# Simple harmonic motion (SHM)

A body exhibits SHM if the time period (T) of an oscillation is not a ected by magnitude. SHM is de ned by WJEC in two ways:
1. SHM occurs when an object moves such that its acceleration is always directed toward a xed point and proportional to its distance from the xed point. (a = −ω2x) 2. The motion of a point whose displacement,x changes with time,t according to x = Asin(ωt + ε), where A, ω and ε are constants. [Variations of this kind are said to be sinusoidal.]
Considering a mass-spring system of a small amplitude as an example of SHM, we can show that:

1.2.3 Resonance
Free oscillations [Natural oscillations] Free oscillations occur when an oscillatory system (such as a mass on a spring, or a pendulum) is displaced and released. [The frequency of the free oscillations is called the system’s natural frequency.]
Forced oscillations These occur when a sinusoidally varying `driving’ force is applied to an oscillatory system, causing it to oscillate with the frequency of the applied force.
Resonance If, in forced vibrations, the frequency of the applied force is equal to the natural frequency of the system (e.g. mass on spring), the amplitude of the resulting oscillations is large. This is resonance. Resonance can be useful such as in circuit tuning and microwave cooking. Resonance should be avoided in some circumstances though, such as bridge design.
1.2.4 Resonance curve

Lightly (under) damped This is achieved by increasing resistive force
Heavily (still under) damped This is achieved by heavily resisting motion. E.g. in a mass-spring system, we could increase drag by moving from air to foil
Critically damped This is provided by an external system that removes energy from the oscillation, critical damping is extremely important in car suspension
Over-damped Over-damped systems actually take longer to reach no amplitude than critically damped
1.2.6 Resonance and damping
Lightly damped Slight reduction in maximum amplitude
Heavily damped Big reduction in max amplitude and slight reduction in natural frequency