#### Question

State the differential equation of linear simple harmonic motion.

#### Solution

State the differential equation of linear S.H.M.

When a particle performs linear SHM. the force acting on the particle is always directed towards the mean position. The magnitude of the force is directly proportional to the magnitude of the displacement of the particle from the mean position. Thus, if `vecF` is the force acting on the particle when its displacement from the mean position is `vecx`

∴ `vecF = -kvecx` ....(1)

where the constant k, the force per unit displacement, is called the force constant. The minus sign indicates that the force and the displacement are oppositely directed.

The velocity of the particle is `(dvecx)/(dt)` and its acceleration is `(d^2vecx)/(dt^2)`

Let m be the mass of the particle

Force = mass x acceleration

∴ vecF = m `(d^2vecx)/(dt^2)`

Hence from Eq. (1), m `(d^2vecx)/(dt^2) = -k vecx`

`:. (d^2vecx)/(dt^2) + k/m vecx = 0` ....(2)

This is the differential equation of linear S.H.M.