Damped & Forced Oscillations, Resonance (CIE A Level Physics)

A student investigates the vertical oscillations of the mass–spring system shown in Fig. 1.1. 

Fig. 1.1

The system is suspended from one end of a thread passing over a pulley. The other end of the thread is tied to a weight. The system is shown in Fig. 1.1 with the mass at the equilibrium position, denoted by the fiducial mark. The spring constant is the same for each spring. 

The table below shows the measurements recorded by the student over the course of 5 repeats of the experiment. 

Table 1.1

Determine the natural frequency of the mass-spring system and the percentage uncertainty in the data. Hence quote the natural frequency with its uncertainty to an appropriate number of significant figures.

The student connects the thread to a mechanical oscillator. The oscillator is set in motion using a signal generator and this causes the mass–spring system to undergo forced oscillations. 

A vertical ruler is set up alongside the mass–spring system as shown in Fig. 1.2. The student measures values of A, the amplitude of the oscillations of the mass as f, the frequency of the forcing oscillations, is varied. 

Fig. 1.2

At the point X, where the mass–spring system is joined to the thread, the amplitude of the oscillations is 4 mm. When the mass–spring system resonates at its natural frequency, amplitude of oscillation is 95 mm. 

On the axes provided, sketch the expected relationship between the amplitude and frequency of oscillation.

Fig. 1.3

The student removes one of the springs, adds more mass to the hanger and then repeats the experiment. 

Sketch a new line on your graph to show the results the student would obtain.

Give a value for the new frequency only, but the new amplitude does not need a specific value.

The relationship between frequency and spring constant can be described as:

A bar magnet of mass 250 g is suspended from the free end of a spring, as illustrated in Fig. 3.1.

Fig. 3.1

The magnet hangs so that one pole is near the centre of a coil of wire.
The coil is connected in series with a resistor and a switch. The switch is open.

The magnet is displaced vertically and then allowed to oscillate.

At time t = 0, the magnet is oscillating freely. At time t = 6.0 s, the switch in the circuit is closed.

The variation with time t of the vertical displacement y of the magnet is shown in Fig. 3.2.

Fig. 3.2

For the oscillating magnet, use data from Fig. 3.2 to determine, to two significant figures:

f = ...................................... Hz [2]

energy = ....................................... J [3]

When the switch is closed, the oscillations are damped.

Explain, with reference to Fig. 3.2, whether this damping is light, critical or heavy.