#### Question

Obtain the differential equation of linear simple harmonic motion.

#### Solution

Differential equation of linear S.H.M:**a.** Let a particle of mass ‘m’ undergo S.H.M about its mean position O. At any instant ‘t’, displacement of particle be ‘x’ as shown in the following figure.

**b.** By definition, F = - kx .….................(1)

where, k is force constant**c.** The acceleration of the particle is given by,

`a=(dv)/(dt)=(d(dx/dt))/dt=(d^2x)/dt^2`

**d.** According to Newton’s second law of motion,

F = ma

`therefore F=m((d^2x)/dt^2)` ...................(2)

**e.** From equations (1) and (2),

`m((d^2x)/dt^2)=-kx`

`therefore (d^2x)/dt^2=-k/mx`

`therefore (d^2x)/dt^2+k/mx=0`.....................(3)

where, `k/m=omega^2=contsant`

`therefore (d^2x)/dt^2+omega^2x=0`.....................(4)

**f.** Equations (3) and (4) represent differential equation of linear S.H.M.