IMAT Physics · Oscillations and Waves

Simple Harmonic Motion and Pendulums

An object pulled back towards a centre by a force proportional to its distance from it swings to and fro with a fixed period: simple harmonic motion.

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Why this matters

Five of the 6 past questions are on this page, two of them from the 2024 ministry paper: spotting which motion is simple harmonic, where the kinetic and potential energy peak, a pendulum's length from its timing, a velocity from a cosine law of motion, and what friction does to a swing.

Concept 1 of 3: Simple harmonic motion: displacement, velocity and acceleration

Pull a mass on a spring aside and let go. The further it is from the centre, the harder it is pulled back, so it speeds up towards the centre, overshoots, slows down, stops at the far end and comes back. That pattern traces a cosine curve in time. The speed is greatest in the middle and zero at the two ends; the acceleration is the opposite, zero in the middle and greatest at the ends.

Definition

Simple harmonic motion (SHM): the acceleration is proportional to the displacement from a fixed centre and always points towards it: $a = -\omega^2 x$ .
Amplitude $A$ : the largest displacement. Period $T$ : the time for one full cycle. Frequency $f = 1/T$ , in hertz.
Angular frequency $\omega = 2\pi f = 2\pi/T$ , in rad/s.
Starting from an end: $x = A\cos(\omega t)$ and $v = -A\omega\sin(\omega t)$ . The speed is greatest, $v_{\max} = A\omega$ , at the centre, and zero at $x = \pm A$ .
Not every repeated motion is SHM. A ball bouncing on the floor feels a constant force (its weight) between bounces, not one proportional to displacement, so it is periodic but not simple harmonic.

SHM from an end position

x = A\cos(\omega t) \qquad v = -A\omega\sin(\omega t) \qquad a = -\omega^2 x \qquad \omega = \frac{2\pi}{T}

$x$ displacement from the centre, in m
$A$ amplitude, in m
$\omega$ angular frequency, in rad/s
$T$ period, in s

Worked example

A point moves along a line with

x = 0.05\cos(4\pi t)

, with

x

in metres and

t

in seconds. Find the amplitude, period, frequency and greatest speed, and the velocity at

t = 0.125\ \text{s}

Practice this conceptself-check · 3 quick reps

Periodic is not the same as simple harmonic

SHM needs a restoring force proportional to the displacement, giving a sine or cosine motion. A bouncing ball, or a puck sliding at steady speed between two walls, repeats itself but is not SHM.

Velocity is zero at the ends, not at the centre

x = \pm A

the object stops for an instant and turns round, so

v = 0

there while the acceleration is largest. When

\cos(\omega t) = \pm 1

\sin(\omega t) = 0

, so the velocity is zero.

Concept 2 of 3: Energy in simple harmonic motion and damping

An oscillator keeps swapping energy between motion and storage. At the centre it is all kinetic; at the ends the object is momentarily still and it is all potential. Without friction the total never changes, so the motion goes on for ever. Friction drains the total away, so the swings get smaller: the motion is damped.

Definition

Total energy of a mass on a spring: $E = \tfrac{1}{2}kA^2$ , constant if there is no friction.
At displacement $x$ : potential energy $\tfrac{1}{2}kx^2$ , kinetic energy $E - \tfrac{1}{2}kx^2$ .
Kinetic energy is greatest at the centre; potential energy is greatest at both ends. For a mass hanging on a vertical spring, measure from the equilibrium position: the total potential energy (elastic plus gravitational) is then greatest at the top and at the bottom.
Energy goes as amplitude squared: double the amplitude, four times the energy.
Damping (friction, air resistance) removes energy, so the amplitude shrinks and the motion stops. The period stays almost the same for light damping. With no friction there is no damping.

Energy of a mass on a spring

E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}m v_{\max}^2

$k$ spring constant, in N/m
$A$ amplitude, in m
$m$ oscillating mass, in kg
$v_{\max}$ speed at the centre, in m/s

Worked example

A 0.50 kg mass on a spring with

k = 200\ \text{N/m}

oscillates with an amplitude of 0.10 m on a smooth table. Find the total energy, the greatest speed, and the kinetic energy when the mass is 0.05 m from the centre.

Practice this conceptself-check · 3 quick reps

Without friction an oscillator never stops

Nothing removes energy from a frictionless oscillator, so it keeps swinging with the same amplitude for ever. Statements that a frictionless pendulum slows down and stops are false; stopping is what damping does.

Concept 3 of 3: The periods of a simple pendulum and a mass on a spring

A longer pendulum takes longer to swing, and stronger gravity pulls it round faster. The mass of the bob does not matter, for the same reason all objects fall together. For a spring, a heavier mass is slower to get moving and a stiffer spring pulls harder, and gravity does not enter at all.

Definition

Simple pendulum (small swings): $T = 2\pi\sqrt{L/g}$ . It does not depend on the mass of the bob or on the amplitude, as long as the swing is small.
Mass on a spring: $T = 2\pi\sqrt{m/k}$ . It does not depend on $g$ , so it is the same on the Moon.
A pendulum passes through the vertical twice in each period, so the time between two passes is $T/2$ .
The bob of a pendulum moves along an arc of a circle. IMAT often takes $g = 10\ \text{N/kg}$ and $\pi^2 \approx 10$ .

Periods of the two standard oscillators

T_{\text{pendulum}} = 2\pi\sqrt{\frac{L}{g}} \qquad T_{\text{spring}} = 2\pi\sqrt{\frac{m}{k}}

$L$ length of the pendulum, in m
$g$ gravitational field strength, in N/kg
$m$ mass on the spring, in kg
$k$ spring constant, in N/m

Worked example

Find the period of a pendulum 0.40 m long, and of a 0.50 kg mass on a spring with

k = 50\ \text{N/m}

. Take

g = 10\ \text{N/kg}

Practice this conceptself-check · 4 quick reps

Passing the centre twice per period

A pendulum crosses its lowest point once going each way, so successive crossings are half a period apart. If a question gives the time between crossings, double it before using

T = 2\pi\sqrt{L/g}

A heavier bob does not change a pendulum's period

Mass cancels out of a pendulum's motion. It does matter for a mass on a spring, where

T \propto \sqrt{m}

. Mixing up the two formulas is a common slip.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Simple harmonic motion: displacement, velocity and acceleration
SHM from an end position
$x = A\cos(\omega t) \qquad v = -A\omega\sin(\omega t) \qquad a = -\omega^2 x \qquad \omega = \frac{2\pi}{T}$
Energy in simple harmonic motion and damping
Energy of a mass on a spring
$E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}m v_{\max}^2$
The periods of a simple pendulum and a mass on a spring
Periods of the two standard oscillators
$T_{\text{pendulum}} = 2\pi\sqrt{\frac{L}{g}} \qquad T_{\text{spring}} = 2\pi\sqrt{\frac{m}{k}}$

Watch out for (5)

Periodic is not the same as simple harmonic
→ Simple harmonic motion: displacement, velocity and acceleration
Velocity is zero at the ends, not at the centre
→ Simple harmonic motion: displacement, velocity and acceleration
Without friction an oscillator never stops
→ Energy in simple harmonic motion and damping
Passing the centre twice per period
→ The periods of a simple pendulum and a mass on a spring
A heavier bob does not change a pendulum's period
→ The periods of a simple pendulum and a mass on a spring

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