A LevelPhysicsCIETopic Questions17. Oscillations17.2 Energy in Simple Harmonic Motion

Syllabus Edition

First teaching 2023

First exams 2025

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Energy in Simple Harmonic Motion (CIE A Level Physics)

Topic Questions

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1a
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Complete the gaps, using the words from the box, in the sentences that describe the relationship between the simple harmonic motion graphs.

cosine     sine     identical     motion     displacement     45°     90°     velocity

 

  • The displacement-time graph is a curve identical to a _____________ graph when the oscillator starts from its equilibrium position.

 

  • Velocity is the rate of change of _____________

 

  • The velocity-time graph is _____________ out of phase with the displacement-time graph

 

  • Acceleration is the rate of change of _____________

 

  • The acceleration-time graph is _____________ to the displacement graph except reflected on the x-axis.

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1b
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State the equation for the total energy of a simple harmonic system.

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1c
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Fig. 1.1 shows the graph of potential, kinetic and total energy of a simple harmonic system for half a period of simple harmonic oscillation. 

 17-1-3c-e-energy-graph-esq-cie-a-level
Fig. 1.1
 

Match the labels to the correct place on the graph by drawing a line between them.

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1d
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Identify, by placing a tick (✓) next to, the correct statements about the key features of the displacement-time graph. 

 
Possible statements about the displacement-time graphs Place a tick (✓) here if the statement is correct

 

The amplitude of oscillations x0 can be found from the maximum value of x

  

 

If the oscillations start at the positive or negative amplitude, the displacement will be at its minimum

  

 

The time period of oscillations T can be found from reading the time taken for one full cycle

  

 

The graph always starts at 0

  

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2a
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Define the total energy for a system oscillating in simple harmonic motion.

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2b
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Fig. 1.1 shows the positions of a simple pendulum in simple harmonic motion.

4-1-4c-question-stem-sl-sq-easy-phy

Fig. 1.1

Identify the position of the pendulum when: 

(i)
Kinetic energy is zero
[1]
(ii)
Potential energy is at a maximum
[1]
(iii)
Kinetic energy is at a maximum
[1]
(iv)
Potential energy is zero
[1]

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2c
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At the equilibrium position, the kinetic energy of the pendulum is 0.324 J. 

Determine the total energy of the system.

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2d
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The amplitude of the pendulum is 5 cm, and the mass of the bob is 20 g.

Calculate the angular speed of the pendulum as it swings through the equilibrium position. 

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3a
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A mass-spring system oscillates with simple harmonic motion. The graph in Fig. 1.1 shows how the potential energy of the spring varies with displacement.

4-1-5a-question-stem-sl-sq-easy-phy

Fig. 1.1

Using Fig. 1.1, determine: 

(i)
the maximum potential energy of the system
[1]
(ii)
the total energy of the system.
[1]

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3b
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Using Fig. 1.1, determine: 

(i)
the amplitude x0 of the oscillation
[1]
(ii)
the potential energy in the spring when the displacement is x = 0.1 m.
[1]

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3c
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Determine the kinetic energy of the mass-spring system at x = 0.1 m.

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3d
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The block used in the same mass-spring system has a mass of 25 g. The maximum kinetic energy of the block is 40 mJ.

Calculate the maximum velocity of the oscillating block

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1a
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A U shaped tube with cross-sectional area A contains a liquid of mass m, density ρ and volume V as shown in Fig. 1.1. When the water of length L is in equilibrium in the tube the levels of water on either side are equal. When the water is pushed down a distance xmax on one side of the tube then the water on the opposite side rises. Upon release the water undergoes simple harmonic motion as the water levels on each side rise and fall about the equilibrium position.

17-2-q1a-h-sq-cie-ial-physics

Fig. 1.1

Obtain an expression for the maximum kinetic energy of the liquid in terms of its cross-sectional area A, density ρ, maximum velocity vmax and length of water L. 

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1b
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Obtain an expression in terms of cross-sectional area A, liquid density ρ, and xmax for the gravitational potential energy Ep (max) of the liquid in the U-shaped tube in Fig. 1.1.

