The equation of motion of a particle executing simple harmonic motion is $4 \frac{\mathrm{d}^2…
The equation of motion of a particle executing simple harmonic motion is $4 \frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dt}^2}+\pi^2 \mathrm{y}=0$ where $y$ is in metres and $t$ is in seconds. The time period of oscillation of the particle is
$1 \mathrm{~s}$
$2 \mathrm{~s}$
$3 \mathrm{~s}$
$4 \mathrm{~s}$
Solution
Equation of motion:
$
\begin{aligned}
& 4 \frac{d^2 y}{d t^2}+\pi^2 y=0 \\
& \frac{d^2 y}{d t^2}+\frac{\pi^2}{4} y=0
\end{aligned}
$
General equation of motion is
$
\frac{d^2 y}{d t^2}+\omega^2 y=0
$
On compairing, $\omega^2=\frac{\pi^2}{2^2}$
$
\omega=\frac{\pi}{2}
$
Time is given as : $T=\frac{2 \pi}{\omega}$
$
\begin{aligned}
& T=\frac{2 \pi}{\pi} \times 2 \\
& T=4 \sec
\end{aligned}
$