Question

The differential equation of all ‘Simple Harmonic Motions’ of given period $\frac{2\pi}{n}$ is

• A. $\frac{d^{2}x}{dt^{2}}+nx=0$
• B. $\frac{d^{2}x}{dt^{2}}+n^{2}x=0$
• C. $\frac{d^{2}x}{dt^{2}}-n^{2}x=0$
• D. $\frac{d^{2}x}{dt^{2}}+\frac{1}{n^{2}}x=0$

Answer 1

Answer: (B)

The displacement $x$ for all $S.H.M.$ is given by
$x=a\,cos\left(nt+b\right)$
$\Rightarrow \frac{dx}{dt}=-na\,sin\left(nt+b\right)$
$\Rightarrow \frac{d^{2}x}{dt^{2}}=-n^{2}a\,cos\left(nt+b\right)$
$\Rightarrow \frac{d^{2}x}{dt^{2}}=-n^{2}x$
$\Rightarrow \frac{d^{2}x}{dt^{2}}+n^{2}x=0$