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Question 1
An object displaced from equilibrium undergoes simple harmonic motion. The necessary relationship between its acceleration and displacement is
A.
is proportional to and directed away from equilibrium.
B.
is proportional to and directed towards equilibrium.
C.
is proportional to and directed towards equilibrium.
D.
is constant and directed towards equilibrium.
Question 2
A horizontal mass-spring oscillator is released from rest at maximum displacement. After one quarter of a period, the mass is at equilibrium. The energy of the oscillator at this instant is
A.
all kinetic and zero elastic potential energy.
B.
zero kinetic and zero elastic potential energy.
C.
half kinetic and half elastic potential energy.
D.
zero kinetic and maximum elastic potential energy.
Question 3
A mass on a spring has period . The mass is replaced by a mass twice as large, using the same spring. The new period is
A.
B.
C.
D.
Question 4
An oscillator satisfies . The graph of acceleration against displacement is
A.
B.
C.
D.
Question 5
A simple pendulum of length has period for small oscillations. The length is changed to at the same location. The new period is
A.
B.
C.
D.
Question 6
An oscillator has angular frequency . Its period is
A.
B.
C.
D.
Question 7
Two oscillators have the same frequency. Oscillator leads oscillator by a phase angle of . The phasor diagram representing this phase relationship is
A.
B.
C.
D.
Question 8
For a particle described by , the particle has maximum positive velocity when the phase angle is
A.
B.
C.
D.
Question 9
A cart of mass is attached to a horizontal spring of spring constant . Friction is negligible.
Determine the period of oscillation of the cart.
[2]
State the effect on the period if the amplitude is doubled while the spring remains within its elastic limit.
[1]
0
Question 10
A pendulum oscillates in air. The maximum displacement reached by the bob decreases after each cycle.
Define damping.
[1]
Explain why the motion is no longer ideal simple harmonic motion.
[2]
0
Question 11
The displacement of a particle in SHM is . At , the particle has displacement and positive velocity. The phase angle is
A.
B.
C.
D.
Question 12
A particle undergoing SHM has amplitude and angular frequency . At an instant when its displacement is , it is moving in the negative direction. Its velocity is
A.
B.
C.
D.
Question 13
An oscillator of mass has angular frequency and amplitude . The kinetic energy when the displacement is is
A.
B.
C.
D.
Question 14
In an ideal simple harmonic oscillator of amplitude , the kinetic energy and potential energy are equal when the magnitude of the displacement is
A.
B.
C.
D.
Question 15
A motion sensor is used to record the acceleration and displacement of a trolley attached to a spring. The graph shows the variation of with .
Outline how the graph shows that the trolley undergoes simple harmonic motion.
[2]
Use the graph to determine the angular frequency of the oscillation.
[2]
0
Question 16
A simple pendulum has length and is released from a small angular displacement. Use .
Calculate the period of the pendulum.
[2]
Explain why the period of an ideal simple pendulum is independent of the mass of the bob.
[2]
0
Question 17
The graph shows the displacement of a small object undergoing simple harmonic motion.
Determine the amplitude and period of the oscillation from the graph.
[2]
Describe the energy changes as the object moves from maximum positive displacement to the equilibrium position.
[2]
0
Question 18
A point moves uniformly in a circle of radius with angular speed . The horizontal projection of the point is observed on a screen.
State the amplitude of the motion of the projection.
[1]
Determine the period of the motion of the projection.
[2]
Outline why the projection undergoes simple harmonic motion.
[1]
0
Question 19
Two oscillators A and B have the same period, . Oscillator B reaches each maximum displacement after oscillator A.
Determine the phase difference between the two oscillators, stating which oscillator lags.
[2]
State the phase difference one cycle later.
[1]
0
Question 20
A motion sensor records the acceleration and displacement of a trolley attached to a horizontal spring. The graph shows plotted against .
State two features of the graph that indicate the motion is simple harmonic.
[2]
Determine the angular frequency and the period of the trolley.
[2]
0
Question 21
The displacement of a particle undergoing simple harmonic motion is recorded as a function of time.
Determine the amplitude of the oscillation.
[1]
Determine the frequency of the oscillation.
[2]
Identify where in the motion the magnitude of the acceleration is greatest.
[1]
0
Question 22
A horizontal mass-spring oscillator moves between positions and . Position is the equilibrium position. The bar chart shows the kinetic energy, elastic potential energy and total mechanical energy at different positions in one oscillation.
Identify the position at which the speed of the mass is greatest.
[1]
Describe the energy changes as the mass moves from position to position .
[2]
The experiment is repeated with significant air resistance. Explain the expected change to the total mechanical energy of the oscillator.
[1]
0
Question 23
A particle undergoing simple harmonic motion has amplitude and angular frequency . Its displacement is described by . At , and the particle is moving in the positive direction.
Determine the phase angle at .
[2]
Determine the velocity of the particle at .
[2]
0
Question 24
An oscillator has amplitude and angular frequency . At one instant its displacement is and it is moving towards the equilibrium position.
Determine the velocity of the oscillator at this instant.
