The model: why it appears everywhere

An oscillation is a repeated motion of a system about a stable position. A periodic motion is a motion that repeats after equal intervals of time. Simple harmonic motion is the neatest periodic model in this topic: idealised, loss-free, and sinusoidal.

Simple harmonic motion is an oscillation in which the acceleration is proportional to the displacement from equilibrium and is directed towards equilibrium. Learn that wording. Then make sure the physics behind it is clear: the further the object moves from the centre, the harder the system accelerates it back.

This model works well because many real systems behave like this, at least for small disturbances. Examples include a mass on a spring, a pendulum at small angle, a vibrating molecule, and the oscillating particles in a wave. More complicated periodic motions can also be built mathematically from sinusoidal motions, which is why the harmonic oscillator appears so often in physics.

The two required conditions

A restoring force is a force that acts on a displaced system in the direction that would return it to its equilibrium position. Simple harmonic motion needs two linked conditions:

• the magnitude of the restoring force, and therefore the acceleration, is directly proportional to the displacement from equilibrium;
• the restoring force, and therefore the acceleration, is directed towards equilibrium, not away from it.

Students often state the second point too loosely. “Proportional to displacement” is not enough on its own: if the acceleration points in the same direction as displacement, the object moves away from equilibrium instead of oscillating.

Use an acceleration–displacement graph as a quick check. For simple harmonic motion, it is a straight line through the origin with a negative gradient. The straight line shows proportionality; the negative gradient shows that acceleration and displacement have opposite signs.

Connections to waves, greenhouse gases and damping

The same oscillator idea comes back later in the wave model. A wave can be treated as many connected particles oscillating locally while energy is transferred through the medium or field. Circular motion gives another useful picture: the shadow, or projection, of uniform circular motion is simple harmonic motion.

Greenhouse gas molecules can be modelled as tiny oscillators because bonds in a molecule can vibrate about stable separations. Infrared radiation can excite these vibrational states. The molecule temporarily stores energy, then re-radiates it in different directions; increasing greenhouse gas concentration increases this absorption and re-radiation, giving the physical basis of the enhanced greenhouse effect.

A damping process is an energy-transfer mechanism that reduces the amplitude of an oscillation with time. Once friction, air resistance or another resistive effect becomes significant, the motion is no longer ideal simple harmonic motion because the amplitude is not constant. It may still be periodic for a while, but it gradually dies away.

The equation and the minus sign

The defining equation is

a = −ω²x, where a is acceleration (m s⁻²), ω is angular frequency (rad s⁻¹, usually written s⁻¹ because the radian is dimensionless) and x is displacement from equilibrium (m).

The minus sign matters. It tells you that acceleration points in the opposite direction to displacement. If the oscillator is to the right of equilibrium, its acceleration is to the left; if it is below equilibrium, its acceleration is upwards. That is the “back towards the centre” condition.

The squared angular frequency matters as well. Since ω² is positive, the negative sign always carries the directional information. The magnitude of the acceleration is ω² times the magnitude of the displacement, so a larger displacement produces a larger acceleration back towards equilibrium.

Reading the equation graphically

From a = −ω²x, a graph of acceleration against displacement has gradient −ω². Experimental data can therefore test whether a system is behaving simply harmonically: plot acceleration against displacement, then look for a straight line through the origin with negative gradient.

In a practical investigation, a motion sensor and data logger can record displacement against time for an oscillator. The software can generate velocity–time and acceleration–time graphs, but the acceleration–displacement plot gives the cleanest check of the defining equation. If you are using ultrasound sensors, keep the motion away from nearby reflecting surfaces, and use small oscillations where the model assumptions are valid.

The language of an oscillator

The equilibrium position is where the net force on the oscillator is zero when it is at rest. For a hanging mass on a spring, this usually isn't the spring's unstretched length. It is the resting position after the weight has stretched the spring.

A displacement is the vector distance of the oscillator from equilibrium, measured in a chosen positive direction. Once that positive direction is chosen, use it consistently for displacement, velocity, acceleration and force.

An amplitude is the maximum displacement from equilibrium during an oscillation. The symbol often used is x₀. Amplitude is a magnitude, so it cannot be negative. Measure it from the middle to one extreme, not from one extreme to the other.

A cycle is one complete repeat of the motion. It ends only when the oscillator has returned to the same position while moving in the same direction. If it starts at equilibrium, for example, then going to one extreme and back through equilibrium is only half the story; the cycle is complete only after the oscillator has reached the other extreme and returned to the starting state.

A time period is the time taken for one complete cycle. A frequency is the number of complete cycles per second.

f = 1/T, where f is frequency (Hz or s⁻¹) and T is time period (s).

For simple harmonic motion, a displacement–time graph is sinusoidal. Read the amplitude from the maximum displacement, and read the period from the time between matching points on successive cycles, such as peak to peak or upward equilibrium crossing to the next upward equilibrium crossing.

Isochronous behaviour and real measurements

An isochronous oscillator has a period that is independent of amplitude. Ideal simple harmonic motion is isochronous. Real systems only behave like this approximately: a pendulum, for instance, has an almost amplitude-independent period only when the angle is small.

In classroom measurements, time several cycles rather than just one. That reduces the effect of reaction time and makes it easier to compare periods when you change one variable, such as mass on a spring or pendulum length.

