Practise › Questions › Simple harmonic motion
Simple harmonic motion questions
One condition defines the most important oscillation in physics. Acceleration proportional to displacement, aimed back at the middle. Everything else, the cosine, the phase relationships, the two maxima, unpacks from that single line.
19 original questions · 55 marks · the simple harmonic motion notes · Periodic motion
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening a mark scheme: the schemes award marks point by point, and the marks are easier to see when you have something of your own to compare against.
State the defining condition for simple harmonic motion.
Mark scheme
The acceleration is directly proportional to the displacement from the equilibrium position (1) and is always directed towards it: a = −ω2x (a ∝ −x) (1).A body oscillates with a frequency of 2.0 Hz. Calculate its angular frequency.
Mark scheme
ω = 2πf = 2π × 2.0 (1)
ω = 12.57 rad s−1 (1)An oscillator has an angular frequency of 10 rad s−1 and an amplitude of 0.050 m. Calculate its maximum speed.
Mark scheme
vmax = ωA = 10 × 0.050 (1)
vmax = 0.50 m s−1 (1)An oscillator has an angular frequency of 7.0 rad s−1. Calculate the magnitude of its acceleration when its displacement is +0.030 m, and state the direction of this acceleration.
Mark scheme
a = ω2x = 7.02 × 0.030 = 1.5 m s−2 (1). It is directed towards the equilibrium position, opposite to the displacement (1).For a body in simple harmonic motion, state the displacement at which (a) the acceleration is greatest and (b) the speed is greatest.
Mark scheme
(a) At maximum displacement, x = ±A, the extremes of the motion (1). (b) At zero displacement, the equilibrium position (1).A vibrating machine panel completes 50 oscillations in 40 s. Calculate the frequency and the period of the oscillation.
Mark scheme
f = 50/40 = 1.25 Hz (1)
T = 1/f = 0.80 s (1)A body performs SHM at 3.0 Hz with an amplitude of 0.020 m. Calculate its maximum acceleration.
Mark scheme
ω = 2πf = 2π × 3.0 (1)
ω = 18.85 rad s−1 (1)
amax = ω2A = 7.11 m s−2 (1)A glider on an air track performs SHM with amplitude 0.10 m and angular frequency 8.0 rad s−1. Calculate its speed when its displacement is 0.060 m.
Mark scheme
v = ±ω√(A2 − x2) (1)
= 8.0 × √(0.102 − 0.0602) (1)
v = 0.64 m s−1 (1)An oscillator moves as x = A cos ωt with A = 0.15 m and ω = 4.0 rad s−1. Calculate its displacement at t = 0.20 s.
An oscillator has an angular frequency of 6.28 rad s−1. Calculate its frequency and its period.
A buoy bobbing on water performs SHM with amplitude 0.12 m and frequency 0.50 Hz. Timing starts as the buoy passes upwards through its equilibrium position, so its displacement is x = A sin ωt. Calculate its displacement at t = 0.40 s.
A loudspeaker cone reproducing a test tone vibrates in SHM at 250 Hz with amplitude 0.35 mm. The manufacturer claims the cone's maximum acceleration exceeds 50g (g = 9.81 m s−2). Deduce whether the claim is correct.
An oscillator is released from rest at its amplitude of 0.080 m and has a period of 1.2 s, so x = A cos ωt. Calculate the time at which its displacement first falls to 0.020 m.
A body performs SHM with amplitude 0.080 m and period 0.50 s. Calculate (a) its angular frequency, (b) its maximum speed and (c) its maximum acceleration.
An object performs SHM of amplitude 0.10 m. Calculate the displacement at which its speed is half of its maximum speed.
Describe the phase relationship between the displacement, velocity and acceleration in SHM.
A student measures the acceleration of an oscillating trolley at three displacements. At x = 0.020 m, a = −0.50 m s−2; at x = 0.040 m, a = −1.00 m s−2; at x = 0.060 m, a = −1.50 m s−2. Deduce whether the motion is simple harmonic and, if it is, determine the angular frequency.
An oscillator of period T passes through its equilibrium position at t = 0, so x = A sin ωt. Show that the time it takes to travel from the equilibrium position to a displacement of A/2 is T/12.
A component in a machine oscillates with SHM. Sensors record a maximum speed of 0.90 m s−1 and a maximum acceleration of 14 m s−2. Determine the angular frequency, the amplitude and the frequency of the oscillation.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise simple harmonic motion one question at a time
The player marks nothing for you. It shows one question, waits, then shows the scheme so you can mark yourself, and brings a question back sooner when it went badly.