[3]
0
Question 25
A particle of mass undergoes simple harmonic motion with amplitude and angular frequency .
Determine the total mechanical energy of the particle.
[2]
Determine the kinetic energy when the displacement is from equilibrium.
[2]
0
Question 26
The displacement of a particle undergoing simple harmonic motion is given by
where is in metres and is in seconds.
Determine the maximum speed of the particle.
[1]
Determine the acceleration of the particle at .
[2]
State the phase difference between displacement and acceleration.
[1]
0
Question 27
A student investigates a vertical mass-spring oscillator. The period is measured for different attached masses . The graph shows plotted against .
Explain why plotting against should produce a straight line.
[1]
Use the graph to determine the spring constant.
[2]
Suggest why changing the amplitude would not change the measured period in the ideal model.
[1]
0
Question 28
A simple pendulum is tested using small angular displacements. The graph shows plotted against the length of the pendulum. Measurements are repeated with two bobs of different mass.
Use the graph to determine the gravitational field strength .
[2]
State the conclusion about the effect of bob mass on the period.
[1]
Explain why the same pendulum would not be expected to follow the graph accurately at large angular amplitudes.
[1]
0
Question 29
Two oscillators, and , undergo simple harmonic motion with the same amplitude and frequency. Their displacement-time graphs are shown.
Determine the period of the oscillations.
[1]
Determine the phase difference between the oscillations and state which oscillator leads.
[2]
State the phase difference between the velocities of the two oscillators.
[1]
0
Question 30
A particle undergoing simple harmonic motion has amplitude . At one displacement, its kinetic energy is equal to its potential energy.
Determine the magnitude of the displacement at which the two energies are equal.
[2]
Explain why this displacement is not half the amplitude.
[1]
0
Question 31
A simplified model treats a bond in a greenhouse gas molecule as an oscillator. The graph shows the absorption of infrared radiation by the molecule as a function of radiation frequency.
Determine the resonant frequency of the molecular oscillator.
[1]
Calculate the period and angular frequency corresponding to this resonance.
[2]
Explain how this oscillator model is related to the enhanced greenhouse effect.
[2]
0
Question 32
The displacement of an oscillator is described by . The graph shows its displacement near .
The amplitude is and . At , and the oscillator is moving in the negative direction. Determine the phase angle .
[2]
Calculate the velocity at .
[1]
Calculate the acceleration at .
[1]
0
Question 33
For an oscillator undergoing simple harmonic motion, the square of its speed is plotted against the square of its displacement .
Use the graph to determine the angular frequency of the oscillator.
[2]
Determine the maximum speed of the oscillator.
[1]
Determine the amplitude of the oscillator.
[1]
Explain the significance of the two possible signs in .
[1]
0
Question 34
A phase-space graph for a particle undergoing simple harmonic motion is shown. The graph plots velocity against displacement . The mass of the particle is .
Determine the amplitude and maximum speed of the particle.
[2]
Determine the angular frequency and period of the motion.
[2]
Calculate the total energy of the oscillator.
[1]
0
Question 35
A trolley is attached to a horizontal spring and moves on a low-friction track. A motion sensor records the acceleration and displacement of the trolley from its equilibrium position. The graph shows the variation of with .
Use the graph to evaluate whether the motion is simple harmonic.
I.
State the graphical feature that shows the acceleration is proportional to displacement.
[1]
II.
Explain the significance of the sign of the gradient.
[2]
III.
Conclude whether the trolley executes simple harmonic motion.
[1]
The magnitude of the gradient of the graph is . Calculate the period of the oscillation.
[2]
Suggest why the points at the largest displacements deviate slightly from the straight line.
[1]
0
Question 36
A simple pendulum of length is displaced through a small angle and released from rest. Air resistance is negligible.
Explain why the pendulum can be modelled as a simple harmonic oscillator for small angular displacements.
I.
Identify the force component that provides the restoring effect.
[1]
II.
Use the small-angle approximation to show that the acceleration has the form required for simple harmonic motion.
[3]
Calculate the time taken for the bob to move from one extreme position to the other extreme position.
[2]
Discuss the effect on the period if the bob is replaced by another bob of twice the mass but the same size.
[1]
0
Question 37
A horizontal spring-block oscillator is pulled to one side and released from rest. The surface is first very smooth and then the experiment is repeated on a rougher surface.
Discuss the energy changes during one complete oscillation on the very smooth surface.
I.
State where the kinetic energy is maximum and where it is zero.
[2]
II.
Describe the variation of elastic potential energy and total mechanical energy.
[2]
On the rougher surface the amplitude decreases after each oscillation. Explain why the motion is no longer ideal simple harmonic motion.
[2]
Sketch the variation of kinetic energy with time for one complete oscillation on the very smooth surface, starting from the instant of release at an extreme position.
[2]
0
Question 38
The shadow of a point moving with uniform circular motion is projected onto a straight horizontal screen. The projected shadow oscillates between two end points.
Compare the circular motion of the point with the simple harmonic motion of its shadow.
I.
State the relationship between one revolution of the point and one oscillation of the shadow.