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1c
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Show that the time period of the oscillation of the liquid in the U shaped tube of Fig. 1.1 is given by the equation

T space equals space 2 pi space square root of fraction numerator l over denominator 2 g end fraction end root

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1d
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Determine the length of the liquid in the pipe, L in centimetres and the corresponding angular velocity, ω when the ratio of maximum displacement and maximum velocity is 0.05 s in Fig. 1.1.

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2a
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A mass-spring system has been set up horizontally on the lab bench, so that the mass can oscillate.

Sketch a velocity-displacement graph of the motion of the block as it undergoes simple harmonic motion.

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2b
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A mass of m = 50 g is attached to the mass-spring system.

The graph in Fig. 1.1 shows the variation with time of the velocity of the block.

q2c_oscillations_ib-sl-physics-sq-medium

Fig. 1.1

Determine the total energy of the system when it is oscillating with a mass of 50 g.

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2c
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Determine the potential energy of the system when 6 seconds have passed.

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2d
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The mass-spring system is now fixed so the mass and spring oscillate vertically. The mass remains at 50 g and the angular velocity and time period the same as before.

Show that the total energy of the system is still the same as in part (b). 

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1a
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A small metal pendulum bob is suspended from a fixed point by a thread with negligible mass. Air resistance is also negligible.

The pendulum begins to oscillate from rest. Assume that the motion of the system is simple harmonic, and in one vertical plane. Fig. 1.1 shows the variation of kinetic energy of the pendulum bob with time.

ib-9-1-sq-q1a-1

Fig. 1.1

Determine the time period of the pendulum.

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1b
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Label a point X on the graph where the pendulum is in equilibrium.

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1c
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The mass of the pendulum bob is 60 × 10–3 kg. 

i)   Determine the maximum kinetic energy of the pendulum bob.

[1]

ii)   Show that the maximum speed of the bob is about 0.82 m s–1.

[3]

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2a
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Fig. 1.1 shows a mass-spring system which undergoes simple harmonic motion.

The mass of 75 g is connected between two identical springs. The mass-spring system rests on a frictionless surface. A force of 0.025 N is needed to compress or extend the spring by 1.0 mm. 

The mass is pulled from its equilibrium position to the right by 0.055 m and then released. The mass oscillates about the equilibrium position in simple harmonic motion.

9-1-hl-sq-medium-3a-mass-spring

Fig. 1.1

Calculate the acceleration of the mass at the moment of release.

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2b
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The mass-spring system from part (a) can be used to model the motion of an ion in a crystal lattice structure. The model is shown in Fig. 1.2.

9-1-hl-sq-medium-3a-ion

Fig. 1.2

The frequency of the oscillation of the ion is 8 × 1012 Hz and the mass of the ion is 6 × 10−26 kg. The amplitude of the vibration of the ion is 2 × 10−11 m.

Estimate the maximum kinetic energy of the ion.

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2c
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For the mass-spring system in part (a) the total energy of the system is 0.077 J.

 

Sketch a graph showing the variation over time of the kinetic energy of the mass and the potential energy of the springs. 

You should use the axes given below, include appropriate values, and show the oscillation over one full period.

9-1-hl-sq-medium-3b-axes

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2d
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The same mass and a single spring from part (a) are attached to a rigid horizontal support. 

Fig 1.3 shows the vertical spring-mass system as it moves through one period.

9-1-hl-sq-medium-3d-diagram

Fig. 1.3

Annotate the diagram to show when:

Ep = max

Ek = max

v = 0

v = max

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3a
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Fig 1.1 shows a graph of the displacement over time of a simple pendulum oscillating in simple harmonic motion. 

ib-hl-emcq-5-q-stem

Fig. 1.1

State the potential energy of the pendulum at points X, Y and Z in terms of maximum and minimum.

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3b
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The amplitude of the pendulum is 60 cm when it swings with a frequency of 0.7 Hz. During this oscillation, the maximum kinetic energy of the pendulum is 23 J.

Determine the mass of the pendulum bob.

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3c
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Calculate the kinetic energy of the pendulum when the pendulum bob is at a distance of 0.4 x subscript 0 from its point of maximum displacement.

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