Question

In the given figure, two elastic rods $P$ and $Q$ are rigidly joined to end supports. A small mass $m$ is moving with velocity $v$ between the rods. All collisions are assumed to be elastic and the surface is given to be smooth. The time period of small mass m will be: $(A=$area of cross section, $Y=$Young's modulus, $L=$length of each rod$)$

• A. $\frac{2L}{v}+2\pi \sqrt{\frac{mL}{AY}}$
• B. $\frac{2L}{v}+2\pi \sqrt{\frac{2mL}{AY}}$
• C. $\frac{2L}{v}+\pi \sqrt{\frac{mL}{AY}}$
• D. $\frac{2L}{v}$

Answer 1

The correct option is A $\frac{2L}{v}+2\pi \sqrt{\frac{mL}{AY}}$

$K_Q=\frac{YA}{L}$
$K_P=\frac{\left(2Y\right)\left(A/2\right)}{L}=\frac{YA}{L}$
$\therefore$ $K_P=K_Q=K=\frac{YA}{L}$

Half oscillation is completed with $P$ and half with $Q.$ But their value of $K$ is same. Hence we can say that in one oscillation one time period is completed with spring.

$T=\left(\frac{2L}{v}\right)+2\pi \sqrt{\frac{m}{K}}=\frac{2L}{v}+2\pi \sqrt{\frac{mL}{AY}}$