AQA A Level Physics

Topic Questions

6.2 Simple Harmonic Motion

1a2 marks

State the conditions necessary for an object to be in simple harmonic motion.

1b3 marks

The defining equation of simple harmonic motion is 

         a = – ω2x 

State the definition of each variable and an appropriate unit for each.

1c4 marks

A student is confirming whether a particular pendulum oscillates with simple harmonic motion. They obtain a graph of their results that represents the equation in part (b). 

Sketch this graph on the axes in Figure 1 and label the axes. State what the maximum and minimum values on the x–axis of the graph represent. 

Figure 1

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1d2 marks

The students calculate the gradient of the graph from part (c) to be – 8.5. 

Calculate the angular frequency of the pendulum.

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2a1 mark

Describe what is meant by the time period of an oscillation.

2b2 marks

A mass on a spring start oscillating from its equilibrium position. Time t = 0 s is measured from where the mass starts moving in the negative direction. 

There are three graphs shown in Table 1 that represents the motion of the mass. The first graph represents the acceleration-time graph of the motion. 

Table 1

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Complete Table 1 to show the correct variable on the y–axis of each graph.

2c1 mark

State how many complete cycles of oscillations of the mass from the graphs in Table 1.

2d3 marks

The mass of the spring is equal to 60 g and the spring has a spring constant of 0.78 N m–1. 

Calculate the total time taken for the oscillations shown in the graphs in Table 1.

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3a3 marks

Fran is experimenting how the displacement x of a simple pendulum consisting of a mass on a string varies with time t in a vacuum. She displaces the pendulum, then starts the timer. Figure 1 is a graph of x against t from a data logger. 

Figure 1

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Calculate the frequency of these oscillations.

3b3 marks

Calculate the length of the string of the pendulum Fran uses in the experiment.

3c4 marks

Using Figure 1, calculate the maximum velocity of the pendulum

3d3 marks

Fran is watching a documentary on the Moon landing and begins to wonder how the time period of the same pendulum would change on the Moon, in comparison to on Earth. 

She concludes that its time period will be smaller because there is no air resistance on the Moon, so the pendulum would oscillate faster. 

State whether or not Fran is correct. Explain your answer clearly.

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4a2 marks

Figure 1 shows a young girl swinging on a garden swing. You may assume that the swing behaves as a simple pendulum. Assume that the effect of air resistance is negligible.

Figure 1

6-2-s-q--q4a-easy-aqa-a-level-physics

The time period, T of her oscillations can be calculated using the equation 

            T = 2π square root of l over g end root

where g is the gravitational field strength and the distance from the top of the chains to the centre of mass of the girl. 

State two assumptions that apply when using this time period equation for the girl on the swing.

4b1 mark

Describe the energy changes that take place on the girl during one complete oscillation, starting at her maximum displacement.

4c3 marks

The girl has a mass of 20 kg. When she first sits on the swing, her centre of mass is raised 320 mm from the ground. Her oscillations reach a maximum height of 1.6 m from the ground. 

Calculate the gravitational potential energy gained by the girl at her maximum amplitude compared to when she first sits on the swing.

4d3 marks

Figure 2 shows the axes for energy (E) against displacement (x) graph for the girl on the swing 

Figure 2

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The amplitude of 1.6 m has already been labelled on the graph. 

For half a cycle of the girls’ oscillations: 

  • Sketch the graph for gravitational potential energy. Label this GPE.
  • Sketch the graph for kinetic energy. Label this KE.
  • Add your value calculated for the gravitational potential energy from part (c).

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5a6 marks

Figure 1 shows a mass m on a spring with spring constant k at different points of its oscillation A, B and C. A is where m is at the equilibrium position, whilst B and C are where m is at the positive and negative amplitude respectively.

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Complete Table 1 to show the position, A, B or C which represents each variable at its maximum. You may state more than one position for each variable. 

                           Table 1

Variable at its Maximum

Position

Acceleration

 

Potential energy

 

Speed

 

Displacement

 

Restoring force

 

Kinetic energy

 

 

5b3 marks

A student measures the time period of the mass on the spring over 10 oscillations to decrease the uncertainty in their results. They find that the mass took 12 seconds to complete 10 oscillations. 

Calculate the frequency of the oscillations.

5c4 marks

The mass m attached to the spring has a mass of 85 g. 

Calculate the value of the spring constant k. State an appropriate unit for your answer.

5d3 marks

Calculate the maximum acceleration of the mass m if it reaches a maximum displacement of 13 cm from the equilibrium position.

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1a4 marks

A mass of 0.42 kg is attached to a spring and the system is made to oscillate with simple harmonic motion (SHM) on a horizontal, frictionless surface. The mass passes through the equilibrium position 200 times per minute.  

