C.1 Simple harmonic motion
Practice exam-style IB Physics questions for Simple harmonic motion, aligned with the syllabus and grouped by topic.
Question 1
An object is displaced a distance from its equilibrium position. What condition is required for the object to undergo simple harmonic motion?
A.
The restoring force is independent of displacement and directed towards equilibrium.
B.
The acceleration is proportional to displacement and directed towards the equilibrium position.
C.
The velocity is proportional to displacement and directed away from equilibrium.
D.
The acceleration is constant and directed towards the equilibrium position.
Question 2
A particle starts at the equilibrium position moving in the positive direction. When has it completed one cycle of oscillation?
A.
When it reaches the negative extreme.
B.
When it first reaches the positive extreme.
C.
When it first returns to equilibrium moving in the negative direction.
D.
When it returns to equilibrium moving in the positive direction.
Question 3
Why are radians used for phase angle in the equations of simple harmonic motion?
A.
Radians make the angular frequency equal to the period.
B.
Radians are required for the arguments of sine and cosine functions in these equations.
C.
Radians make the amplitude dimensionless.
D.
Radians remove the need for an initial phase angle.
Question 4
The acceleration of an oscillator varies with displacement as , where SI units are used. What is the period of the motion?
A.
$0.25 ext{ s}$
B.
$25 ext{ s}$
C.
$0.39 ext{ s}$
D.
$1.57 ext{ s}$
Question 5
A mass attached to a spring has period . The mass is replaced by a mass four times as large, using the same spring. What is the new period?
A.
$T/2$
B.
$4T$
C.
$2T$
D.
$T/4$
Question 6
A simple pendulum of length has period for small oscillations. The pendulum is moved to a planet where the gravitational field strength is , with the same length and small amplitude. What is the period on the planet?
A.
$T/2$
B.
$4T$
C.
$T/4$
D.
$2T$
Question 7
For a vertical mass–spring oscillator, what is the equilibrium position?
A.
The position where the spring has its unstretched length.
B.
The position where the mass would remain at rest if released from rest.
C.
The position where gravitational potential energy is maximum.
D.
Either extreme position of the oscillation.
Question 8
Two oscillators have the same period . Oscillator P reaches maximum positive displacement a time before oscillator Q. What is the phase difference between P and Q?
A.
$\pi/6$
B.
$2\pi/3$
C.
$\pi/3$
D.
$\pi/2$
Question 9
The displacement of an oscillator is . What is the phase relationship between velocity and displacement?
A.
Velocity lags displacement by $\pi$.
B.
Velocity leads displacement by $\pi/2$.
C.
Velocity is always opposite in direction to displacement.
D.
Velocity is in phase with displacement.
Question 10
The graph shows the displacement of a particle undergoing simple harmonic motion against time.
Determine the amplitude.
[1]
Determine the period.
[1]
Calculate the frequency.
[1]
Question 11
A mass–spring oscillator performs ideal simple harmonic motion. At what position is the kinetic energy of the mass a maximum?
A.
At the positive extreme only.
B.
Halfway between equilibrium and an extreme.
C.
At the negative extreme only.
D.
At the equilibrium position.
Question 12
An oscillator is significantly damped by air resistance. Which statement describes the effect of the damping?
A.
Mechanical energy is transferred from the oscillator to the surroundings.
B.
The amplitude remains constant while the period becomes zero.
C.
The motion becomes ideal simple harmonic motion.
D.
The restoring force is always removed from the system.
Question 13
The displacement of an oscillator is . At , the oscillator is at and moving in the positive direction. What is a possible value of ?
A.
$\pi/2$
B.
$0$
C.
$\pi$
D.
$3\pi/2$
Question 14
An oscillator has amplitude and angular frequency . What is the speed when its displacement is from equilibrium?
A.
$0.30 ext{ m s}^{-1}$
B.
$0.40 ext{ m s}^{-1}$
C.
$0.26 ext{ m s}^{-1}$
D.
$0.10 ext{ m s}^{-1}$
Question 15
For an ideal oscillator, the displacement is half the amplitude. What fraction of the total energy is kinetic energy?
A.
$1/2$
B.
$3/4$
C.
$1/4$
D.
$1$
Question 16
The amplitude of an ideal oscillator is doubled while its mass and angular frequency remain unchanged. What happens to the total energy?
A.
It doubles.
B.
It becomes half as large.
C.
It becomes four times larger.
D.
It is unchanged.
Question 17
A student records the acceleration and displacement of a cart attached to a spring.
State the two features of an acceleration–displacement graph that show the cart is undergoing simple harmonic motion.
[1]
The gradient of the graph is . Determine the period of the motion.
