Simple harmonic motion is the repetitive movement back and forth through an equilibrium point so that the maximum displacement on one side of this point is equal to the maximum displacement on the other side - the restoring force (Fr) is directly proportional to the displacement force (the force that propels the mass away from the center).
Simple harmonic motion can be modeled as a sine wave.
Fact: According to Hooke’s Law, the restoring force is always directed toward the equilibrium position and is directly proportional to the distance from it. That is:
where Fr is the restoring force, x is the displacement, and k is a constant.
The negative sign is customarily added to signify that the restoring force due to the spring is in the opposite direction to the force which caused the displacement.
Fact: Whenever a mass (m) is subject to such a restoring force and is caused to oscillate back and forth, the time for one complete oscillation - the period (T), will be given by:
With simple harmonic motion, it follows then that
x can also be represented as l * Ɵ where l is the length of the pendulum and Ɵ is the amplitude:
Now, when Ɵ is very small (< 10° - 15°),
Therefore,