A horizontal spring block system of mass M executes simple harmonic motion. When the block is passing through its equilibrium position, an object of mass m is put on it and the two move together. Find the new amplitude and frequency of vibration. Given, k is the spring constant of the system.

Explanation

Let amplitude of vibration a.
`v_0=aomega=asqrt(k/M)`

`v=a^'omega^'=a^'sqrt((k)/(M+m))`
By momentum conservation between (1) and (2)
`Mv_0=(M+m)v`
`Masqrt(k/M)=(M+m)a^'sqrt((k)/(M+m))`
`a^'/a=(sqrtM)/(M+m)impliesa^'=asqrt((M)/(M+m))`
`omega^'/omega=f^'/f=sqrt((M)/(M+m))impliesf^'=sqrt((M)/(M+m))`