Two masses `m_(1)` and `m_(2)` are connected by a spring of force constant k and are placed on a frictionless horizontal surface. Initially the spring is stretched through a distance `x_(0)`, when the system is released from rest. Find the distance moved by two masses before they again comes to rest.

Explanation

Blocks again come to rest when spring is compressed by `x_(0)`. Since no external force is acting on the system, so there is no change in the position of c.m. of the system i.e., `Delta x_(cm)=0`.
Let mass `m_(1)` displaces by `Delta x_(1)` and `m_(2)` displaces by `Delta x_(2)`, then
We have `Delta x_(1)+Delta x_(2)=2x_(0)` ...... (i)
and `Delta x_(cm)=(m_(1)Delta x_(1)+m_(2)Delta x_(2))/(m_(1)+m_(2))`
As `Delta x_(cm)=0 therefore (m_(1)Delta x_(1)+m_(2)Delta x_(2))/(m_(1)+m_(2))=0` ......(ii)
After solving equation (i) & (ii), we get
`Delta x_(1)=(2m_(2)x_(0))/(m_(1)+m_(2)), Delta x_(2)=(2m_(1)x_(0))/(m_(1)+m_(2))`