Displacement, velocity and acceleration graphs for SHMEdexcel A-Level Physics: Subtopic test

1. 1
   A trolley on a horizontal spring oscillates with simple harmonic motion. Its amplitude is 0.080 m and its period is 0.50 s. At t = 0 it is at its maximum positive displacement, so its displacement is x = 0.080 cos ωt. A student plots graphs of its displacement, velocity and acceleration against time.

   (a)

   At which time is the gradient of the displacement–time graph zero for the first time after the trolley is released?

   [1 mark]

   • A0.125 s
   • B0.375 s
   • C0.250 s
   • D0.0625 s

   (b)

   What is the velocity of the trolley at t = 0.125 s?

   [1 mark]

   • Azero
   • B1.01 m s⁻¹ in the positive direction
   • C0.080 m s⁻¹ in the negative direction
   • D1.01 m s⁻¹ in the negative direction

   (c)

   Calculate the maximum acceleration of the trolley.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A mass on a vertical spring is released from rest at its maximum upward displacement of 0.050 m at t = 0. Upward is taken as positive, and the period of the oscillation is 0.80 s. A graph of its velocity against time is plotted.

   (a)

   What is the gradient of the velocity–time graph at t = 0.40 s?

   [1 mark]

   • AIt is at its maximum positive value
   • BIt is zero
   • CIt is at its maximum negative value
   • DIt is positive but less than its maximum value

   (b)

   At which time is the velocity of the mass at its most negative value?

   [1 mark]

   • A0 s
   • B0.20 s
   • C0.40 s
   • D0.60 s

   (c)

   Calculate the maximum speed of the mass and state the first time at which it occurs.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The displacement x of a pendulum bob from its equilibrium position is given by x = 0.060 cos(πt), where x is in metres and t is in seconds, so the period is 2.0 s. A graph of displacement against time is plotted for one complete cycle.

   (a)

   Calculate the velocity of the bob at t = 0.50 s and explain how this value could be found from the displacement–time graph.

   [3 marks]

   (b)

   For the first complete cycle, state the times at which the velocity of the bob is zero and the times at which its speed is a maximum, and calculate the maximum speed.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A mass of 0.25 kg oscillates on a spring with an amplitude of 0.12 m and a period of 1.6 s. It is released from rest at its maximum positive displacement at t = 0. A student draws graphs of displacement, velocity and acceleration against time for two complete cycles.

   (a)

   Describe the shapes of the displacement–time, velocity–time and acceleration–time graphs for the mass, including the maximum values and how the graphs are related by their gradients.

   [6 marks]

   (b)

   Use the equations of motion for SHM to calculate the displacement, velocity and acceleration of the mass at t = 0.20 s, and show that your values are consistent with the gradient relationships between the graphs.

   [6 marks]

   Total for question 4: 12 marks

End of questions