Mass-spring systems and simple pendulums

A mass–spring system is undergoing simple harmonic motion (SHM). The maximum velocity of the mass is 0.39 m s^–1 and the period of oscillation is 0.81 s

Draw, on Figure 4, a graph of acceleration against displacement for the system. Label each axis with a suitable unit and scale.

Label, with a P, a point on your graph where the mass has 50% of its maximum

A pendulum is oscillating with simple harmonic motion.

Which row gives the momentum of the pendulum bob and its gravitational potential energy (GPE) when the acceleration of the pendulum bob is zero?

A spring with a spring constant of 50 N m⁻¹ is attached to a rigid horizontal bar. An object of mass m is attached to the bottom of the spring.

When the bar is made to move vertically up and down with a frequency of 2.6 Hz, the mass–spring system undergoes resonance. Ignore the effects of damping.

What is m ?

A simple pendulum of length 8.0 cm oscillates on Earth with a frequency f₁. An identical pendulum on the Moon oscillates with a frequency f₂. The acceleration due to gravity is 1.6 m s⁻² on the surface of the Moon.

This question is about the vertical oscillations of a test tube in a liquid.

Figure 10 shows some wire wound tightly around a test tube. The ends of the wire are secured with tape.

P and Q are points marked on the wire, as shown in the enlarged view in Figure 10. A travelling microscope with a vernier scale is used to measure the distance between P and Q. The microscope reading at P is 9.26 mm.

Figure 11 shows the microscope reading at Q.

Show that the wire has a thickness of approximately 1.2 mm.

The external diameter of the test tube is 18.0 mm.

Deduce the length of wire between P and Q.

A student investigates vertical oscillations of the test tube in a liquid.

Figure 12 shows the test tube and wire floating at rest in a beaker of dark-coloured oil. The part of the test tube below the surface cannot be seen clearly through the oil.

The test tube is then pushed downwards as far as possible, as shown in Figure 13.

The test tube is now released from the position shown in Figure 13. The test tube oscillates vertically with simple harmonic motion of period T.

A horizontal fiducial mark is used to reduce uncertainty in the measurement of T. The fiducial mark is to be placed level with one of the positions A, B, C or D shown in Figures 12 and 13.

Identify the correct position for the fiducial mark. Go on to explain why you chose this position.

When the total mass of the test tube, wire and tape is constant, it can be shown that:

When the test tube oscillates in the oil, T is 0.587 s.

The test tube is cleaned and placed in an identical beaker containing milk. When the test tube oscillates in the milk, T is 0.534 s.

Determine the density of the milk.

Due to damping, only a small number of oscillations occurs after the test tube is released.

The student wants to produce a greater number of oscillations for a particular liquid without changing T.

State and explain two ways in which the apparatus can be modified to achieve this.

When a mass M is suspended from a spring, the spring extends by a distance x. M is displaced vertically and, when released, it oscillates with a period T. M is removed and suspended from a different spring. The spring extends by a distance M is again displaced vertically and released.

What is the new period of oscillations of M ?

A mass–spring system and a simple pendulum have identical periods of oscillation T when at the surface of the Earth. Both are taken to planet X where the acceleration due to gravity is What are the periods of the mass–spring system and the simple pendulum on X?

A mass, attached to two springs, oscillates horizontally between P and Q. The motion of the system is simple harmonic.

Which quantity has its magnitude at a minimum value when the mass is at Q ?

A teacher sets up a demonstration to show the relationship between circular motion and simple harmonic motion (SHM). She places a block on a turntable at a point 0.25 m from its centre, as shown in Figure 9.

The turntable rotates with an angular speed of 1.8 rad s^–1 and the block does not slip.

Calculate the time taken for the turntable to complete one revolution.

Figure 10 shows a plan view of the turntable and block. The turntable rotates in a clockwise direction.

Draw an arrow on Figure 10 to show the direction of the resultant force on the block.

The mass of the block is 0.12 kg.

Calculate the magnitude of the resultant force on the block.

Describe, with reference to one of Newton’s laws of motion, the evidence that a resultant force is acting on the block.

The teacher adjusts the angular speed of the turntable so that the block completes one rotation every 2.50 s. She sets up a simple pendulum above the centre of the turntable so that it swings in phase with the movement of the block.

