6.3 Forced Vibrations & Resonance

Figure 1 shows a string XY supporting a heavy pendulum P and four pendulums, A, B, C, and D of smaller mass. 

 Figure 1

Pendulum P is set in oscillation perpendicular to the plane of the diagram. 

Which of the pendulums, A to D oscillates with the largest amplitude? Explain your answer.

Figure 2 shows the displacement-time graph for pendulum P. 

Figure 2

On the empty axes in Figure 2, sketch the displacement-time graph for the pendulum with the largest amplitude.

Explain the shape of your graph in part (b) in terms of frequency, amplitude and phase. 

Both pendulums C and D appear to oscillate in phase with pendulum P, whereas the others do not. 

Compare the phases of oscillation of A with P, and B with P and explain these differences qualitatively in terms of displacement, velocity, and energy transfers. 

You may sketch graphs to help illustrate your answer.

Figure 1 shows an apparatus for investigating forced vibrations and resonance of a mass-spring system. 

Figure 1

Figure 2 shows the displacement-time graph when the system is resonating. 

Figure 2

The rotating wheel causes the mass to perform forced vibrations. Under certain conditions, the system may demonstrate resonance. 

The mass of the spring used in the experiment was 0.44 kg and the additional mass placed on the spring was 0.5 kg. 

When the rotating wheel stops, Figure 3 shows how the amplitude of the oscillations of the mass subsequently varies with time. 

Figure 3

Explain whether the graph in Figure 3 supports the suggestion that the amplitude of the damped oscillations varies exponentially with time. Show your reasoning clearly.

Determine the ratio  

The mass-spring system is now suspended from a horizontal support rod that can be made to oscillate vertically, as shown in Figure 4, with amplitude 10 mm at several different frequencies. 

Figure 4

The response of the masses suspended from the spring to the vertical oscillations of the support roads varies with frequency. 

Discuss in as much detail as you can, the motion of the mass when the support rod oscillates at a frequency of 

The quality of your written answer will be assessed in this question.

A student investigates the vertical oscillations of the mass–spring system shown in Figure 1. 

Figure 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 Figure 1 with the mass at the equilibrium position. The spring constant is the same for each spring. 

The table below shows the measurements recorded by the student. 

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 Figure 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. 

Figure 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.

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

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

You may wish to use the equation f =

A bar magnet is suspended on a spring as shown in Figure 1. A pole of the magnet is located near to one end of a solenoid. 

Figure 1

The observations made are summarised in Table 1. 

Table 1

Explain these observations.

Figure 2a shows a long bar magnet suspended by a coiled spring from a rigid support along with the displacement-time graph of the resulting oscillations. 

Figure 2a

The amplitude of vertical oscillations of the magnet can be measured by observing the motion of a pointer attached to it as the pointer moves over a fixed vertical scale. 

The lower end of the magnet is now suspended in a cylinder of aluminium as shown in Figure 2b. This produces damping of the vertical oscillations. 

Figure 2b

In Figure 3, a flat horizontal coil has been placed around the S pole of the long magnet and the damping cylinder removed. 

Figure 3

When there is an alternating current in the coil, the magnet vibrates under forced oscillations. 

Explain how the magnet in Figure 3 is made to perform forced oscillations.

The graph, Figure 4, shows how the amplitude of the oscillations varies with f, frequency of the alternating current. 

Figure 4

Add a second line to Figure 4 to show how  varies with f when the damping cylinder is now replaced in its original position as in Figure 2b.    

Explain briefly the difference between a free and a forced oscillation.

Figure 1 below shows a simple pendulum that consists of a large mass at the end of a long string. A, B and C are positions of the pendulum as it oscillates in the air. A and C are the extreme positions of the motion and B is the centre of the motion. 

Figure 1

State, and explain, at which position the damping is greatest.

State the effect on the motion of the pendulum when: 

            (i)         A longer string is used. 

            (ii)        A greater mass of the same size is used.     

