Question

A body is executing S.H.M. When its displacement from the mean position is 4 cm and 5 cm, the corresponding velocity of the body is 10 cm/sec and 8 cm/sec. Then the time period of the body is:

• A. $2\pi \sec$
• B. $\pi /2\sec$
• C. $\pi \sec$
• D. $3\pi /2\sec$

Answer 1

The correct option is C $\pi \sec$

When the displacement from the mean position is $4cm$

Potential energy=$\frac{1}{2}{4}^{2}u=8u=8\left(m{\omega }^2\right)$

Kinetic energy$=\frac{1}{2}m{v}^2=\frac{1}{2}{10}^{2}m=50m$

Total energy$=8m{\omega }^2+50m$

When the displacement from the mean position is $5cm$

Potential energy=$\frac{1}{2}{5}^{2}u=12.5u=12.5\left(m{\omega }^2\right)$

Kinetic energy$=\frac{1}{2}m{v}^2=\frac{1}{2}{8}^{2}m320m$

Total energy$=12.5m{\omega }^2+32m$

Now as per conservation of energy

$=8m{\omega }^2+50m$$=12.5m{\omega }^2+32m$

$⟹\omega =2$

And $\omega =\frac{2\pi }{T}=2⟹T=\pi s$