Two particles of mass `m_(1) " and " m_(2)` are connected by a spring of natural length L and force constant k. The masses are brought close enough so as to compress the spring completely and a string is used to tie the system. Assume that length of spring in this position is close to zero. The system is projected with a velocity `V_(0)` along the positive x direction. At the instant it reaches origin at time t = 0, the string snaps and the spring starts opening.
(a) Show that the mass `m_(1) (or m_(2))` will are perform SHM in the reference frame attached to the centre of mass of the system. Find the time period of oscillation.
(b) Write the amplitude of `m_(1) " and " m_(2)` as a function of time. lt( c) Write the X co ordinates of `m_(1) " and " m_(2)` as a function of time

Answer

(a) T=2pi sqrt((m_(1)m_(2))/((m_(1)+m_(2))k))`
` (b) (m_(2)L)/(m_(1)+m_(2))=A_(1)`
`(c ) X_(1)=V_(0^(t))-A_(1)(1-cos omegat);`
`X_(2)=V_(0^(t))-A_(2)(1-cos omegat)
Where
`A_(1)=(m_(2)L)/(m_(1)+m_(2)); A_(2)=(m_(1)L)/(m_(1)m_(2)); omega=sqrt((k(m_(1)+m_(2)))/(m_(1)m_(2)))`