`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 performes 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 zero 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
`rArr a=(kx)/(m_(1)+m_(2)) ` (For `(m_(1)+m_(2)`+spring ) system)
`rArr T=2pisqrt(x/a) =2pisqrt((x(m_(1)+m_(2)))/(kx))=2pisqrt((m_(1)+m_(2))/k)`
`rArr ` 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 (3) is incorrect.
After `x=(m_(2)g)/k, m_(1)` moves towards right till the total kinetic energy aquired does not converted to potential energy.
Hence (4) is also incorrect.
Hence (2) is the answer.