Two point masses `m_1` and `m_2` are connected by a spring of natural length `l_0`. The spring is compressed such that the two point masses touch each other and then they are fastened by a string. Then the system is moved with a velocity `v_0` along positive x-axis. When the system reached the origin, the string breaks `(t=0)`. The position of the point mass `m_1` is given by `x_1=v_0t-A(1-cos omegat)` where A and `omega` are constants. Find the position of the second block as a function of time. Also, find the relation between A and `l_0`.

Answer

(i) `v_(0)t + A(m_(1))/(m_(2))(1-cosomegat)` , (ii) `((m_(1))/(m_(2)) + 1) A`

Explanation

(i) Two massse `m_(1)` and `m_(2)` are connected by a spring of length `l_(0)`. The spring is in compressed position. It is held in this position by a string. When the string snaps, the spring force is brought into operation. The spring force is an internal force w.r.t masses-spring system. No external force is applied on the system. The velocity of centre of mass will not change.
Velocity of centre of mass `= v_(0)`
`:.` Location/x -coordinate of centre of mass of time
`t = v_(0)t`
`:. barv = (m_(1)x_(1) + m_(2)x_(2))/(m_(1) + m_(2))`
`rArr v_(0)t = (m_(1)[v_(0)t - A(1-cosomegat)]+m_(2)x_(2))/(m_(1) + m_(2))`
`rArr (m_(1)+m_(2))v_(0)t=m_(1)[v_(0)t-A(1-cosomegat)]+m_(2)x_(2))`
`rArr m_(1)v_(0)t + m_(2)v_(0)t = m_(1)v_(.0)t - m_(1)A(1-cosomegat)] + m_(2)x_(2)`
`rArr m_(2)x_(2) = m_(2)v_(0)t + m_(1)A(1-cosomegat)`
`rArr x_(2)=v_(0)t + (m_(1)A)/(m_(2))(1-cosomegat)"......"(i)`
To express `l_(0)` in terms of A.
`:. x_(1) = v_(0)t - A(1-cosomegat) :. (dx_(1))/(dt^(2)) = -Aomega^(2) sinomegat`
`:. (d^(2)x^(2))/(dt^(2)) = - Aomega^(2) cosomegat "........"(ii)`
`x_(1)` is displacement of `m_(1)` at time t.
`:. (d^(2)x_(1))/(dt^(2)) =` acceleration of `m_(1)` at time t.
When the spring attains its natural length `l_(0)`, then acceleration is zero and `(x_(2) - x_(1)) = l_(0))`
`:. x_(2) x_(1) = l_(0)` , Put `x_(2)` from (i)
`rArr [v_(0)t + (m_(1)A)/(m_(2)) (1-cosomegat)] - [v_(0)t - A(1-cosomegat)] = l_(0)`
`rArr l_(0) = ((m_(1))/(m_(2)) + 1)A(1-cosomegat)`
When `(d^(2)x_(1))/(dt^(2)) = 0, cosomegat = 0` from (ii).
`:. l_(0) = ((m_(1))/(m_(2)) + 1)A`.