SHM acceleration

Intermediate

Proportional and opposite to displacement.

Key relationa = −ω²x

a = −ω²x, maximal at the ends (ω²A), zero at the centre, always directed to the centre.

Worked example

A = 0.05 m, ω = 20: a_max = 20 m/s².

Common mistake

Same sign as −x.

In more detail

The acceleration in SHM is a(t) = −ω²x, which is largest at the extreme positions and zero at equilibrium. It always points towards equilibrium.

Designing seats and restraints to handle accelerations of oscillating machinery.
Measuring vibrations with accelerometers in engineering.
Vibration engineers check that the acceleration on a sensitive instrument stays below a limit when the mount oscillates.

Where is the acceleration zero?

At equilibrium, where displacement is zero.

Why is the sign negative?

Because the acceleration always points back towards equilibrium, opposing the displacement.

Where is the acceleration largest?

At the extremes of the motion, where the displacement is greatest and the restoring force is strongest.