The simple pendulum is then attached to a thread with a second pendulum attached to it, as shown in Figure 2.

Figure 2

P is a heavy pendulum, the frequency   of which can be varied. Q is a lighter pendulum of fixed frequency . As the frequency of oscillation of P is increased by shortening the thread, the amplitude of the oscillation of Q changes. 

Sketch a graph to show the relationship between the amplitude of the oscillation of Q and the frequency  of P.

Resonance can either be useful or a problem. For example, bridges can become dangerous when external forces from wind or pedestrians cause the bridge to resonate. 

Describe: 

For each example identify what is oscillating and what causes these oscillations.

To celebrate the Millennium in the year 2000, a footbridge was constructed across the River Thames in London. After the bridge was opened to the public it was discovered that the structure could easily be set into oscillation when large numbers of pedestrians were walking across it. 

To solve the problem the engineers decided to use damping mechanisms — giant shock absorbers to limit the bridge's response to external forces. They decided to use two systems: viscous dampers, similar to car shock absorbers, and tuned mass dampers. 

A tuned mass damper is a large mass stiffened by springs. Figure 1 shows the tuned mass dampers which were fitted to the bridge. They are tuned to the natural frequency of oscillation of the bridge. 

Figure 1

Discuss how the tuned mass dampers reduce the amplitude of the oscillations of the bridge and explain why they must be very heavily damped.

The Tacoma Narrows Bridge was a suspension bridge in the U.S state of Washington built 1940. Not long after its construction, the bridge began to oscillate from heavy winds. 

Under what condition would this become particularly hazardous? Explain your answer.

Other than installing dampers or shock absorbers, suggest two other measures which engineers might adopt in order to reduce the size of the oscillations of a bridge.    

An electric motor in a washing machine drives a rotating drum by means of a rubber belt attached to pulleys, one on the motor shaft and one on the drum shaft, as shown in Figure 1. 

Figure 1

When the motor rotates at a particular speed, it causes a flexible metal panel in the washing machine to vibrate loudly. 

Explain why this happens.

In order to reduce the vibrations of the flexible metal panel on the machine, damping measures are added to the motor. 

Explain why damping would reduce the vibrations of the flexible metal panel on the washing machine.

The drums of an automatic washing machine are damped by suspending it from the casing by springs, at the top and bottom, as shown in Figure 2. 

Figure 2

The inner drum rotates within the outer drum at variable speeds according to the washing programme. The natural period of oscillation of the system is 1.2 s. 

When the washing machine enters the spin part of its programme, it starts from rest building up rotational speed gradually. 

As the speed increases, the system is observed to oscillate with increasing amplitude which reaches a maximum value of 4.0 cm before decreasing again at higher speeds. 

Sketch a graph showing how the amplitude varies with the frequency of rotation of the drum. 

Label the axes of the graph and give the axes a suitable scale.

State and explain the effect on the oscillation of running the machine: 

(i)         Without the block of concrete fixed to the drum. 

(ii)        With added springs on each side of the drum.

When a tuning fork is struck with a rubber hammer, a pure sound of fixed frequency is produced. Figure 1 shows a tuning fork connected to a wooden sounding box. 

Figure 1

There are two main observations:

• The sounding box amplifies the sound produced when the tuning fork is struck.
• The sound lasts for a shorter time than if the tuning fork were to be struck identically but without the sounding box. 

Explain these observations.

The tuning fork is then placed in a cylinder of water and is struck again. Figure 2 shows how the displacement of one of the forks varies with time. 

Figure 2

The amplitude of the oscillation decreases with each cycle. 

Explain why this effect is observed.

It is suggested that the decrease in amplitude is exponential. 

Use Figure 2 to determine if this is approximately true.

Figure 3 shows how the amplitude of the oscillating fork varies over a range of forcing frequencies. 

Figure 3

Curve A shows the results of the investigation using the apparatus in air. The student repeats the investigation with the oscillating mass in a cylinder of water. Curve B shows these results. 

Describe and explain the differences between the two sets of results.