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/3

\(2 mu - mu = 3 mv\)

\(\Rightarrow v = \frac{u}{3}\)

\(\frac{1}{2}mu^2 = \frac{3GMm}{2R} - \frac{GMm}{R} \)

\(u = \sqrt{\frac{Gm}{R}}\)

\(\Delta U(r) = \frac{GM (3m)}{2R^3}A^2\)

\(\Delta U(r) = \frac{1}{2} 3m.v^2\)

\(\Rightarrow \frac{GM}{2R^3}A^2 = \frac{1}{2} \frac{1}{9} \frac{GM}{R}\)

\(\frac{A^2}{R^2} = \frac{1}{9}\)

\(\Rightarrow A = \frac{R}{3}\)