4.1 Oscillations

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.

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

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

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.

   (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.

    (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.