A wagon of mass `M` can move without friction along horizontal rails. A simple pendulum consisting of a sphere of mass `m` is suspended from the ceiling of the wagon by a string of length l. At the initial moment the wagon and the pendulum are at rest and the string is deflected through an angle `alpha` from the vertical. Find the velocity of the wagon when the pendulum passes through its mean position.

Answer

A, B

Explanation

`h=l(1-cos alpha)` …(i)
`P_m=P_M` (in opposite directions)
`:. sqrt(2K_mm)=sqrt(2K_MM)`
or `K_m/K_M=M/m` …(ii)
`K_m+K_M=mgh=mgl(1-cos alpha)` …(iii)

Solving these three equations we get,
`K_M=((m)/(m+M))mgl(1-cos alpha)`
`:. 1/2Mv_M^2=((m)/(M+m))mgl(1-cos alpha)`
`:. v_m=2msi n(alpha)/(2)sqrt((gl)/(M(M+m))`