4.1 Oscillations

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

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.

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.

Some grains of salt are placed onto the disk. 

The amplitude of the oscillation is increased gradually from zero.

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

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

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