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, but make sure the physics is clear too. The further the object moves from the centre, the harder the system accelerates it back.

This model works because many real systems behave this way, at least for small disturbances: 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, so the harmonic oscillator appears all over 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 make the second point too vaguely. “Proportional to displacement” on its own doesn’t do the job: if the acceleration points in the same direction as displacement, the object runs away from equilibrium instead of oscillating.

Use an acceleration–displacement graph as a quick test. 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 later supports 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. The motion may still be periodic for a while, but it gradually dies away.

The equation and the minus sign

The defining equation is

That minus sign is doing the real work. It tells you that acceleration points in the opposite direction to displacement. Put the oscillator to the right of equilibrium, and its acceleration is to the left; put it below equilibrium, and its acceleration is upwards. In other words, the acceleration is directed “back towards the centre”.

The squared angular frequency matters as well. Since $\omega^2$ is positive, the negative sign is always what gives the direction. The size of the acceleration is $\omega^2$ times the size of the displacement, so moving further from equilibrium produces a larger acceleration back towards equilibrium.

Reading the equation graphically

From $a = -\omega^2 x$, an acceleration–displacement graph has gradient $-\omega^2$. So, in an experiment, you can test for simple harmonic behaviour by plotting acceleration against displacement and checking for a straight line through the origin with a negative gradient.

In a practical investigation, a motion sensor and data logger can record displacement against time for an oscillator. The software can then produce velocity–time and acceleration–time graphs, while the acceleration–displacement plot gives the clearest check of the defining equation. If you’re 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

An equilibrium position is the position where the net force on the oscillator is zero when it is at rest. For a hanging mass on a spring, this is usually not the unstretched length of the spring. It is the resting position after the weight has stretched the spring.

A displacement is a vector distance 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 $x_0$. Since amplitude is a magnitude, it is not 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 moving in the same direction. For example, if it starts at equilibrium, moves to one extreme and comes back through equilibrium, that is only half the story; the cycle is complete only after it has visited the other extreme and returned to its starting state.

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

A displacement–time graph for simple harmonic motion is sinusoidal. The amplitude is the maximum displacement. The period is the time between matching points on successive cycles, such as peak to peak or an upward equilibrium crossing to the next upward equilibrium crossing.

Isochronous behaviour and real measurements

An isochronous oscillator is an oscillator whose period 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. This 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

Period, frequency and angular frequency are related by

$T = 1/f = 2\pi/\omega$.

This equation connects oscillations with circular motion. In one complete cycle, the angle changes by $2\pi$ radians, so an oscillator with angular frequency $\omega$ takes $2\pi/\omega$ seconds to complete one full oscillation.

An angular frequency gives the rate at which the phase of an oscillation changes. It is often written in $\mathrm{rad\ s^{-1}}$, but the radian is the ratio of arc length to radius, so it is dimensionless. For that reason, you may also see the unit written as $\mathrm{s^{-1}}$.

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. It’s not just a neat demonstration: the sine and cosine functions used for circular motion are the same ones used for SHM.

The circular model is useful in wave physics as well. You can represent a particle in a sinusoidal wave using a rotating vector, whose projection gives the particle’s displacement. That is why period, frequency, angular frequency and phase keep appearing in the wave model.

Why a spring gives SHM

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

For a spring,

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

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

The period equation

For a mass–spring system,

$T = 2\pi\sqrt{m/k}$.

A larger 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 investigate this relationship directly. Change only the mass while using the same spring, or change only the spring constant while keeping the same mass. For a fixed spring, plotting suitable transformed variables, such as $T^2$ against $m$, should produce a straight line.

Small-angle pendulum motion

A simple pendulum is an ideal oscillator: a small mass hanging from a light, inextensible string, 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 bob acts along the arc of the swing. When angles are small and measured in radians,

Also,

so the restoring acceleration becomes proportional to displacement.

The small-angle form is

$a = -(g/l)x$, where $g$ is gravitational field strength ($\mathrm{m\,s^{-2}}$ or $\mathrm{N\,kg^{-1}}$).

Comparing this with $a = -\omega^2x$ gives the pendulum period

$T = 2\pi\sqrt{l/g}$.

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 larger period, while a stronger gravitational field gives a smaller period.

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

Energy swaps through the cycle

Kinetic energy is the energy an object has because of its motion. 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. The energy just changes form.

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.

A simple pendulum follows the same pattern, 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 how far through its cycle an oscillator has got, using an angle. It is measured in radians. One full cycle is $2\pi$ rad, half a cycle is $\pi$ rad, and a quarter cycle is $\pi/2$ rad.

A phase difference is the angle by which one oscillation leads or lags another oscillation with the same frequency. On two displacement–time graphs with the same period, the graph that reaches each peak first leads in phase.

The circular-motion model makes this easy to see. Picture each oscillator as a rotating radius. If the two radii stay separated by a constant angle, that angle is the phase difference. Their projections produce two sinusoidal motions, shifted in time.

Radians are used in phase calculations because the sine and cosine functions in the SHM equations need arguments in radians. Degrees are fine when drawing a pendulum angle, but they can’t be substituted directly into SHM phase equations unless they are converted.

Displacement and velocity equations

An oscillator does not have to start at equilibrium. For a starting point anywhere in the cycle,

Differentiate displacement with respect to time and you get velocity:

The velocity is greatest at equilibrium and zero at the extremes. Because the velocity is in cosine form while displacement is in sine form, velocity is a quarter cycle out of phase with displacement.

A useful velocity equation with no time term is

$v = \pm \omega\sqrt{x_0^2 - x^2}$

Keep the $\pm$ sign. At the same displacement, the particle may be moving in either direction, so the equation gives two possible velocities unless the direction is specified.

Acceleration and the defining equation

Differentiate velocity and you get acceleration, which brings us back to the defining equation:

$a = -\omega^2 x$

The kinematics fit together neatly: displacement is sinusoidal, velocity is shifted by a quarter cycle, and acceleration is opposite to displacement.

A computer model can use the defining equation as a second-order differential equation, then update acceleration, velocity and displacement in small time steps. If a resistive force term is added to the model, the amplitude decreases. That helps show why damped motion is no longer ideal SHM.

Quantitative energy equations

At equilibrium, the oscillator has maximum speed and maximum kinetic energy. At the extremes, it has maximum potential energy and zero kinetic energy. The total energy is

The potential energy at displacement $x$ is

Use energy conservation to find the kinetic energy:

So

$E_k = \frac{1}{2}m\omega^2(x_0^2 - x^2)$

The energy–displacement graphs are parabolas. Potential energy is minimum at equilibrium and maximum at the extremes. Kinetic energy is maximum at equilibrium and zero at the extremes. The point where kinetic and potential energies are equal is not at half the amplitude; it occurs when $x^2$ is half of $x_0^2$, so it lies closer to the extremes than many students first expect.