C.1 Simple harmonic motionIB Physics SL: Revision notes

Simple harmonic motion (SHM) occurs when the acceleration of an object is

• proportional to its displacement from a fixed equilibrium position, and
• always directed towards that equilibrium position.

This happens whenever the restoring force is proportional to displacement, for example a mass on a spring obeying Hooke's law, or a pendulum swinging through a small angle.

The defining equation of SHM is

a = −ω²x

The minus sign shows the acceleration is always opposite to the displacement. The constant ω is the angular frequency. A test for SHM from data: calculate a/x for each reading; if it is constant and negative, the motion is simple harmonic. The maximum acceleration occurs at maximum displacement: a_max = ω²x₀.

• Displacement x: distance and direction from equilibrium.
• Amplitude x₀: the maximum displacement.
• Time period T: time for one complete oscillation.
• Frequency f: oscillations per second (Hz).
• Angular frequency ω.

They are linked by T = 1/f = 2π/ω. In SHM the period does not depend on the amplitude (it is isochronous).

Mass–spring system: T = 2π√(m/k). The period increases with mass and decreases with spring stiffness; it does not depend on g, so it is the same horizontally or vertically.

Simple pendulum: T = 2π√(l/g), valid for small angles (below about 10°). The period depends on length and g but not on the mass of the bob. Rearranging gives g = 4π²l/T², a standard method for measuring g.

In an undamped oscillation total energy is constant but changes form twice each cycle:

• at maximum displacement: speed zero, energy all potential (elastic for a spring; gravitational for a pendulum);
• at equilibrium: speed maximum, energy all kinetic;
• in between: a mixture.

If friction or air resistance acts, energy is dissipated as thermal energy and the amplitude gradually decreases, while the period stays almost unchanged.