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Simple harmonic motion

Recognise and solve the SHM equation x¨=−n2x\ddot{x} = -n^2 x; derive the velocity formula v2=n2(A2−x2)v^2 = n^2(A^2 - x^2) using a=vdvdxa = v\frac{dv}{dx}; write displacement in the form x=Asin⁡(nt+α)x = A\sin(nt + \alpha) or x=Acos⁡(nt+α)x = A\cos(nt + \alpha); find amplitude, period, maximum speed and maximum acceleration; solve SHM problems in context.

Worked examples

Reading the parameters from a displacement equation

Straightforward

Problem

A particle has displacement x=6sin⁡(2t−π/3)x = 6\sin(2t - \pi/3) m. Find the (a) amplitude, (b) period, (c) maximum speed, and (d) maximum acceleration.

Deriving and using the velocity–displacement formula

Moderate

Problem

A particle satisfies x¨=−4x\ddot{x} = -4x and starts from rest at x=5x = 5 m. Find its speed when it is at x=3x = 3 m.

Finding displacement as a function of time from initial conditions

Challenging

Problem

A particle starts at the centre of its oscillation (x=0x=0) moving in the positive direction with speed 1010 m/s, in SHM with period π\pi s. Write xx as a function of tt.

Practise

Q1·Straightforward
A particle undergoes simple harmonic motion satisfying x¨=−9x\ddot{x} = -9x. State the value of nn, where x¨=−n2x\ddot{x} = -n^2 x.
Q2·Straightforward
A particle in SHM has displacement x=4sin⁡(3t)x = 4\sin(3t) m. State the amplitude (in metres).
Q3·Straightforward
A particle in SHM has displacement x=4sin⁡(3t)x = 4\sin(3t) m. Find the period (in seconds). Give your answer to two decimal places.
Q4·Straightforward
A particle moves in SHM with x=3cos⁡(2t)x = 3\cos(2t) m. Find its maximum speed (in m/s).
Q5·Moderate
A particle moves in SHM with amplitude 55 m and period 44 s. Find the speed (in m/s) when x=3x = 3 m. Give your answer to two decimal places.
Q6·Moderate
A particle satisfies x¨=−16x\ddot{x} = -16x and starts from rest at x=3x = 3 m. Find its speed (in m/s) when x=1x = 1 m. Give your answer to two decimal places.
Q7·Moderate
A particle has displacement x=5sin⁡ ⁣(2t+π6)x = 5\sin\!\left(2t + \dfrac{\pi}{6}\right) m. Find the velocity (in m/s) at t=0t = 0. Give your answer to two decimal places.
Q8·Straightforward
A particle satisfies x¨+25x=0\ddot{x} + 25x = 0. Find the period of its motion (in seconds). Give your answer to two decimal places.
Q9·Moderate
A particle in SHM has period π2\dfrac{\pi}{2} s and amplitude 33 m. Find the maximum acceleration (in m/s²).
Q10·Challenging
A particle starts at x=0x = 0 moving in the positive direction with SHM: n=2n = 2, amplitude A=4A = 4 m. Find the first positive time tt (in seconds) at which x=2x = 2 m. Give your answer to two decimal places.
Q11·Challenging
A particle in SHM has amplitude 1010 m and maximum speed 55 m/s. Find the period (in seconds). Give your answer to two decimal places.
Q12·Straightforward
A particle in SHM has x=3cos⁡(4t)x = 3\cos(4t) m. Find the acceleration when t=π8t = \dfrac{\pi}{8} s.
Q13·Challenging
A particle moves in SHM satisfying x¨=−n2x\ddot{x} = -n^2 x. Prove that v2=n2(A2−x2)v^2 = n^2(A^2 - x^2), where AA is the amplitude (the maximum displacement from the centre).

✎ Work this one through on paper — proofs are self-assessed.