Force law of simple harmonic motion
Consider a physical system consisting of a mass attached to the end of a spring, where mass is free to move on a horizontal, frictionless track.
When the mass is displaced a small distance xm from equilibrium, the spring exerts a force on m given by the Hooke's law F = -kxm. If the mass then released from rest, it oscillates.
If we apply Newton's second law to the motion of the mass in the x-direction, we get
F = -kx(t) = ma(t).
Recalling that
a(t) = - ω2x(t),
we obtain = kx(t) = -mω2x(t),