A small bar A resting on a smooth horizontal plane is attached by threads to a point P and by means of weightless pulley, to a weight B possessing the same mass as the bar itself. The bar is also attached to a point O by means of a light non-deformed spring of length `l_0=50cm` and stiffness `k=5mg//l_0`, where `m` is the mass of the bar. The thread PA having been burned, the bar starts moving to the right. Find its velocity at the moment when it is breaking off the plane.

Explanation

Let `theta` be the angle between spring and vertical at the instant when block A breaks off the plane `(N=0)`.
`implies cos theta=(l_0)/(l_0+x)` (i)
`N+kx cos theta=mg`
`implies kx cos theta=mg` (as `N=0`) (ii)
Let d is the distance covered by A and B till this instant and V is the speed acquired by A and B.
(same because they are connected)
From (i) and (ii), using `k=(5mg)/(l_0)`
`implies (5mg)/(l_0)xx(l_0)/(l_0+x)=mg`
`impliesx=l/4l_0impliesd=sqrt((l_0+x)^2-l_0^2)=(3l_0)/(4)`
Using energy conservation:

`mgd=2(1/2mv^2)+1/2kx^2`
`impliesmg(3l_0)/(4)=mv^2+1/2(5mg)/(l_0)(l_0^2)/(16)`
`implies v^2=(3gl_0)/(4)-(5)/(32)gl_0impliesv=sqrt((19gl_0)/(32))`