The kinetic energy of the mass as it passes through the equilibrium position is 500 mJ. There are two points where the restoring force acting on the mass is at its maximum. 

Show that the distance between these points is approximately 29 cm.

1b2 marks

Sketch a graph to show how the velocity of the mass varies with time. Label the graph with any suitable values.

1c2 marks

Find the distance of the mass from the equilibrium position when the speed of the block is 0.8 m s–1

1d3 marks

The experiment is moved to planet X. The gravitational acceleration on planet X is gx. It is known that  ​​g subscript x over g= 2. Another change is that three more identical springs are placed parallel to the original spring. 

Outline, without calculations, how these changes affect the frequency with which the mass oscillates.

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2a3 marks

An experiment is carried out on Planet M using a pendulum of length l and a block of mass m attached to a spring with spring constant k. The block moves horizontally on a frictionless surface. A motion sensor with a lightbulb is placed above the equilibrium position of the block. Every time the block passes the equilibrium position, the lightbulb lights up.  

Figure 1

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The pendulum and the block are displaced from their equilibrium positions and are made to oscillate with simple harmonic motion. The pendulum bob completes 150 full oscillations in seven minutes and the lightbulb lights up once every 0.70 seconds.  

The mass of the block is 320 grams. 

Show that the value of kis approximately 6 N m-1

2b2 marks

Show that l = fraction numerator 4 m g over denominator k end fraction

2c2 marks

The volume of planet M is the same as the volume of the Earth and the density of Planet M is twice of the density of Earth. Calculate the value of l.

2d4 marks

When the acceleration of the pendulum bob is at its maximum, the angle that the string makes with the horizontal is 68°. Find the maximum speed reached by the pendulum bob.

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3a3 marks

A buoy, floating in a vertical tube, generates energy from the movement of water waves on the surface of the sea. When the buoy moves up, a cable turns a generator on the sea bed producing power. When the buoy moves down, the cable is wound in by a mechanism in the generator and no power is produced. 

Figure 1

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The motion of the buoy can be assumed to be simple harmonic. 

A wave of amplitude 3.8 m and wavelength 28 m, moves with a speed of 3.2 m s–1. Calculate the maximum vertical speed of the buoy.

3b3 marks

Use the axes provided in Figure 2 to sketch a graph to show the variation with time of the generator output power. Label the time axis with a suitable scale. 

Figure 2

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3c6 marks

Determining the best location for wave generators, such as the one in Figure 1, is very important. The graph in Figure 3 gives an indication of the relationship between the amplitude of ocean waves and their period. 

Figure 3

6-2-s-q--q3c-hard-aqa-a-level-physics

Discuss the use of wave generators in terms of the potential electrical power output. In your answer you should: 

  • Explain how wave height and wave period affect the energy within a wave
  • Describe how different energy losses in the system might affect the power output

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4a2 marks

A skateboarder is using a piece of equipment called a ‘bowl’, as shown in Figure 1. The bowl is frictionless and hemispherical, with a constant radius of curvature, r. The displacement of the skater from the resting position is x. 

Figure 1

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The bowl has a radius of 4 m. Determine the period of oscillation of the skateboarder.

4b4 marks

Sketch a graph, using the axes provided in Figure 2, to show how the position, velocity and acceleration of the skateboarder vary with time. Include any necessary values. 

Figure 2

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4c2 marks

Determine the frequency of a skateboarder on a similar bowl that is constructed on a different planet P with a gravitational acceleration, gbegin mathsize 14px style p end style = begin mathsize 14px style g over 3 end style, where g is the acceleration due to gravity on Earth.

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5a3 marks

A small metal pendulum bob is suspended at rest from a fixed point with a length of thread of negligible mass. Air resistance is negligible. The pendulum begins to oscillate. Figure 1 shows the variation of kinetic energy of the pendulum bob with time.

Figure 1

6-2-s-q--q5a-hard-aqa-a-level-physics

(i)
Label on the graph with the letter X a point where the speed of the pendulum is half that of its initial speed.
(ii)
Calculate, in m, the length of the thread.
5b3 marks

Figure 2 shows how the kinetic energy of the pendulum varies with displacement.

Figure 2

6-2-s-q--q5b-hard-aqa-a-level-physics

(i)
Sketch on the diagram above a graph to show how the potential energy of the pendulum varies with displacement.
(ii)
Calculate the mass of the pendulum bob.
5c2 marks

Write down the equation that models the variation of position with time for the simple harmonic motion of this pendulum.