[1]
II.
Explain how the radius and angular speed of the circular motion relate to the amplitude and angular frequency of the shadow.
[2]
The point completes revolutions in . Calculate the angular frequency of the shadow.
[2]
Explain why the shadow has zero acceleration at the centre of its motion.
[1]
0
Question 39
A mass of undergoes simple harmonic motion. The graph shows the kinetic energy and potential energy as functions of displacement .
Determine the total energy of the oscillator.
[1]
Use the potential-energy graph to determine the angular frequency of the oscillator.
[2]
Calculate the speed of the mass when .
[2]
0
Question 40
A computer model represents an oscillator by the equation , where is the phase angle. At one instant the model displays the state of the oscillator.
| Quantity / unit | Value |
|---|---|
| Amplitude, x₀ / m | 0.090 |
| Angular frequency, ω / s⁻¹ | 12 |
| Displacement, x / m | 0.054 |
| Velocity direction | positive |
The model gives , and , with the velocity positive. Determine the phase angle at this instant.
[2]
Calculate the velocity at this instant.
[2]
Determine the fraction of the total energy that is potential energy at this instant.
[1]
0
Question 41
A student investigates vertical oscillations of different masses attached to the same spring. The student measures the time for ten oscillations for each mass and plots against mass .
Use the mass-spring model to explain the expected relationship between and .
I.
Derive an expression for in terms of and the spring constant .
[2]
II.
State what the gradient of the graph represents.
[1]
The gradient of the best-fit line is . Calculate the spring constant.
[2]
Evaluate one reason why the graph may have a small positive vertical intercept.
[2]
0
Question 42
The bonds in a carbon dioxide molecule can vibrate about stable separations. A simplified model treats one vibrational mode as a small oscillator that can absorb infrared radiation.
Discuss why a vibrating molecular bond can be approximated as a simple harmonic oscillator for small displacements.
I.
State the restoring condition that must be satisfied.
[2]
II.
Explain why the approximation is expected to fail for large molecular displacements.
[1]
Explain how absorption and re-radiation by many such molecular oscillators can contribute to the enhanced greenhouse effect.
[2]
Suggest one limitation of modelling the molecule as an ideal undamped oscillator.
[1]
0
Question 43
An oscillator has displacement described by . A data logger records that at the oscillator is at and moving in the negative direction. The amplitude is and the period is .
Evaluate the phase information for this oscillator.
I.
Calculate the angular frequency.
[1]
II.
Determine the possible values of the phase angle at from the displacement information.
[2]
III.
Use the direction of motion to choose the correct value of .
[2]
Calculate the velocity of the oscillator at .
[2]
Explain why radians rather than degrees must be used in the argument of the sine function for the standard equations of simple harmonic motion.
[1]
0
Question 44
A particle of mass undergoes simple harmonic motion with amplitude and angular frequency .
Use the kinematic equations to discuss the speed of the particle at different displacements.
I.
Calculate the maximum speed.
[1]
II.
Calculate the speed when the displacement is .
[2]
III.
Explain why two velocities are possible at this displacement.
[1]
Calculate the total mechanical energy of the oscillator.
[2]
Determine the kinetic energy of the particle when and comment on its relation to the speed found in (a)(ii).
[2]
0
Question 45
A horizontal oscillator is modelled by . The total energy is and the amplitude is .
Evaluate how the energy is shared between kinetic and potential forms.
I.
Determine the displacement at which the kinetic energy equals the potential energy.
[3]
II.
Calculate the potential energy at .
[1]
Sketch, on the same axes, the variation of kinetic energy and potential energy with displacement from to .
[2]
Explain why the points where the two curves cross are not at half the amplitude.
[1]
0
Question 46
A particle undergoes simple harmonic motion according to , where is in metres and is in seconds.
Use the equation to explain the motion of the particle.
I.
State the amplitude, angular frequency and phase angle at .
[3]
II.
Determine the acceleration of the particle when .
[1]
Calculate the first time after at which the particle passes through equilibrium moving in the negative direction.
[2]
Explain why the velocity is a quarter cycle out of phase with the displacement.
[1]
0
Question 47
Two identical oscillators, P and Q, have the same amplitude and frequency. Their displacement-time graphs are sinusoidal. Oscillator Q reaches each maximum displacement after oscillator P. The period of both oscillators is .
Compare the phase of the two oscillators.
I.
Calculate the phase difference between P and Q in radians.
[2]
II.
State which oscillator leads and explain your answer.
[2]
At one instant P is at positive maximum displacement. Determine the displacement and direction of motion of Q at that instant, as fractions of the amplitude.
[2]
Explain why a constant phase difference requires the two oscillators to have the same frequency.
[1]
0
Question 48
A small glider of mass on an air track is attached to a spring. The glider is released from rest at . The spring constant is .
Evaluate the motion using the SHM model.
I.
Calculate the angular frequency of the glider.
[2]
II.
Write a displacement equation of the form for the motion.
[2]
Calculate the speed of the glider when it first reaches .
[2]
Compare the kinetic energy and elastic potential energy at .
[2]
0
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