The definition of simple harmonic motion, displacement, velocity and acceleration as functions of time, energy in SHM, and damping.

What this key area is asking

The SQA wants you to define simple harmonic motion by $a = -\omega^2 y$, use the equations for displacement, velocity and acceleration as functions of time, analyse the interchange of kinetic and potential energy in an oscillation, and describe the effect of damping.

Defining SHM

The restoring acceleration always pulls the object back to equilibrium, which is what makes the motion oscillate. The angular frequency $\omega = 2\pi f = \frac{2\pi}{T}$ sets how fast the oscillation runs. A mass on a spring and a simple pendulum (for small swings) both satisfy this defining relationship, which is the starting point for every SHM problem.

Displacement, velocity and acceleration

These follow directly from the calculus of the previous area. The speed is greatest at the centre ($y = 0$, $v_{\max} = \omega A$) and zero at the extremes; the acceleration is greatest at the extremes ($y = A$, $a_{\max} = \omega^2 A$) and zero at the centre. Reading where in the cycle a quantity peaks is a common exam skill.

Energy in SHM

The kinetic energy is $E_k = \tfrac{1}{2}m\omega^2(A^2 - y^2)$ and the potential energy is $E_p = \tfrac{1}{2}m\omega^2 y^2$, so their sum $\tfrac{1}{2}m\omega^2 A^2$ does not depend on position. A graph of energy against displacement shows two parabolas crossing, with the constant total as a horizontal line, a frequently examined diagram.

Damping

A lightly damped oscillator keeps oscillating but with shrinking amplitude; heavy (over)damping stops it returning past equilibrium at all; critical damping brings it back to rest in the shortest time without overshooting. Real oscillators are always damped to some degree, and engineers choose the damping deliberately, for example critical damping in a car's suspension or a measuring instrument's needle.

Examples in context

A mass on a spring and a simple pendulum are the textbook examples, both obeying $a = -\omega^2 y$ for small displacements. Car suspension uses dampers (shock absorbers) tuned near critical damping so the car settles quickly after a bump without bouncing. A vibrating guitar string and the balance wheel of a watch are oscillators whose timekeeping depends on a steady period. Seismometers use damped masses to record ground motion without the needle oscillating wildly.

Try this

Q1. Write the defining relationship for simple harmonic motion. [1 mark]

• Cue. $a = -\omega^2 y$.

Q2. State where in the cycle an oscillator has its maximum speed. [1 mark]

• Cue. At the centre (equilibrium position), where displacement is zero.

Q3. State the effect of damping on an oscillation. [1 mark]

• Cue. It reduces the amplitude over time as energy is lost to resistive forces.