`m_1 ` & `m_2` are connected with a light inextensible string with `m_1` lying on smooth table and `m_2` hanging as shown in figure. `m_1` is also connected to a light spring which is initially unstretched an the system is released from rest.

Options

A. system performs SHM with angular frequency given by `sqrt((K(m_(1)+m_(2)))/(m_(1)m_(2)))`
B. system performs SHM with angular frequency given by `sqrt((K)/(m_(1)+m_(2)))`
C. tension in string will be 0 when the system is released
D. maximum displacement of `m_(1)` will be `(m_(2)g)/(K)`

Answer

B

Explanation

After the system is released, `m_(2)` moves down . The extension in the spring becomes:
`(m_(2)g)/(k)(m_(2)g=kx_(0))`,
which is the new equilibrium position of the system. For small `x`: restoring force on the system is F=kx
`Rightarrow a=(kx)/(m_(1)+m_(2)) `(For ` (m_(1)+m_(2)+`spring)system)
`T=2pisqrt((x)/(a))=2pisqrt((x(m_(1)+m_(2))/(kx))=2pisqrt((m_(1)+m_(2))/(k))`
`Rightarrow` Angular frequency =`omega=(2pi)/(T)=sqrt((k)/(m_(1)+m_(2)))`
F.B.D.of `m_(1)` and `m_(2)`just after the system is released :
From above: `T=(m_(1)+m_(2))/(m_(1)+m_(2))g`
Hence (c) is incorret.
After `x=(m_(2)g)/(k)`:`m_(1)` moves towards right till the total kinetic energy acquired does not converted to potential energy.
Hence(D) is also incorrect.
Hence (B)is the answer.