Simple Harmonic Motion

We know that vibratory or oscillatory motion is a repeated back and forth motion along the same path. Simple Harmonic Motion is a special kind of vibratory motion.

Simple Harmonic Motion is one in which the restoring force F_rest is directly proportional to the displacement of the oscillating object from the mean position.

We know that vibration occurs about a mean (or equilibrium) position. At equilibrium point, the net force acting on the body is zero. When the body is at a small distance from the equilibrium point, there is a net force acting the body. This net force pushes the body to the equilibrium point. This force is called restoring force.

Mathematically,                               F_rest ∝ -x

Or equivalently,                                                a ∝ -x

Conditions for Simple Harmonic Motion

In order to understand the necessary conditions of SHM, we take an example.

Ball-bowl example of SHM

Consider the figure. A ball in a bowl is at equilibrium position, O. When we displace the ball to a side from the equilibrium position to the maximum displacement A, the restoring force pushes it back to the equilibrium position reducing its displacement. Thus the force is opposite to the displacement.

Reaching the equilibrium position, it does not stop due to inertia. It goes towards B. Since it goes away from the equilibrium position, the restoring force gradually develops with the increase in the displacement in the opposite direction. It reaches B at maximum displacement where the restoring force is also maximum but in the opposite direction to the displacement. From here it again starts towards O, decreasing its displacement as well as restoring force. Therefore,

F_rest ∝ -x           …           (1)

Where the negative sign shows the restoring force is always towards the equilibrium position and opposite to the direction of displacement x.

Since F ∝ a, we can also say that

a ∝ -x              …            (2)

Please note that (1) and (2) are different proportionalities and the proportionality constants in the two cases will be different.