Simple Harmonic Motion (DP IB Physics: SL): Exam Questions

A pendulum undergoes small-angle oscillations.

Outline the equation that defines simple harmonic motion.

Sketch a graph to represent the change in amplitude, against time for one swing of the pendulum. Start the time at zero seconds.

The time period of 10 oscillations is found to be 12.0 s.

Determine the frequency when the bob is 1.0 cm from its equilibrium position.

The student wants to double the frequency of the pendulum swing. The time period, T of a simple pendulum is given by the equation:

T = 2π

where L is the length of the string and g is the acceleration due to gravity

Deduce the change which would achieve this.

State and explain whether the motion of the objects in graphs I, II and III are simple harmonic oscillations

Explain why, in practice, a freely oscillating pendulum cannot maintain a constant amplitude.

The motion of an object undergoing SHM is shown in the graph below.

For this oscillator, determine: 

(i) The amplitude, A.

[1]

(ii) The period, T.

[1]

(iii) The frequency, f.

[1]

Using the graph from part (c), state a time in seconds when the object performing SHM has: 

(i) Maximum positive velocity.

[1]

(ii) Maximum negative acceleration.

[1]

(iii) Maximum potential energy.

[1]

A ball of mass 44 g on a 25 cm string oscillating in simple harmonic motion obeys the following equation:

a = −ω²x

Demonstrate mathematically that the graph of this equation is a downward sloping straight line that goes through the origin.

The graph below shows the acceleration, a, as a function of displacement, x, of the ball on the string.

The angular speed, ω, in rad s⁻¹, is related to the frequency, f, of the oscillation by the following equation:

For the ball on the string, determine the period, T, of the oscillation.

The ball is held in position X and then let go. The ball oscillates in simple harmonic motion.

Explain the change in acceleration as the ball on the string moves through half an oscillation from position X.

You can assume the ball is moving at position X.

Describe the energy transfers occurring as the ball on the string completes half an oscillation from position X.

A smooth glass marble is held at the edge of a bowl and released. The marble rolls up and down the sides of the bowl with simple harmonic motion.

The magnitude of the restoring force which returns the marble to equilibrium is given by:

 Where  is the displacement at a given time, and  is the radius of the bowl.

Outline why the oscillations can be described as simple harmonic motion.

Describe the energy changes during the simple harmonic motion of the marble.

As the marble is released it has potential energy of 15 μJ. The mass of the marble is 3 g.

Calculate the velocity of the marble at the equilibrium position.

Sketch a graph to represent the kinetic, potential and total energy of the motion of the marble, assuming no energy is dissipated as heat. Clearly label any important values on the graph.

An object is attached to a light spring and set on a frictionless surface. It is allowed to oscillate horizontally. Position 2 shows the equilibrium point.

(i) Sketch a graph of acceleration against displacement for this motion.

[2]

(ii) On your graph, mark positions 1, 2 and 3 according to the diagram.

[1]

The mass begins its motion from position 1 and completes a full oscillation.

(i) Sketch a graph of velocity against time to show this.

[2]

(ii) On your graph, add labels to show points 1, 2 and 3

[2]

At the point marked Y on the graph, the potential energy of the block is E_P. The block has mass m, and the maximum velocity it achieves is v_max.

Determine an equation for the potential energy at the point marked X.

Give your answer in terms of v_max , and m.

The graph shows how the displacement x of the mass varies with time t.

Determine the frequency of the oscillations. 

An experiment is carried out on Planet Z using a simple pendulum and a mass-spring system. The block moves horizontally on a frictionless surface. A motion sensor is placed above the equilibrium position of the block which lights up every time the block passes it. 

The pendulum and the block are displaced from their equilibrium positions and oscillate with simple harmonic motion. The pendulum bob completes 150 full oscillations in seven minutes and the bulb lights up once every 0.70 seconds. The block has a mass of 349 g.

Show that the value of the spring constant k is approximately 7 N m⁻¹. 

The volume of Planet Z is the same as the volume of Earth, but Planet Z is twice as dense. 

For the experiment on Planet Z

(i) Show that the length of the pendulum,

[2]

(ii) Calculate the value of .