[1]
Question 18
A mass of is attached to a spring of spring constant and set into small oscillations.
Calculate the period of the oscillation.
[1]
State and explain the effect on the period of doubling the amplitude, assuming ideal simple harmonic motion.
[1]
Question 19
A horizontal mass–spring oscillator moves without damping.
State where the speed of the mass is zero.
[1]
Describe the changes in kinetic energy, potential energy and total mechanical energy as the mass moves from an extreme position to equilibrium.
[1]
Question 20
A point moves with uniform circular motion. Its horizontal projection is observed on a screen.
Describe the motion of the projection.
[1]
State the relation between the angular speed of the circular motion and the period of the projected motion.
[1]
State what determines the amplitude of the projected motion.
[1]
Question 21
An oscillator has amplitude and angular frequency .
Calculate the maximum speed.
[1]
Calculate the speed when .
[1]
Explain why two possible velocities can correspond to the same displacement.
[1]
Question 22
The graph shows the displacement of a damped oscillator against time.
Describe how the amplitude changes with time.
[1]
State why this motion is not ideal simple harmonic motion.
[1]
Describe the energy change responsible for the observed motion.
[1]
Estimate the period from the graph.
[1]
Question 23
A video analysis of a pendulum gives the speed of the bob at several labelled positions during one swing.
| Position label | Angular displacement / ° | Speed / m s^-1 |
|---|---|---|
| A | -18 | 0.00 |
| B | -9 | 0.72 |
| C | 0 | 1.01 |
| D | +9 | 0.72 |
| E | +18 | 0.00 |
Identify the labelled position where kinetic energy is maximum.
[1]
Identify the labelled positions where gravitational potential energy is maximum.
[1]
Explain why the bob has the same speed at two positions equally far from equilibrium on opposite sides, neglecting damping.
[1]
Question 24
Two SHM displacements are represented by rotating vectors of equal angular frequency. The angle from vector P to vector Q in the direction of rotation is . What is the phase of Q relative to P?
A.
Q leads P by $\pi/4$.
B.
Q leads P by $3\pi/4$.
C.
Q is in antiphase with P.
D.
Q lags P by $3\pi/4$.
Question 25
A pendulum has length and oscillates with a small angular amplitude.
Explain why the motion is approximately simple harmonic only for small angular amplitudes.
[1]
Calculate the period of the pendulum using .
[1]
Question 26
A molecule in a greenhouse gas can vibrate about a stable bond separation.
Explain why the vibration can be modelled as an oscillator for small displacements.
[1]
Outline how absorption and re-radiation of infrared radiation by such molecules contributes to the enhanced greenhouse effect.
[1]
Question 27
A student investigates the period of a mass–spring system by changing the attached mass.
Suggest why timing 10 oscillations gives a more reliable period than timing one oscillation.
[1]
State a graph that should be plotted to determine the spring constant .
[1]
Explain how can be found from the graph.
[1]
Question 28
Two displacement–time graphs for oscillators P and Q of the same frequency are shown.
Determine which oscillator leads in phase.
[1]
The time shift between corresponding maxima is and the period is . Calculate the phase difference in radians.
[1]
State why phase difference is expressed in radians in SHM equations.
[1]
Question 29
The displacement of a particle is , with in metres and in seconds. At , and the velocity is positive.
Determine .
[1]
Calculate the velocity at .
[1]
Question 30
A oscillator has angular frequency and amplitude .
Calculate the total energy.
[1]
Calculate the potential energy when .
[1]
Determine the kinetic energy at this displacement.
[1]
Question 31
For an oscillator of amplitude , the kinetic energy equals the potential energy at two positions.
Determine the magnitude of the displacement at these positions in terms of .
[1]
State whether these positions are closer to equilibrium or to the extremes.
[1]
Question 32
Two oscillators have displacements
and .
State the phase difference between the oscillators.
[1]
State which oscillator leads.
[1]
Express the time lead in terms of the period .
[1]
Question 33
A motion sensor records the acceleration and displacement of a trolley attached to a spring. The graph shows against .
State how the graph shows that the motion is simple harmonic.
[1]
Determine the angular frequency from the graph.
[1]
Calculate the period of the oscillation.
[1]
Suggest one reason why points at large displacements may deviate from the straight line.
[1]
Question 34
A student measures the period of a vertical spring with different masses attached. The data are plotted as against mass .
Explain why plotting against should give a straight line.
[1]
Use the gradient of the line to determine the spring constant.
[1]
Suggest the significance of a small positive intercept on the axis.
[1]
Question 35
A pendulum is timed for several lengths. The graph shows against length .
