How Restoring Force Sets Simple Harmonic Motion

A linear restoring force creates simple harmonic motion

Simple harmonic motion is the special oscillation produced when acceleration is proportional to displacement from equilibrium and points in the opposite direction:

$$a=-\omega^2x.$$

For an ideal mass-spring system, Hooke's law gives $F=-kx$. Combining it with $F=ma$ produces

$$\omega=\sqrt{\frac{k}{m}}, \qquad T=2\pi\sqrt{\frac{m}{k}}.$$

The general position is

$$x(t)=A\cos(\omega t+\phi).$$

Differentiation gives $v(t)=-A\omega\sin(\omega t+\phi)$ and $a(t)=-A\omega^2\cos(\omega t+\phi)$. Position and acceleration are opposite in sign; velocity is one quarter-cycle out of phase with position.

Fresh example with a different phase

A $0.50\,\mathrm{kg}$ mass on an $18\,\mathrm{N/m}$ spring passes through equilibrium at $t=0$ moving in the positive direction. Its amplitude is $0.10\,\mathrm{m}$. Then

$$\omega=\sqrt{\frac{18}{0.50}}=6.0\,\mathrm{rad/s}.$$

Because $x(0)=0$ and $v(0)>0$, a convenient form is

$$x(t)=0.10\sin(6.0t)\,\mathrm{m},$$ $$v(t)=0.60\cos(6.0t)\,\mathrm{m/s},$$ $$a(t)=-3.6\sin(6.0t)\,\mathrm{m/s^2}.$$

The period is $T=2\pi/6.0=1.05\,\mathrm{s}$.

Energy moves between two forms

In an ideal oscillator, the total mechanical energy is constant:

$$E=\tfrac12kA^2=\tfrac12kx^2+\tfrac12mv^2.$$

Spring potential energy is greatest at the turning points, where speed is zero. Kinetic energy is greatest at equilibrium, where the speed magnitude is $A\omega$.

Damping, driving, spring mass, friction, and a nonlinear restoring force change this ideal model.