 1.70 s, 2.55 s, 3.40 s, 4.25 s, 5.10 s, 5.95 s, (v) 0.40 s, 1.30 s, 2.10 s, 3.00 s, 3.80 s, 4.70 s, 5.50 s 6 D 7 A 8 (a) 0.050 m, (b) 4.0 Hz, (c) 0.25 s, (d) 1.26 m s−1, (e) 31.6 m s−2

5 Discussion Questions 1 Describe an example of a free oscillation. Explain why in practice a free oscillation cannot have a constant amplitude. 2 A mass oscillates with simple harmonic motion and the variation of velocity with displacement is as shown in Fig. 2.1. Fig. 2.1 (a) Explain why there are two values of velocity for zero displacement. (b) Explain why there are two values of displacement for zero velocity. 3 In order to check the speed of a camera shutter, the camera was used to photograph the bob of a simple pendulum moving in front of a horizontal scale. The extreme positions of the bob were at the 600 mm and 700 mm marks. The photograph showed that while t he shutter was open the bob moved from the 650 mm mark to the 675 mm mark. If the period of the pendulum was 2.0 s, calculate the time the shutter remained open. 4 (a) In some harbours, the rise and fall of the water level is simple harmonic. Explain what simple harmonic means. (b) In one harbour, the equation for the depth h of water is 25.0 3.0sin 45600th =+  where h is given in metres and t is the time in seconds. For this harbour, calculate (i) the maximum depth of water, (ii) the minimum depth of water, (iii) the time interval between high- and low-water, (iv) the two values of t at which the water is 5.0 m deep, (v) the length of time for each tide during which the depth of water is more than 7.0 m. v/ms−1 x/m x0 −x0 ωx0 − ωx0

6 5 Two helical springs are attached to a trolley of mass 590 g, as shown in Fig. 5.1. Fig. 5.1 The trolley is displaced along the axis of the springs and then released. The trolley undergoes simple harmonic motion. The variation of the kinetic energy Ek of the trolley with displacement x from its equilibrium position is shown in Fig. 5.2 Fig. 5.2 (a) Use data from Fig. 5.2 to calculate the frequency of oscillation of the trolley. (b) The trolley loses energy so that its maximum kinetic energy is reduced by 40 mJ. Use Fig. 5.2, without any further calculation, to determine the amplitude of the oscillations. Show your construction on Fig. 5.2.

7 6 A block of wood of mass m floats in still water, as shown in Fig. 6.1. Fig. 6.1 When the block is pushed vertically downwards into the water, without totally submerging it, and is then released, it bobs up and down in the water with a frequency f given by the expression mf 2821= where f is measured in Hz and m in kg. Surface water waves of speed 0.90 m s−1 and wavelength 0.30 m are then incident on the block. These cause resonance in the up-and-down motion of the block. (a) Explain what is meant by the term resonance. (Hint: consider the context of the question) (b) Calculate (i) the frequency o