State the relationship between and predicted by the simple pendulum model.
[1]
Determine from the gradient.
[1]
Explain why the measurements at the largest amplitude may give periods greater than predicted.
[1]
Suggest one improvement to reduce random uncertainty in .
[1]
Question 36
Two oscillators X and Y are represented by the displacement–time graph. Both have the same period.
Determine the period from the graph.
[1]
Determine the time difference between corresponding maxima.
[1]
Calculate the phase difference in radians.
[1]
State which oscillator leads.
[1]
Question 37
A particle undergoing SHM has total energy , mass and amplitude .
Determine the angular frequency.
[1]
Calculate the maximum speed.
[1]
State the displacement at which potential energy is maximum.
[1]
Question 38
A computer model of an undamped oscillator updates displacement, velocity and acceleration in small time steps using .
Explain why the sign of the acceleration changes as the particle passes through equilibrium.
[1]
State the phase difference between acceleration and displacement.
[1]
State the phase difference between velocity and displacement.
[1]
Question 39
The graph shows kinetic energy and potential energy against displacement for a mass–spring oscillator of known mass.
Determine the amplitude of the oscillator from the graph.
[1]
Determine the total energy of the oscillator.
[1]
Use the total energy to determine the angular frequency.
[1]
Identify the displacement where kinetic and potential energies are equal.
[1]
Question 40
A sensor records speed at different displacements for an oscillator. The data are plotted as against .
Show that the graph should be a straight line for SHM.
[1]
Determine from the gradient.
[1]
Determine the amplitude from the graph.
[1]
Suggest why using loses information about the motion.
[1]
Question 41
The graph shows the displacement of an oscillator described by .
Determine and from the graph.
[1]
Determine .
[1]
Use the displacement and slope at to choose the correct initial phase angle from the graph.
[1]
Question 42
A horizontal spring obeying Hooke’s law is attached to a trolley on a low-friction track.
Show that the trolley satisfies the defining condition for simple harmonic motion.
[1]
Explain how the period of the trolley depends on mass and spring constant, and describe the qualitative energy changes during one complete oscillation.
[1]
Question 43
A simple pendulum is used as a timing device.
Explain why a pendulum can be treated as an approximate simple harmonic oscillator for small angular displacements.
[1]
Discuss how the period is affected by length, gravitational field strength, bob mass and amplitude.
[1]
Question 44
Uniform circular motion can be used to visualize simple harmonic motion.
Describe how the projection of uniform circular motion produces simple harmonic motion.
[1]
Explain how this model helps to interpret amplitude, period, angular frequency and the sinusoidal form of displacement–time graphs.
[1]
Question 45
An oscillator is tested at different amplitudes. The table gives amplitude and measured total mechanical energy .
| x0 / m | ET / J |
|---|---|
| 0.020 | 0.0032 |
| 0.030 | 0.0073 |
| 0.040 | 0.0127 |
| 0.050 | 0.0202 |
| 0.060 | 0.0340 |
State the expected relationship between and for constant and .
[1]
Identify a graph that should be linear if the oscillator follows the SHM energy model.
[1]
Use the gradient of this graph to determine .
[1]
Suggest one reason why the largest-amplitude data point may not fit the model.
[1]
Question 46
A student claims that a pendulum is isochronous because all measured periods are the same, regardless of amplitude.
Define amplitude and period for an oscillating pendulum, and state what is meant by isochronous.
[1]
Evaluate the student’s claim, including experimental and physical limitations of the pendulum model.
[1]
Question 47
An oscillator has displacement . At , and . The amplitude is and .
Determine the possible phase angles from the initial displacement and select the one consistent with the initial velocity.
[1]
Calculate the acceleration at , the next time after when the oscillator first passes through equilibrium, and the speed at .
[1]
Question 48
A mass undergoes ideal SHM with angular frequency and amplitude .
Derive expressions for total energy, potential energy and kinetic energy as functions of displacement.
[1]
Explain, using these expressions, the shapes of energy–displacement graphs and determine where kinetic and potential energies are equal.
[1]
Question 49
Two particles A and B execute SHM with the same amplitude and angular frequency. Their displacements are
and .
Compare the phase of B with the phase of A and express the time shift in terms of the period.
[1]
Compare their displacements, velocities and accelerations at , using the SHM equations.
[1]
Question 50
A sensor records the motion of a mass–spring oscillator. A proposed model is
,
where is in metres and is in seconds. The mass is .
Determine the period, maximum speed and total energy predicted by the model.
[1]
Evaluate whether this model would remain valid if the oscillator were released with a much larger amplitude, considering both kinematic and energy aspects of SHM.
[1]
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