What does the gradient of a velocity–time graph give?

The acceleration at that point.

Where is the gradient of the displacement–time graph steepest?

At the equilibrium position, where the speed is a maximum.

How does the velocity graph compare with the displacement graph?

It is a quarter of a cycle ahead.

How does the acceleration graph compare with the displacement graph?

It is inverted (half a cycle out of phase), since a = −ω²x.

What is the maximum gradient of the velocity–time graph?

The maximum acceleration, Aω².

What is the velocity when the acceleration is a maximum?

Zero, at the ends of the oscillation.

What is the acceleration when the velocity is a maximum?

Zero, at the equilibrium position.

How can you tell whether an oscillating object is speeding up?

Its velocity and acceleration have the same sign.

Do the displacement, velocity and acceleration graphs have the same period?

Yes, all three have the same period T = 2π/ω.

Graphs of simple harmonic motionEdexcel International A Level Physics: Flashcards

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What does the gradient of a displacement–time graph give?

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What does the gradient of a displacement–time graph give?: The velocity at that point.
What does the gradient of a velocity–time graph give?: The acceleration at that point.
What is the gradient of the displacement–time graph at maximum displacement?: Zero, so the velocity is zero.
Where is the gradient of the displacement–time graph steepest?: At the equilibrium position, where the speed is a maximum.
What is the shape of the displacement–time graph for SHM?: A cosine (or sine) curve.
How does the velocity graph compare with the displacement graph?: It is a quarter of a cycle ahead.
How does the acceleration graph compare with the displacement graph?: It is inverted (half a cycle out of phase), since a = −ω²x.
What is the maximum gradient of the displacement–time graph?: The maximum velocity, Aω.
What is the maximum gradient of the velocity–time graph?: The maximum acceleration, Aω².
What is the velocity when the acceleration is a maximum?: Zero, at the ends of the oscillation.
What is the acceleration when the velocity is a maximum?: Zero, at the equilibrium position.
How can you tell whether an oscillating object is speeding up?: Its velocity and acceleration have the same sign.
Do the displacement, velocity and acceleration graphs have the same period?: Yes, all three have the same period T = 2π/ω.

Exam questions on Graphs of simple harmonic motion

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.
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
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.
Calculate the gradient of the velocity–time graph at t = 0 and state what this gradient represents.2 marks
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.
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

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).