5d2 marks

Calculate the maximum force upon the pendulum

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1a2 marks
(i)
Give an equation for the frequency, f, of the oscillations of a simple pendulum in terms of its length, l, and the acceleration due to gravity, g.
(ii)
State the condition under which this equation applies.
1b2 marks

A simple pendulum consists of a 35 g mass tied to the end of a light string 650 mm long. The mass is drawn to one side until it is 10 mm above its rest position, as shown in Figure 1. 

When released it swings with simple harmonic motion. 

Figure 1

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Calculate the frequency of the pendulum.

1c4 marks

Calculate: 

(i)         The maximum speed of the mass during the first oscillation. 

(ii)        The initial amplitude of the oscillations.         

1d3 marks

The pendulum is left to oscillate. 

Figure 2

6-2-s-q--q1d-medium-aqa-a-level-physics

On the axes in Figure 2, sketch a graph to show how the kinetic energy of the simple pendulum varies with time over two complete cycles of the motion. 

Start your graph from the time the pendulum is 10 mm above its rest position. You are not required to mark a scale on either axis.

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2a4 marks

The equation that describes simple harmonic motion is given by 

         a = negative omega squaredx  

(i)         State the meaning of each symbol. 

(ii)        Explain the significance of the negative sign.

2b3 marks

The bob of a simple pendulum, of mass 35 g, swings with an amplitude of 61 mm. It takes 51.2 s to complete 20 oscillations. 

Calculate the length of the pendulum.

2c3 marks

Calculate the magnitude of the restoring force that acts on the bob when at its maximum displacement.

2d4 marks

Figure 1 shows a graph of displacement against time for the pendulum. 

Sketch, on Figure 1, graphs of acceleration against time for the pendulum and the total potential energy against time for the pendulum. 

Figure 1

6-2-s-q--q2d-medium-aqa-a-level-physics

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3a2 marks

Describe the energy changes which take place during one complete oscillation of a vertical mass-spring system, starting when the mass is at its highest point.

3b3 marks

A spring, which obeys Hooke’s law, hangs vertically from a fixed support and requires a force of 3.0 N to produce an extension of 60 mm. A mass of 0.70 kg is attached to the lower end of the spring. 

The mass is pulled down a distance of 10 mm from the equilibrium position and then released. 

Show that the frequency of the simple harmonic vibrations is 1.3 s–1.

3c3 marks

Sketch the displacements of the mass against time on Figure 1, starting from the moment of release and continuing for two full oscillations. Show appropriate time and distance scales on the axes. 

Figure 1

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3d3 marks

(i)         State at which point in the cycle the mass has its maximum acceleration. 

(ii)        Calculate the acceleration of the mass at that point.

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4a3 marks

Figure 1 shows one cycle of the displacement-time graphs for two mass-spring systems X and Y that are performing simple harmonic motion. 

Figure 1

6-2-s-q--q4a-medium-aqa-a-level-physics

The springs used in oscillators X and Y have the same spring constant. 

Using information from Figure 1, determine the relation between the mass used in oscillator Y and that in oscillator X.

4b2 marks

Explain briefly how would you use one of the graphs in Figure 1 to confirm that the motion is simple harmonic.

4c3 marks

Figure 2 shows how the potential energy of oscillator Y varies with displacement. 

Figure 2

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Draw on Figure 2: 

(i)        A graph to show how the kinetic energy of the mass used in oscillator Y varies with its displacement. Label this A. 

(ii)       A graph to show how the kinetic energy of the mass used in oscillator X varies with its displacement. Label this B.

4d3 marks

Use data from the graphs to determine the spring constant of the springs used.

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5a3 marks

A trolley rests on a horizontal surface attached to two identical stretched springs, as shown in Figure 1. 

Each spring has a spring constant of 40 N m–1, can be assumed to obey Hooke’s law, and to remain in tension as the trolley moves. 

Figure 1

6-2-s-q--q5a-medium-aqa-a-level-physics

The trolley is displaced to the right by 50 mm and is then released. The time period T of the oscillation of the trolley is given by

            T =2 straight pi square root of fraction numerator straight m over denominator 2 straight k end fraction end root

where m = mass of the trolley and k = spring constant of one spring. 

When the time period is 0.57s, calculate the value of m, the mass of the trolley.

5b4 marks
(i)
Calculate the magnitude of the maximum resultant force of the trolley at the moment of release. 
(ii)
State the direction of the trolley’s acceleration at this moment. 
5c2 marks

The oscillating trolley performs simple harmonic motion. 

State the two conditions which have to be satisfied to show that a body performs simple harmonic motion.

5d3 marks

Calculate the maximum kinetic energy of the trolley.

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