12.1 Simple Harmonic Oscillations

Oscillation means repeated back-and-forth motion around an equilibrium position. The period T is the time for one complete cycle. The frequency f is cycles per second. The amplitude A is the maximum displacement from equilibrium. Displacement x is signed: it records both distance from equilibrium and direction.

IB questions often test whether students confuse amplitude with full travel distance in one oscillation. For a mass moving from +A to -A and back to +A, the path length in one cycle is 4A, but the amplitude is still A. Keep geometric definitions separate from path totals.

A system oscillates only if displacement away from equilibrium creates a response that pushes it back. In a spring, that response is elastic force. In a pendulum, that response is the tangential component of weight. In both cases, no restoring tendency means no sustained oscillatory behavior.

The spring relation gives one of the most important SHM period formulas. Increasing mass slows the oscillation (longer period), while increasing stiffness speeds it up (shorter period). That balance is physically intuitive: heavier bodies resist acceleration, while stiffer springs produce larger restoring forces for the same displacement.

In SHM, displacement-time is sinusoidal, velocity-time is also sinusoidal but phase-shifted by a quarter cycle, and acceleration-time is exactly inverted relative to displacement-time. The acceleration-displacement graph is especially diagnostic: it must be a straight line through the origin with negative slope.

That slope equals -omega^2, so the line is not only a qualitative shape check but a quantitative measurement route. If you extract slope from an a-x graph, you can recover omega and then period using T = 2pi/omega. This is a high-value exam technique because it links graph reading directly to timing results.