A thin hoop of mass M and radius r is placed on a horizontal plane. At the initial instant, the hoop is at rest. A small washer of mass m with zero initial velocity slides from the upper point of the hoop along a smooth groove in the inner surfaces of the hoop. Determine the velocity u of the centre of the hoop at the moment when the washer is at a certain point A of the hoop, whose radius vector forms an angle `phi` with the vertical (figure). The friction between the hoop and the plane should be neglected

Answer

B, C

Explanation

Let `v_r` be the velocity of washer relative to centre of hoop and v the velocity of centre of hoops. Applying conservation of linear momentum and mechanical energy we have,
`m(v_rcosphi-v)=Mv` …(i)
`mgr(1+cosphi)`
`=1/2Mv^2+1/2m(v_r^2+v^2-2v v_rcosphi)` ...(ii)
Solving Eqs. (i) and (ii), we have,
`v=m cos phi sqrt((2gr(1+cosphi))/((M+m)(M+sin^2phi)))`