C.1 Simple harmonic motion

Start with the restoring force

Simple harmonic motion occurs when the restoring force is proportional to displacement from equilibrium and always points back toward equilibrium: $F\propto -x$. For constant mass this gives $a\propto -x$.

Locate equilibrium

Measure $x$ from the equilibrium position, not from an arbitrary origin. At equilibrium $x=0$, so the restoring force and acceleration are zero; away from equilibrium, the acceleration points opposite to the displacement.

Check the model

A spring–mass system is an SHM model when the spring force is linear. A simple pendulum approximates SHM only for small angular displacements, where $\sin\theta\approx\theta$ in radians.

Common trap

“Periodic” motion alone is not enough. The defining condition is the restoring acceleration $a=-\omega^2x$, including the opposite direction and proportional dependence.

Write the definition

Simple harmonic motion is defined by $a=-\omega^2x$, where $x$ is displacement from equilibrium and $\omega$ is angular frequency. The acceleration is proportional to displacement and points in the opposite direction.

Interpret the minus sign

If the particle is displaced to positive $x$, acceleration is negative; if it is displaced to negative $x$, acceleration is positive. At equilibrium, $x=0$ and $a=0$, although the particle may have maximum speed there.

Connect frequency to acceleration

A larger $\omega$ gives a larger acceleration for the same displacement. The equation also shows why increasing amplitude does not change the period of ideal SHM: acceleration scales with the displacement.

Common trap

Do not write $a=+\omega^2x$ or measure $x$ from an arbitrary origin. The displacement must be relative to equilibrium, and the sign must restore the particle toward it.

Name each quantity

The equilibrium position is the central position where the resultant restoring force is zero. Displacement $x$ is the signed distance from equilibrium. Amplitude $x_0$ is the maximum magnitude of displacement.

Connect time measures

The period $T$ is the time for one complete cycle. Frequency $f$ is the number of cycles per second, so $T=1/f$. Angular frequency is $\omega=2\pi f=2\pi/T$, measured in radians per second.

Keep amplitude and displacement distinct

Amplitude is a non-negative fixed maximum for an ideal oscillation; displacement changes continuously between $-x_0$ and $+x_0$. The sign of displacement identifies the side of equilibrium.

Common trap

Do not call the distance travelled in one cycle the amplitude. Amplitude is measured from equilibrium to an extreme position, not from one extreme to the other.

Use the three equivalent measures

For an oscillation, $T$ is the period, $f$ is frequency and $\omega$ is angular frequency. They are related by $T=1/f=2\pi/\omega$, or $\omega=2\pi f$.

Choose the given quantity

If the number of cycles per second is given, use $T=1/f$. If angular frequency is given, use $T=2\pi/\omega$. Keep frequency in hertz, period in seconds and angular frequency in rad s$^{-1}$.

Connect to oscillator models

For a simple pendulum in the small-angle limit, $T=2\pi\sqrt{l/g}$; for a mass–spring system, $T=2\pi\sqrt{m/k}$. These model-specific equations determine the period before the general conversions are applied.

Common trap

Do not use $\omega= f/2\pi$. Angular frequency is larger than frequency by a factor of $2\pi$, because one complete cycle corresponds to $2\pi$ radians.

Conserve the total energy

In ideal SHM, total mechanical energy remains constant while energy shifts between kinetic energy and the oscillator’s potential energy, such as elastic or gravitational potential energy.

At an extreme position

At $x=\pm x_0$, the particle is instantaneously at rest, so kinetic energy is zero and potential energy is maximum. The restoring acceleration has maximum magnitude and points toward equilibrium.

At equilibrium

At $x=0$, the potential energy is minimum for the oscillator and the speed, hence kinetic energy, is maximum. During each quarter-cycle, energy transfers from one form to the other.

Common trap

Do not say that energy disappears at an extreme or that “potential energy” is sufficient when the system is a spring. Name elastic/spring potential energy when that is the stored form.

Read the displacement graph

For a displacement–time graph, the gradient gives instantaneous velocity. At an extreme displacement the gradient is zero, so velocity is zero; at equilibrium the gradient has maximum magnitude, so speed is maximum.

Use the defining relation

Acceleration follows $a=-\omega^2x$: it is zero at equilibrium and has maximum magnitude at the extremes, with sign opposite to displacement. The acceleration–time graph is therefore inverted relative to the displacement graph.

Track the cycle

At each time, combine the graph’s displacement sign with its gradient. This gives both the direction of motion and the acceleration direction; do not infer velocity from height alone.

Common trap

A zero gradient means zero velocity, not zero acceleration. At an extreme position the object reverses direction, so velocity is zero while acceleration is greatest toward equilibrium.

Describe one complete cycle

In ideal SHM, total mechanical energy is constant. As the particle moves from an extreme position to equilibrium, stored potential energy changes into kinetic energy; from equilibrium to the opposite extreme, kinetic energy changes back into potential energy.

Use quarter-cycle checkpoints

At an extreme, speed and kinetic energy are zero while the relevant potential energy is maximum. At equilibrium, speed and kinetic energy are maximum while that potential energy is minimum. The energy pattern repeats every cycle.

Connect the model

A circular-motion picture can help visualize phase: the projected coordinate oscillates between two extremes and passes equilibrium twice per cycle. Use it as a representation of the oscillation, while the energy explanation remains based on the SHM position and speed.

Common trap

Do not claim that the energy transfer stops at equilibrium. The particle has maximum speed there, so the transfer reverses direction as it continues toward the next extreme.