Calculate the length of the simple pendulum.

A lamp is used to project shadow images of the block and pendulum bob on a screen. Both shadows appear to move with SHM across the screen. Figure 11 shows the images on the screen at one instant.

Initially the shadows move in phase with the same amplitude.

Air resistance affects the motion of the pendulum.

Suggest the effect this has on the amplitude relationship and the phase relationship between the moving shadows.

A simple pendulum and a mass−spring system each have a time period T on the Earth. They are taken to the surface of a planet where the acceleration due to gravity is

What are the time periods of the pendulum and the mass−spring system on this planet?

A particle of mass m is oscillating with simple harmonic motion. The period of the oscillation is T and the amplitude is A.

What is the maximum kinetic energy of the particle?

A particle of mass m undergoes simple harmonic motion with amplitude A and frequency

What is the total energy of the particle?

A mass of 0.90 kg is suspended from the lower end of a light spring of stiffness 80 N m⁻¹.

When the mass is displaced vertically and released, it undergoes vertical oscillations of small amplitude.

What is the frequency of the oscillations?

The period of a simple pendulum is doubled when the pendulum length is increased by 1.8 m.

What is the original length of the pendulum?

A simple pendulum performs oscillations of period T in a vertical plane.

Figure 1 shows views of the pendulum at the equilibrium position and at the instant of release. Figure 1 also shows a rectangular card marked with a vertical line.

The card can be used as a fiducial mark to reduce uncertainty in the measurement of T.

Annotate Figure 1 to show a suitable position for the fiducial mark. Explain why you chose this position.

The period of the pendulum is constant for small-amplitude oscillations. Figure 2 shows an arrangement used to determine the maximum amplitude that can be considered to be small, by investigating how T varies with amplitude.

Describe a suitable procedure to determine A_R, the amplitude of the pendulum as it is released. You may add detail to Figure 2 to illustrate your answer.

Figure 3 shows some of the results of the experiment.

Estimate, using Figure 3, the expected percentage increase in T when A_R increases from 0.35 m to 0.70 m. Show your working.

In another experiment the pendulum is released from a fixed amplitude. The amplitudes A_n of successive oscillations are recorded, where n = 1, 2, 3, 4, 5....

Table 1 shows six sets of readings for the amplitude A₅.

Determine the result that should be recorded for A₅. Go on to calculate the percentage uncertainty in this result.

Table 2 shows results for A_n and the corresponding value of ln(A_n / m) for certain values of n.

Complete Table 2.

Plot on Figure 4 a graph of ln(A_n / m) against n.

It can be shown that

where A₀ is the amplitude of release of the pendulum δ is a constant called the damping factor.

Explain how to find δ from your graph. You are not required to determine δ.

The graph shows the variation of displacement d with time t for a particle moving with simple harmonic motion of period T.

Which graph shows the variation of kinetic energy E_k of the particle with time?

Two pendulums A and B oscillate with simple harmonic motion. The time period of A is 2.00 s and the time period of B is 1.98 s.

A and B are released in phase.

What is the number of oscillations of A before A and B are next in phase?

The frequency of oscillation of a vertical spring is f when the mass hanging from the spring is m.

What is the relationship between f and m ?

An object of mass 0.15 kg performs simple harmonic motion. It oscillates with amplitude 55 mm and frequency 0.80 Hz

What is the maximum value of its kinetic energy?

Which graph shows how the gravitational potential energy E_p of a simple pendulum varies with displacement s from the equilibrium position?

The vibration generator is connected to a signal generator. The signal generator is used to vary the frequency of vibration of the metal rod. When the signal generator is set at 2.0 Hz, the mass attached to spring A oscillates with a maximum amplitude of 2.5 × 10^–2 m and has a maximum kinetic energy of 54 mJ.

Deduce the spring constant k for spring A and the mass m suspended from it.

Calculate the frequency at which the mass attached to spring B oscillates with maximum amplitude.

Figure 11 shows how the amplitude of the oscillations of the mass varies with frequency for spring A.

The investigation is repeated with the mass attached to spring B immersed in a beaker of oil.

A graph of the variation of the amplitude with frequency for spring B is different from the graph in Figure 11.

Explain two differences in the graph for spring B.

Which statement is not correct?

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