Graphs of simple harmonic motionEdexcel International A Level Physics: Subtopic test

1. 1
   A mass hangs from a spring and oscillates vertically with simple harmonic motion, with an amplitude of 5.0 cm and a period of 2.0 s. A data logger plots the displacement of the mass against time, taking upwards as positive. At t = 0 the mass is at its maximum upward displacement, so the displacement–time graph is a cosine curve.

   (a)

   At which time is the gradient of the displacement–time graph equal to zero?

   [1 mark]

   • A0.50 s
   • B1.5 s
   • C1.0 s
   • D0.75 s

   (b)

   What is the maximum magnitude of the gradient of the graph, and where does the mass have this speed?

   [1 mark]

   • A0.16 m s⁻¹, at a displacement of ±5.0 cm
   • B0.16 m s⁻¹, at the equilibrium position
   • C0.025 m s⁻¹, at the equilibrium position
   • D0.050 m s⁻¹, at a displacement of ±5.0 cm

   (c)

   Use the gradient of the displacement–time graph to describe how the velocity of the mass changes between t = 1.0 s and t = 2.0 s.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For the same oscillating mass (amplitude 5.0 cm, period 2.0 s, released from maximum upward displacement at t = 0), the data logger also plots velocity against time, with upwards as positive. The velocity is given by v = −0.157 sin(πt), where v is in m s⁻¹ and t is in s, so the velocity–time graph is a negative sine curve.

   (a)

   How does the velocity–time graph compare with the displacement–time graph for the same oscillation?

   [1 mark]

   • AThe velocity graph is a quarter of a cycle ahead of the displacement graph
   • BThe velocity graph is half a cycle ahead of the displacement graph
   • CThe two graphs are in phase
   • DThe velocity graph is a quarter of a cycle behind the displacement graph

   (b)

   At which time does the velocity–time graph have its steepest gradient?

   [1 mark]

   • A0.50 s
   • B1.5 s
   • C0.75 s
   • D1.0 s

   (c)

   Calculate the gradient of the velocity–time graph at t = 0 and state what this gradient represents.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A trolley oscillates horizontally between two springs with simple harmonic motion. Its amplitude is 8.0 cm and its period is 1.6 s. It is at its maximum positive displacement at t = 0, so the graph of displacement against time is a cosine curve.

   (a)

   Describe how the gradient of the displacement–time graph changes during one complete oscillation, and say what this shows about the velocity of the trolley at t = 0, 0.40 s and 0.80 s.

   [3 marks]

   (b)

   Calculate the maximum gradient of the displacement–time graph and the maximum gradient of the velocity–time graph, stating what each represents.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student displays three graphs on a data logger for a trolley oscillating between two springs: displacement against time, velocity against time and acceleration against time. The amplitude is 0.12 m and the period is 0.80 s, and the trolley is released from its maximum positive displacement at t = 0.

   (a)

   Explain how the three graphs are related to each other, including their phase relationships.

   [6 marks]

   (b)

   Calculate the velocity and the acceleration of the trolley at t = 0.10 s, and use their signs to explain whether the trolley is speeding up or slowing down at that instant.

   [6 marks]

   Total for question 4: 12 marks

End of questions