Mass-spring systems & simple pendulums

On the International Space Station (ISS), astronauts measure their mass once a month using a Body Mass Measurement Device (BMMD).

The BMMD is constructed from a large spring attached to the floor of the ISS, with a platform and handles attached to the spring. The spring is compressed and the astronaut puts his body onto the platform and holds onto the handles, as shown.

(Source: https://www.nasa.gov/content/nasa-astronaut-rick-mastracchio-3)

The spring is released and the astronaut and platform oscillate with simple harmonic motion.

An astronaut used the BMMD. The frequency of oscillation was 0.34 Hz.

Determine the mass of the astronaut.

spring constant = 350 N m^–1 mass of platform = 5.7 kg

The distance between the upper and lower points of the first oscillation is 0.29 m.

Calculate the magnitude of the maximum acceleration of the astronaut.

Calculate the speed of the astronaut 3.5 s after the start of the oscillations.

A student used springs to attach a trolley between two fixed supports, as shown.

When displaced horizontally, the trolley oscillated with simple harmonic motion.

To determine the time period T of oscillation of the trolley, the student displaced the trolley from its equilibrium position and released it. As she released the trolley, she started a stopwatch. She stopped the stopwatch when the trolley had returned to its starting point.

Explain how the procedure used by the student to determine T could have been improved.

The mass of the trolley was M. The student added a small mass m to the trolley and determined the new value of T. She repeated the procedure for a range of values of m.

She plotted a graph of T² against m.

Explain how she could use her graph to determine a value for M.

*(a) Describe how the principle of energy conservation and the principle of momentum conservation apply to this situation.

Consider the situation from the instant the spring gun is released to the instant the bob reaches its maximum height.

When the modelling clay hits the pendulum bob, the pendulum swings to one side.

Show that the time taken for the pendulum bob to move from its lowest position to its highest position is about 0.6 s.

effective length of pendulum = 1.25 m

The pendulum bob was then attached to a spring of stiffness 0.12 N m^–1. When the bob was displaced vertically and released, it oscillated with a time period equal to that of the pendulum.

Calculate the mass of the pendulum bob.

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 simple pendulum consisting of a thread and a bob is set up next to a horizontal rod.

The bob is displaced to the left and released. When the bob reaches the equilibrium position the thread strikes the horizontal rod. For half of the cycle, only the lower part of the pendulum moves.

The diagram shows the swing of the pendulum.

The diagram below shows the dimensions of the pendulum.

Determine the frequency of the oscillations of the pendulum.

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

The student measured the length of the spring as he added masses to the holder. The rule was held as shown to measure the distance between the top and bottom coils of the spring. He determined the extension for each value of total mass on the holder. He did this by subtracting the original length of the spring from each extended length.

Explain whether this method would produce accurate values for the extensions of the spring.

Explain how the student could modify his method in order to obtain more accurate values for the extensions of the spring.

In another experiment, the student displaced the mass vertically each time a mass was added to the spring. He used a stopwatch to determine the period of vertical oscillations of each mass.

The student used his data to plot a graph of T² against m as shown.

The student expected the graph to be a straight line through the origin. He thought that there may be systematic error due to reaction time.

Give an example of another possible systematic error in this experiment.

Another student suggests that to reduce the uncertainty in the value for the period, a data logger connected to a light gate could be used to measure time.

Comment on the student’s suggestion.

Determine a value for the stiffness of the spring.

When determining the period of oscillation for each mass, the student measured the time for 20 oscillations. He repeated this measurement to obtain a mean time for 20 oscillations.

Explain how the student’s procedure contributed to the accuracy of the measurement.

A pendulum of length l with a bob of mass m oscillates with frequency f.

What is the frequency of a pendulum of length 4l with a bob of mass 2m?

A student measured the deflection of a mass attached to the end of a thin strip of metal. The strip was clamped to a bench at one end as shown.

