Forced vibrations, damping & resonance

The photograph shows the Petronas Twin Towers in Kuala Lumpur, which are the world’s tallest twin buildings.

Each tower is supported by 16 cylindrical concrete columns.

Calculate the stress at the base of each cylindrical concrete column.

mass of each tower = 3.010⁸kg diameter of each column = 2.4 m

Tall buildings sway in the wind. The design of the Petronas Towers permits safe movement of the towers.

The towers were designed to permit a maximum horizontal displacement of 0.90 m at the top of the towers. The natural frequency of each tower is 0.17 Hz.

Calculate the maximum velocity at the top of one of the towers when the tower is oscillating at its natural frequency.

The twin towers were linked together to improve the response of the building to strong winds.

The graph below shows how the maximum displacement x varies with the frequency f for a single tower and for two linked towers.

Explain the behaviour of a single tower compared with the behaviour of the linked towers.

A ‘jumperoo’ is a harness, suspended by a vertical spring, into which a baby can be placed, as shown.

The jumperoo is adjusted so that the baby’s feet are a few centimetres above the floor. If the baby is then displaced downwards and released, he oscillates vertically.

The graph shows how the acceleration a of the baby depends upon the displacement x of the baby from its equilibrium position.

For safety reasons, it is suggested that the maximum velocity of the baby should not

Assess whether it is safe for the baby to oscillate in the jumperoo with an amplitude of 22 cm.

The amplitude of the oscillations quickly decreases, so the baby has to push down on the floor to maintain the oscillations.

When the baby pushes at a particular frequency, the amplitude of oscillation increases to a maximum.

A baby of greater mass is placed in the jumperoo.

This baby pushes on the floor at a frequency that produces a maximum amplitude of oscillation.

Explain how this frequency compares with the frequency of pushing of the original baby.

A calculation is not necessary.

The suspension system in a car includes a spring attached to each wheel as shown.

The car, of mass 1100 kg, is stationary. Each spring is compressed by 152 mm due to a quarter of the weight of the car. Each spring is well within both the limit of proportionality and the elastic limit.

State what is meant by within the elastic limit.

Show that the stiffness of each spring is about 18 000 N m^–1.

A force is applied to the car which results in a further small compression of each spring. The force is then removed, and the body of the car oscillates with simple harmonic motion.

Determine the frequency of the oscillations.

State the conditions for simple harmonic motion.

The oscillations are heavily damped by a piston in the suspension system. The piston moves within a cylinder filled with oil, as shown. The oil has a high viscosity.

Explain why using oil of high viscosity will produce heavy damping.

A student investigated the horizontal oscillations of a trolley between two springs, using the apparatus shown.

The student displaced the trolley from its equilibrium position. She then released the trolley and started a stopwatch. She stopped the stopwatch when the trolley had completed one oscillation.

Describe how the method used by the student could be improved to determine a more accurate value of the time period.

The student displaced the trolley 6.0 cm from the equilibrium position. She recorded the velocity of the oscillating trolley using a sensor connected to a data logger.

The output from the data logger is shown below.

The time-base of the data logger output was set to 250 ms div^–1.

Determine the maximum velocity of the trolley.

Explain the shape of the graph.

Determine the effective spring constant k of the oscillating trolley system.

mass of trolley = 0.87 kg

The student displaced the pendulum through a small angle so that it oscillated. She determined the time period T as 2.50 s.

Calculate the length of the pendulum.

Explain why the amplitude of oscillation of the pendulum did not stay constant.

The student recorded how the amplitude of oscillation varied over time.

It is suggested that the relationship between amplitude A and time t is

where A₀ is the initial amplitude of the oscillation and k is a constant. Explain why a graph of lnA against t would give a straight line.

The table shows the student’s data.

Plot a graph of lnA against t on the grid opposite. Use the additional column to show your processed data.

Determine values for A₀ and k.

A garden ornament consists of a metal flamingo suspended from a spring as shown. The spring is hung from a support using the hook.

The mass of the flamingo is 65 g. When the flamingo is suspended vertically the spring extends by 8.5 cm.

The flamingo is pulled downwards by a small extra displacement and then released. The flamingo undergoes simple harmonic motion vertically.

The instructions state that the flamingo will oscillate with a frequency of 2.5 Hz.

Deduce whether this statement is correct.

After being set into vertical oscillation, the flamingo comes to rest after a short time.

Explain why the flamingo comes to rest.

In a slight breeze the flamingo swings from side to side and behaves as a simple pendulum.

Show that the period of oscillation of the flamingo pendulum is about 2.2 s.

pendulum length = 1.25 m

The amplitude of oscillation of the flamingo pendulum is 7.5 cm.

Calculate the maximum velocity of the flamingo pendulum.

A damped mass-spring system is driven into oscillation. The graph shows the amplitude of oscillation as the driving frequency is varied.

The damping is decreased.

Which row of the table describes what happens to the maximum amplitude of oscillation and the driving frequency at which this occurs?

A damped mass-spring system is driven into oscillation. The graph shows the amplitude of oscillation as the driving frequency is varied.

The damping is decreased.

Which row of the table describes what happens to the maximum amplitude of oscillation and the driving frequency at which this occurs?

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

A student investigated the behaviour of a spring under tension. The spring was hung vertically with a mass holder attached.

The position of the bottom of the mass holder was recorded. The spring was stretched by adding masses to the mass holder and the new positions were recorded. The extension of the spring each time was calculated.

The student produced the following table.

Criticise the student’s table.

The student used her data to plot a graph as shown.

Determine a value for the force constant k of the spring.

Two identical springs are joined in series and a bar magnet is hung from one end as shown.

The bar magnet is displaced a small distance vertically from its equilibrium position and released.

Calculate the frequency at which the system oscillates.

mass of magnet = 120 g spring constant of each spring = 22 N m⁻¹

*(d) Identical bar magnets are suspended from identical springs, with the North pole of each magnet inside a coil of wire as shown. The two coils are connected together with conducting leads.

Magnet A is displaced so that it oscillates vertically. The North pole of magnet A moves into and out of the coil of wire with simple harmonic motion. As this motion continues, magnet B starts to oscillate. The amplitude of oscillation of magnet B increases over time.

Explain why magnet B starts to oscillate with an increasing amplitude.