Simple Harmonic Motion (HL IB Physics)

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 is , and the density of the water is .

Construct an equation to find the restoring force, , for the motion and simplify as much as possible.

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.

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 

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:

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

For the shadow of the model Moon:

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.

The needle carrier has a mass of 23.9 g, and the needle has a mass of 0.7 g. The needle moves a distance of 36 mm between its extremities of movement.

Assuming that the fabric requires a negligible force for the needle to penetrate it:

For the needle-carrier system:

[2]

A metal pendulum bob of mass 7.5 g is suspended from a fixed point by a length of thread with negligible mass. The pendulum is set in motion and oscillates with simple harmonic motion. 

The graph shows the kinetic energy of the bob as a function of time.

Calculate the length of the thread.

For the simple pendulum:

Show that the amplitude of the oscillation is around 0.6 m.

A steel spring with an unstretched length of 33 cm is attached to a fixed point and a mass of 35 g is attached and gently lowered until equilibrium is reached and the spring has a length of 37.5 cm. The spring is then stretched elastically to a length of 42 cm and released. 

Design a plan to investigate if the oscillation is simple harmonic motion.

For the stretching of the spring:

A student with mass 68 kg hangs from a bungee cord with a spring constant, k = 270 N m⁻¹. The student is pulled down to a point where the cord is 4.0 m longer than its unstretched length, and then released. The student oscillates with SHM.

For the student:

A second student wants to do the bungee jump, however, they would like a greater number of bounces in their five minute session. 

Evaluate the possibilities for facilitating the student's wishes.

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: 

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

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

a = −ω²x

Show 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.

Deduce mathematically why the oscillations of the marble are in simple harmonic motion.

Describe the energy changes that take place 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.

(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.

(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. 

A ball within a ball-spring system oscillates about an equilibrium point. 

Explain how the ball-spring system demonstrates simple harmonic motion.

The ball oscillates with simple harmonic motion. On the axes provided, taking the oscillation as beginning at the positive amplitude when t = 0, sketch for the ball's motion:

In a separate experiment, the ball is oscillating from the equilibrium position. The graph shows the displacement of the ball over time.

Determine the maximum velocity of the ball.

Using the graph from part (c) for the motion of the ball 

[1]

Determine the total energy in the oscillation of the wooden block.

Using the information from part (a), sketch on the axes the kinetic, potential, and total energies of the oscillating wooden block as they vary with displacement.

The investigation from part (a) is repeated. The same wooden block and spring are used, but a second identical spring is added in parallel. 

Determine how this change will affect the fractional uncertainty in the mass of the wooden block. 

In a separate experiment, the mass-spring system from part (a) is set to oscillate in simple harmonic motion. 

The graph shows the variation of displacement with time t of the mass on the spring.

For the new experiment on the mass-spring system

[2]

A 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.

The mass-spring system can be used to model the motion of an ion in a crystal lattice structure. 

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.

For the oscillations

[4]

(ii)   Estimate the maximum kinetic energy of the ion.

[3]

For the mass-spring system

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

The length of the spring with the mass attached is 64 mm. The mass is pulled downwards until the length of the spring is 76 mm. The mass is released, and the vertical mass-spring system performs simple harmonic motion.

For the new mass-spring system

[2]

The diagram shows the vertical spring-mass system as it moves through one period.

Annotate the diagram to show when:

A ball is displaced through a small distance x from the bottom of a bowl and then released. 

The frequency of the resulting oscillation is 1.5 Hz and the maximum velocity reaches 0.36 m s⁻¹. r is the radius of the bowl.

For the oscillating ball:

[2]

For the ball 

[2]

Sketch the graphs showing how the displacement, velocity and acceleration of the ball vary with time. You should include any relevant values.

The ball was replaced by a ball of the same size, but with a greater mass. 

Outline what effect this would have on the period of the oscillation.

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

Considering the motion of both the spring and the pendulum, show that the length of the pendulum,

Calculate the value of .

[2]

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.