Simple Harmonic Oscillations (OCR A Level Physics): Structured Questions

One end of a spring is fixed to a support.

A toy car, which is on a smooth horizontal track, is pushed against the free end of the spring. 

The spring compresses. The car is then released. The car accelerates to the right until the spring returns back to its original length.  

  The car moves with simple harmonic motion as the spring returns to its original length. The acceleration of the car is given by the expression a = $-\left(\frac{\mathit{k}}{\mathit{m}}\right)\mathrm{x}$, where m is the mass of the car, k is the force constant of the spring and x is the compression of the spring.

Use the data below to calculate the time t it takes for the spring to return to its original length after the car is released.

• mass of car m = 80 g

• force constant k of the spring = 60 N m^–1.

t = ....................................................... s [4]

The arrangement in (a) is used to propel the toy car along a smooth track.

i) Point A is at the top of the track.   The launch speed of the car is now adjusted until the car just reaches A with zero speed. The height of A is 0.20 m above the horizontal section of the track.  

All the elastic potential energy of the spring is transferred to gravitational potential energy of the car.  

Calculate the initial compression x of the spring.

x = ...................................................... m [3]

ii) At a specific speed, the car leaves point A horizontally and lands on the track at point B. The horizontal distance between A and B is D.

Air resistance has a negligible effect on the motion of the car between A and B.

1. Explain how the time of flight between A and B depends on the speed of the car at A.

[2]

2. Explain how the distance D depends on the speed of the car at A.

[2]

This question is about a simple pendulum made from a length of string attached to a mass (bob).

For oscillations of small amplitude, the acceleration a of the pendulum bob is related to its displacement x by the expression

$a=-\left(\frac{g}{L}\right)x$

where g is the acceleration of free fall and L is the length of the pendulum.

The pendulum bob oscillates with simple harmonic motion.

(i) Show that the period T of the oscillations is given by the expression

$T^2=\frac{4{\pi }^2}{g}L$

[3]

(ii) A student notices that the amplitude of each oscillation decreases over time.

Explain this observation and state what effect this may have on T.

[2]

Describe with the aid of a labelled diagram how an experiment can be conducted and how the data can be analysed to test the validity of the equation $T^2=\frac{4{\pi }^2}{g}L$ for oscillations of small amplitude.

Another student conducts a similar experiment in the laboratory to investigate the small amplitude oscillations of a pendulum of a mechanical clock. Each 'tick' of the clock corresponds to half a period.

(i) Show that the length of the pendulum required for a tick of 1.0 s is about 1 m.

[2]

(ii) If the pendulum clock were to be used on the Moon, explain whether this clock would run on time compared with an identical clock on the Earth.

[2]

A buoy, floating in a vertical tube, generates energy from the movement of water waves on the surface of the sea. When the buoy moves up, a cable turns a generator on the seabed, producing power. When the buoy moves down, the cable is wound in by a mechanism in the generator and no power is produced.

The equipment is set up as shown in Fig. 21.1.

Fig. 21.1

The vertical motion of the buoy can be assumed to be simple harmonic with the same amplitude as the wave.

State the conditions necessary for an object to be in simple harmonic motion.

A wave of amplitude 3.8 m and wavelength 28 m is travelling at 3.2 m s⁻¹.

Calculate the maximum vertical speed of the buoy caused by the movement of the wave.

maximum vertical speed = .................................................. m s⁻¹

On the axes below, sketch a graph to show the variation of the generator output power with time.

Label the time axis with a suitable scale.

Determining the best location for wave generators, such as the one in Fig. 21.1, is very important. The graph in Fig. 21.2 gives an indication of the relationship between the height of ocean waves and their period.

Fig. 21.2

Discuss the use of wave generators in terms of the potential electrical power output.

In your answer, you should:

• explain how wave height and wave period affect the energy within a wave

• describe how different energy losses in the system might affect the power output

Two coastal locations, A and B, are being considered for the installation of wave generators.

At location A, the waves have a shorter period and a smaller height.

At location B, the waves have a longer period and a larger height.

Using your understanding of wave energy and power transfer, evaluate which location is more suitable for wave generators and justify your conclusion.