Consider the situastion shown in figure. Show that if that blocks are displaced slightly in opposite directions and released, they will execute simple harmonic motion. Calculate the time period.

Answer

`T=2pisqrt(m/(2k))`

Explanation

The centre of mass of the system should not change during the motion.
So,if the block m on the left moves towards right a distance x the block on the right moves towards left a distance x so total compression of the spring is 2x.
By energy method
`1/2k(2x)^2+1/2mv^2+1/2mv^2=C`
`rarr mv^2+2kx^2=C`
Taking derivative of both sides with respect to t.
`m2v(dv)/(dt)+2kxx2x(dx)/(dt)=0`
`:. ma+2kx=0`
[because `v=(dx)/(dt) and a=(dv)/(dt)]`
`rarr -a/x=(2k)/momega^2`
`rarr omega=sqrt((2k)/m)`
`rarr`Time period
`T=2pisqrt(m/(2k))`