Obtain the differential equation of linear simple harmonic motion.

Explanation

For linear S.H.M.,
Restorign force `(F) prop -x.`
i.e., `F= -kx " " …(i) `
Where k is force constant and x is displacement of body from mean position.
Now, by Newton's second law of motion,
Force= mass `xx` acceleration
F = ma
Since acceleraion `(a)=(dv)/(dt) ` velocity `(v)=(dx)/(dt)`
`a=(d)/(dt)((dx)/(dt))=(d^(2)x)/(dt^(2))`
`therefore F=m xx (d^(2)x)/(dt^(2)) " " `...(ii)
From equation (i) and (ii), we get
`m(d^(2)x)/(dt^(2))+kx=0`
`therefore m(d^(2)x)/(dt^(2))+kx=0 " " ` ...(iii)
`(d^(2)x)/(dt^(2))+(k)/(m)x=0 " " ` ....(iv)
Putting `(k)/(m)=omega^(2)`, where `omega` is angular velocity.
As `(k)/(m)` is positive quantity, new constant `omega^(2)` is also positive.
`omega` may be positive or negative.
`therefore (d^(2)x)/(dt^(2))+omega^(2)x=0`
which is the required differential of linear S.H.M.