Question

If the differential equation for a simple harmonic motion is $\frac{d^{2}y}{dt^{2}}+2y=0,$ the time-period of the motion is

• A. $\pi\sqrt{2}$ sec
• B. $\frac{\sqrt{2}s}{\pi}$ sec
• C. $\frac{\pi}{\sqrt{2}}$ sec
• D. $2\pi$ sec

Answer 1

Answer: (A)

$\pi\sqrt{2}$ sec

Answer 2

The differential equation of simple harmonic motion is
$\frac{d^{2}y}{dt^{2}}+2y=0$ or $\frac{d^{2}y}{dt^{2}}=-2y\,...\left(i\right)$
Standard equation of simple harmonic motion is
$\frac{d^{2}y}{dt^{2}}=-\omega^{2}y\,...\left(ii\right)$
Comparing e $\left(i\right)$ and $\left(ii\right)$,
$\omega^{2}=2$ or $\omega=\sqrt{2}$
As we know, $\omega=\frac{2\pi}{T}$
$\therefore$ Time period, $T=\frac{2\pi}{\omega}=\frac{2\pi}{\sqrt{2}}=\pi\sqrt{2}\,sec$