Linking linear repetition to angular repetition

The period, frequency and angular frequency are linked by

T = 1/f = 2π/ω.

This equation connects oscillations with circular motion. In one complete cycle, the angular change is 2π radians, so an oscillator with angular frequency ω takes time 2π/ω to complete one full oscillation.

An angular frequency is the rate at which the phase of an oscillation changes. It is often written in rad s⁻¹, but a radian is a ratio of arc length to radius, so it has no dimension; that’s why the unit may also be written as s⁻¹.

Circular motion as a picture of SHM

Picture a point moving at constant speed around a circle. Shine light from the side, and the point’s shadow on a screen moves backwards and forwards in one dimension. That shadow undergoes simple harmonic motion. This is not just a neat demonstration: the same sine and cosine functions describe both circular motion and SHM.

The circular model also helps in wave physics. A rotating vector can represent a particle in a sinusoidal wave, and the projection of that vector gives the particle’s displacement. So the same quantities — period, frequency, angular frequency and phase — keep appearing in the wave model.

Why a spring gives SHM

A mass–spring system is an oscillator made from a mass attached to a spring, with the spring supplying the restoring force. When the spring follows Hooke’s law and resistive effects are negligible, the system behaves as an exact simple harmonic oscillator.

For a spring,

Fₛ = −k**x, where Fₛ is the spring force on the mass (N) and k is the spring constant (N m⁻¹).

The minus sign tells you the spring force acts in the opposite direction to the displacement from equilibrium. Combine that with Newton’s second law,

Fnet = m**a, where Fnet is the net force on the mass (N) and m is mass (kg),

and you get the SHM form: acceleration is proportional to displacement and points back towards equilibrium.

The period equation

For a mass–spring system,

T = 2π√(m/k).

A heavier mass oscillates more slowly because it has more inertia. A stiffer spring oscillates more quickly, since it gives a larger restoring force for the same displacement. In this ideal model, the period does not depend on amplitude.

You can test this relationship quite directly. Change only the mass while keeping the same spring, or change only the spring constant while keeping the same mass. For a fixed spring, plotting suitable transformed variables, such as T² against m, should produce a straight-line relationship.

Small-angle pendulum motion

A simple pendulum is an ideal oscillator: a small mass hanging from a light, inextensible string and swinging in a vertical plane. It shows approximate simple harmonic motion only when the angular displacement is small.

For a pendulum, the component of weight that restores the motion acts along the arc of the swing. For small angles, measured in radians, sin θ ≈ θ, where θ is the angular displacement from the vertical (rad). Since θ ≈ x/l, where l is the pendulum length from pivot to the centre of the bob (m), the restoring acceleration is proportional to the displacement.

The small-angle form is

a = −(g/l)x, where g is gravitational field strength (m s⁻² or N kg⁻¹).

Compare this with a = −ω²x to get the pendulum period

T = 2π√(l/g).

The bob’s mass does not appear in the equation. In the ideal model, changing the bob’s mass does not change the period. A longer pendulum has a longer period. A larger gravitational field strength gives a shorter period.

At larger amplitudes, the small-angle approximation fails, so the pendulum is no longer accurately isochronous. Pendulum clocks therefore need small, controlled swings to keep reliable time.

Energy swaps through the cycle

Kinetic energy is the energy an object has because it is moving. Potential energy is stored energy linked to position or configuration, such as elastic potential energy in a spring or gravitational potential energy in a pendulum. Total mechanical energy is the sum of kinetic and potential energies in the oscillator.

In ideal simple harmonic motion, total mechanical energy stays constant. Only the form of the energy changes.

For a horizontal mass–spring oscillator:

• at either extreme, the mass is momentarily at rest, so kinetic energy is zero and elastic potential energy is maximum;
• at equilibrium, speed is maximum, so kinetic energy is maximum and elastic potential energy is minimum;
• between these positions, energy is continuously transferred between kinetic and potential forms.

For a simple pendulum, the same idea applies, but with gravitational potential energy instead of elastic potential energy. At the highest points of the swing, the bob is momentarily at rest and gravitational potential energy is maximum. At the lowest point, speed and kinetic energy are maximum.

During one complete oscillation, there are two speed maxima: one crossing equilibrium in each direction. So kinetic energy reaches a maximum twice per oscillation, and potential energy also reaches a maximum twice per oscillation. The energy variation has twice the frequency, and half the period, of the displacement oscillation.

If damping is present, energy is transferred from the oscillator to the surroundings, usually as thermal energy and sound. The amplitude decreases, and the total mechanical energy of the oscillator is no longer constant.

Describing where the oscillator is in its cycle

A phase angle tells you the stage an oscillator has reached in its cycle, measured as an angle in radians. One complete cycle is 2π rad, half a cycle is π rad, and a quarter cycle is π/2 rad.

A phase difference is the angle by which one oscillation leads or lags another oscillation with the same frequency. If two displacement–time graphs have the same period and one gets to each peak earlier, that oscillation leads in phase.

The circular-motion model shows this neatly. Picture each oscillator as a rotating radius. When the two radii keep a fixed angle between them, that angle is the phase difference. Project the radii onto an axis and you get two sinusoidal motions shifted in time.

We use radians in phase calculations because the sine and cosine functions in the SHM equations take arguments in radians. Degrees are fine when drawing a pendulum angle, but don’t substitute them directly into SHM phase equations unless you convert them first.