The Mass-Spring Oscillator

A spring resists being stretched or compressed, and Hooke's law says that resistance grows in direct proportion to displacement: F = -kx. The minus sign means the force always points back toward equilibrium. Plug that into Newton's second law, F = ma, and you get ma = -kx, so a = -(k/m)x. Acceleration is proportional to displacement and points opposite to it. That single relationship is the definition of simple harmonic motion, and it is why the mass never just stops at equilibrium — it overshoots, gets pulled back, overshoots again.

This equation also explains the intuition. A stiffer spring (larger k) pulls harder at any given displacement, so it accelerates the mass back faster, shrinking the period. A heavier mass (larger m) resists that acceleration more, so it takes longer to complete a cycle. The period comes out to T = 2 π √(m/k), built entirely from how strongly the force reacts to displacement versus how much the mass resists changing speed.

Hang the same spring vertically and gravity adds a constant downward pull, mg, that does not depend on x. That constant force just stretches the spring a bit further before things balance, sliding the equilibrium point down. Measured from that new equilibrium, the restoring force is still exactly -kx, so the motion is still simple harmonic with the same period. This sets up the next idea in the unit: once you accept that period depends only on m and k, you can ask how energy trades between kinetic and spring potential as the mass swings through that equilibrium point.