Question

A tunnel is dug in the earth across one of its diameter. Two masses $m$ and $2m$ are dropped from the two ends of the tunnel. The masses collide and stick each other. They perform $SHM,$ the amplitude of which is $(R=$ radius of earth)

• A. $R$
• B. $R/2$
• C. $R/3$
• D. $2R/3$

Answer 1

Answer: (C)

$R/3$

Let v is the velocity at centre then,

$K_i+U_i=K_f+U_f$

or $0-\frac{GMm}{R}=\frac{1}{2}m{v}^2-\frac{3}{2}\frac{GMm}{R}$

$v=\sqrt{\frac{GM}{R}}$

From conservation of linear momentum,

$2mv-mv=(2m+m)v$

$v^{\prime }=\frac{v}{3}=\frac{1}{3}\sqrt{\frac{GM}{R}}$

Again applying energy conservation,

$K_i+U_i=K_f+U_f$

$\frac{1}{2}3m\left[\frac{1}{3}\sqrt{\frac{GM\left(3m\right)}{R}}\right]-\frac{3}{2}\frac{GM\left(3m\right)}{R}=0-3m\left[\frac{GM}{R^3}\right(1.5R^2-0.5r^2\left)\right]$

Solving this equation:

$r=\frac{R}{3}$

Hence, $\frac{R}{3}$ is the amplitude of oscillations.