[2]

The angle that the pendulum string makes with the horizontal is 81.4 ° when the acceleration of the pendulum bob is at a maximum. 

Determine the maximum speed reached my the pendulum bob.

Compare and contrast how performing the experiment on Planet Z, rather than on Earth, affects the period of the oscillations of the pendulum and the mass-spring system.

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

The time period of the mass is given by the equation:

 T = 2π

(i) Calculate the spring constant of a spring attached to a mass of 0.7 kg and time period 1.4 s.

 [1]

(ii) Outline the condition under which the equation can be applied.

[1]

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

A new mass of m = 50 g replaces the 0.7 kg mass and is now attached to the mass-spring system.

The graph shows the variation with time of the velocity of the block.

Determine the total energy of the system with this new mass.

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

A volume of water in a U-shaped tube performs simple harmonic motion.

State and explain the phase difference between the displacement and the acceleration of the upper surface of the water.

The U-tube is tipped and then set upright, to start the water oscillating. Over a period of a few minutes, a motion sensor attached to a data logger records the change in velocity from the moment the U-tube is tipped. Assume there is no friction in the tube.

Sketch the graph the data logger would produce.

The height difference between the two arms of the tube , and the density of the water .

Construct an equation to find the restoring force, , for the motion.

The time period of the oscillating water is given by  where  is the height of the water column at equilibrium and g is the acceleration due to gravity.

If  is 15 cm, determine the frequency of the oscillations.

The diagram shows a flat metal disk placed horizontally, that oscillates in the vertical plane.

The graph shows how the disk's acceleration, a, varies with displacement, x.

Show that the oscillations of the disk are an example of simple harmonic motion.

Some grains of salt are placed onto the disk. 

The amplitude of the oscillation is increased gradually from zero.

At amplitude A_Z, the grains of salt are seen to lose contact with the metal disk. 

(i) Determine and explain the acceleration of the disk when the grains of salt first lose contact with it.

[3]

(ii) Deduce the value of amplitude A_Z.

[1]

For the amplitude at which the grain of salt loses contact with the disk: 

(i) Deduce the maximum velocity of the oscillating disk.

[2]

(ii) Calculate the period of the oscillation.

[1]

For a homework project, some students constructed a model of the Moon orbiting the Earth to show the phases of the Moon. 

The model was built upon a turntable with radius r, that rotates uniformly with an angular speed ω.

The students positioned LED lights to provide parallel incident light that represented light from the Sun. 

The diagram shows the model as viewed from above.

The students noticed that the shadow of the model Moon could be seen on the wall.

At time t = 0,  θ = 0 and the shadow of the model Moon could not be seen at position E as it passed through the shadow of the model Earth. 

Some time later, the shadow of the model Moon could be seen at position X

For this model Moon and Earth 

(i) Construct an expression for θ in terms of ω and t

[1]

(ii) Derive an expression for the distance EX in terms of r, ω and t

[1]

(iii) Describe the motion of the shadow of the Moon on the wall

[1]

The diameter, d, of the turntable is 50 cm and it rotates with an angular speed, ω, of 2.3 rad s⁻¹.

For the motion of the shadow of the model Moon, calculate: 

(i) The amplitude, A.

[1]

(ii) The period, T.

[1]

(iii) The speed as the shadow passes through position E.

[2]

The defining equation of SHM links acceleration, a, angular speed, ω, and displacement, x.

For the shadow of the model Moon: 

(i) Determine the magnitude of the acceleration when the shadow is instantaneously at rest.

[2]

(ii) Without the use of a calculator, predict the change in the maximum acceleration  if the angular speed was reduced by a factor of 4 and the diameter of the turntable was half of its original length.

[1]

The needle carrier of a sewing machine moves with simple harmonic motion. The needle carrier is constrained to move on a vertical line by low friction guides, whilst the disk and peg rotate in a circle. As the disk completes one oscillation, the needle completes one stitch.

The sewing machine completes 840 stitches in one minute. Calculate the angular speed of the peg.

Label, on the diagram, the position of the peg at the point of maximum velocity, and the point of maximum contact force of the peg on the slot.