What does the minus sign in a = -\omega^2 x show?

The acceleration is always in the opposite direction to the displacement, towards equilibrium.

Where is the speed maximum and what is its value?

At equilibrium, v{max} = \omega A.

Where is the acceleration maximum and what is its value?

At the extremes of the motion, a{max} = \omega^2 A.

How is the v–t graph obtained from the x–t graph?

The velocity is the gradient of the displacement–time graph.

How is the a–t graph obtained from the v–t graph?

The acceleration is the gradient of the velocity–time graph.

What is the phase relationship between a and x?

Antiphase: they are half a cycle (π rad) out of phase.

What does a graph of a against x look like for SHM?

A straight line through the origin with negative gradient -\omega^2.

Simple harmonic motionAQA A-Level Physics: Flashcards

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State the condition for simple harmonic motion.

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State the condition for simple harmonic motion.: The acceleration is directly proportional to the displacement from equilibrium and directed towards equilibrium: $a \propto -x$ .
State the defining equation of SHM.: $a = -\omega^2 x$
What does the minus sign in $a = -\omega^2 x$ show?: The acceleration is always in the opposite direction to the displacement, towards equilibrium.
How is the angular frequency related to the frequency and period?: $\omega = 2\pi f = 2\pi/T$
What is the relationship between time period and frequency?: $T = 1/f$
Give the equation for displacement when the motion starts at maximum displacement.: $x = A\cos(\omega t)$
Give the equation for the speed at displacement x.: $v = \pm\omega\sqrt{A^2 - x^2}$
Where is the speed maximum and what is its value?: At equilibrium, $v_{max} = \omega A$ .
Where is the acceleration maximum and what is its value?: At the extremes of the motion, $a_{max} = \omega^2 A$ .
How is the v–t graph obtained from the x–t graph?: The velocity is the gradient of the displacement–time graph.
How is the a–t graph obtained from the v–t graph?: The acceleration is the gradient of the velocity–time graph.
What is the phase relationship between a and x?: Antiphase: they are half a cycle (π rad) out of phase.
What does a graph of a against x look like for SHM?: A straight line through the origin with negative gradient $-\omega^2$ .

Exam questions on Simple harmonic motion

A buoy bobs vertically in a regular swell with simple harmonic motion of amplitude 0.40 m and time period 5.0 s.
Calculate the maximum acceleration of the buoy.2 marks
The cone of a loudspeaker moves backwards and forwards with simple harmonic motion at a frequency of 250 Hz and an amplitude of 0.50 mm. The cone is at its maximum positive displacement at t = 0.
Calculate the displacement of the cone at t = 0.50 ms.2 marks
A particle moves along a straight line with simple harmonic motion. Its displacement x, in metres, from its equilibrium position at time t, in seconds, is given by x = 0.060 cos(8.0πt).
Determine the amplitude, the frequency and the time period of the motion.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).