Simple harmonic motion

Simple harmonic motion (SHM) is an oscillation in which the acceleration is always directed towards a fixed central point and is proportional to the displacement from it: a = −ω²x. A mass bouncing on a spring and a pendulum swinging through a small angle both move this way. Because of that one rule, the displacement, velocity and acceleration all vary like sine or cosine waves, and the time period does not depend on the amplitude.

The condition for simple harmonic motion

AQA A-level Physics gives the condition for SHM as a ∝ −x and the defining equation as:

• x is the displacement from the equilibrium (central) position.
• a ∝ x: the further from the centre, the bigger the acceleration.
• The minus sign: the acceleration always points back towards the centre, opposite to the displacement. The force causing it is called the restoring force.
• ω is the angular frequency in rad s⁻¹, linked to the frequency and the period by ω = 2πf = 2π ÷ T.

Here x is the displacement in metres and y is the acceleration in m s⁻², for an oscillator with ω = 10.5 rad s⁻¹ and amplitude 0.050 m. If plotting acceleration against displacement gives a straight line through (0, 0) that slopes downwards, the motion is simple harmonic, and the slope is −ω².

Simple harmonic motion equations

These are all in AQA A-level Physics section 3.6.1.2. If the oscillation starts from the centre instead of from an end, use x = A sin ωt. Your calculator must be in radians.

Example 1: a mass on a spring

A mass on a spring oscillates with an amplitude of 0.050 m and a period of 0.60 s. Find its maximum speed, its maximum acceleration, and its speed when it is 0.025 m from the centre.

Halfway out, it is still moving at 87% of its top speed: speed falls slowly near the centre and quickly near the ends.

Example 2: displacement at a given time

For the same oscillator, released from x = +0.050 m at t = 0, find the displacement at t = 0.10 s.

After one sixth of a period it is exactly halfway in, because cos(π/3) = ½.

SHM graphs: displacement, velocity and acceleration

In both graphs x is the time in seconds. The first shows the displacement in metres, the second the velocity in m s⁻¹, over two full periods of the oscillator in Example 1. AQA wants you to see that each graph comes from the gradient of the one before:

• Velocity is the gradient of the displacement-time graph. It is zero at the ends (where the x graph is flat) and greatest at the centre (where the x graph is steepest).
• Acceleration is the gradient of the velocity-time graph. It has the same shape as the displacement graph, but flipped upside down, because a = −ω²x.
• The velocity is a quarter of a cycle out of step with the displacement, and the acceleration is half a cycle (exactly opposite).

Check your understanding.

Simple harmonic systems: springs and pendulums

AQA A-level Physics studies two systems in section 3.6.1.3:

• Mass on a spring: m is the mass in kg and k the spring constant in N m⁻¹. A heavier mass or a weaker spring gives a longer period. See the spring constant equation.
• Simple pendulum: l is the length in m and g the gravitational field strength. Changing the mass of the bob makes no difference to the period, and it is only SHM for small angles, where the small-angle approximation sin θ ≈ θ holds. See small angle approximation.

Example 3: two quick periods

A 0.25 kg mass on a spring with k = 40 N m⁻¹ has T = 2π√(0.25 ÷ 40) = 0.497 s. A pendulum with a period of 2.0 s needs a length l = gT² ÷ 4π² = 9.81 × 2.0² ÷ 4π² = 0.994 m, close to a metre.

Watch both systems and change their length, mass and spring constant in the Pendulum and SHM simulator; the pendulum simulation guide has experiments to try.

Energy in simple harmonic motion

As the oscillator moves, energy is transferred back and forth between kinetic energy and potential energy (elastic for a spring, gravitational for a pendulum). With no damping, the total stays constant:

• At the centre: kinetic energy is greatest and potential energy is least.
• At the ends: kinetic energy is zero and potential energy is greatest.
• The total energy is ½mω²A², so it depends on the square of the amplitude.

Kinetic energy and potential energy each go through two full cycles for every one oscillation, because the speed is greatest twice per cycle (once each way through the centre).

Damping

Real oscillations lose energy to their surroundings, usually through friction or air resistance, so the amplitude decreases. This is damping:

• Light damping: the amplitude falls gradually over many oscillations; the period stays almost the same.
• Critical damping: the system settles back to equilibrium as quickly as possible with no overshoot, as in a car's suspension.
• Heavy damping: it returns very slowly, without oscillating.

Common mistakes

• Forgetting the minus sign in a = −ω²x. It is what makes the force a restoring force.
• Calculator in degrees. ωt is in radians.
• Mixing up where maximum speed and maximum acceleration happen. Speed is greatest at the centre; acceleration at the ends.
• Using f instead of ω. Convert first: ω = 2πf.
• Saying the pendulum period depends on mass. It depends only on length and g.
• Using T = 2π√(l/g) for big swings. It only applies to small angles.

Practise with a short timed paper

Simple harmonic motion is in the second year of A-level Physics. The daily A-level Physics: Year 2 paper is a short timed paper of 8 questions, with a new one every day.

Read next

Physics · Waves and lightPendulum simulation: period, energy and damping, liveSwing a virtual pendulum or a mass on a spring, change the length, angle, gravity and damping, and watch the period, the graphs and the energy change.