The student varied the force on the end of the strip by changing the mass attached.

The deflection was measured each time when the mass was in its equilibrium position.

The student obtained the following graph of deflection against force.

State why the mass will oscillate with simple harmonic motion when it is displaced slightly from its equilibrium position and released.

The student then investigated the oscillations of the mass on the metal strip. The student fixed different numbers of 10 g masses to the end of the metal strip.

The student noticed that the smaller the mass the higher the frequency of the oscillations. He estimated that the maximum number of oscillations he could count was two per second. He decided that the smallest mass he should use was 50 g.

Determine whether 50 g is the smallest mass he should use.

You may assume that the system acts in the same way as a mass on a spring.

A student measured the deflection of a mass attached to the end of a thin strip of metal. The strip was clamped to a bench at one end as shown.

The student varied the force on the end of the strip by changing the mass attached.

The deflection was measured each time when the mass was in its equilibrium position.

The student obtained the following graph of deflection against force.

State why the mass will oscillate with simple harmonic motion when it is displaced slightly from its equilibrium position and released.

The student then investigated the oscillations of the mass on the metal strip. The student fixed different numbers of 10 g masses to the end of the metal strip.

The student noticed that the smaller the mass the higher the frequency of the oscillations. He estimated that the maximum number of oscillations he could count was two per second. He decided that the smallest mass he should use was 50 g.

Determine whether 50 g is the smallest mass he should use.

You may assume that the system acts in the same way as a mass on a spring.

A student carried out an experiment with a pendulum hung from a fixed support. The fixed support was a distance h above floor level as shown.

As the student was unable to measure the length of the pendulum directly, she measured the distance d from the bob to the floor.

To determine the period T of the pendulum, the student used the following method:

• release the bob from its highest position and start a stopwatch • stop the stopwatch when the bob reaches the same position again.

Criticise the student’s method for measuring the period.

The student used her data to plot a graph of T² against d as shown below.

Determine a value for the acceleration due to gravity g.

A student carried out an experiment with a pendulum hung from a fixed support. The fixed support was a distance h above floor level as shown.

As the student was unable to measure the length of the pendulum directly, she measured the distance d from the bob to the floor.

To determine the period T of the pendulum, the student used the following method:

• release the bob from its highest position and start a stopwatch • stop the stopwatch when the bob reaches the same position again.

Criticise the student’s method for measuring the period.

The student used her data to plot a graph of T² against d as shown below.

Determine a value for the acceleration due to gravity g.

Show that the length of the wire supporting the sphere is about 10m.

A student is using a simple pendulum to determine a value for the acceleration of free fall g.

She measures the length l of the pendulum four times with a metre rule and records the following values.

She calculates the mean length l_m of the pendulum using the following method:

Calculate a more accurate value for l_m.

Determine the time period of the oscillations of this pendulum, using your^m calculated value for l_m.

She sets the pendulum into oscillations with small amplitude and uses a stopwatch to determine the time period.

The student releases the pendulum at A and simultaneously starts the stopwatch. She measures the time taken for 5 oscillations and divides the value by 5. She repeats the procedure twice and calculates a mean time period.

Explain two modifications to the student’s method that would improve the value obtained for the time period.

A student set up a “seconds pendulum”. This is a simple pendulum for which the time taken to move from the bob’s highest position on one side to its highest position on the opposite side is 1.00 s.

Calculate the length L required for the pendulum to be a “seconds pendulum”.

The student set the pendulum into oscillation. She used a stopwatch to check the accuracy of the pendulum’s period T.

Describe the procedure the student should have used to obtain an accurate value for T.

Another student suggested that the uncertainty in the measurement of the time period of the pendulum could be reduced by using a light gate and a data logger. The data logger would record the time between successive interruptions of the light beam. Both the data logger and the stopwatch have a resolution of 0.01 s.

Comment on the student’s suggestion of using a data logger rather than a stopwatch.

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.