A small body A is fixed to the inside of a thin rigid hoop of radius R and mass equal to that of the body A. The hoop rolls without slipping over a horizontal plane, at the moments when the body A gets into the lower position, the centre of the hoop moves with velocity `v_0`(figure). At what values of `v_0` will the hoop move without bouncing?

Explanation

Clearly the tendency of bouncing of the hoop will be maximum when the small body A, will be at the highest point of the hoop during its rolling motion. Let the velocity of C.M. of the hoop equal v at this position. The static friction does no work on the hoop, so from conversation of mechanical energy, `E_1=E_2`
or, `0+1/2mv_0^2+1/2mR^2(v_0/R)^2-mgR=1/2m(2v)^2+1/2mv^2+1/2mR^2(v/R)^2+mgR`
or, `3v^2=v_0^2-2gR` (1)
From the equation `F_n=mw_n` for body A at final position 2:
`mg+N^'=momega^2R=m(v/R)^2R` (2)
As the hoop has no acceleration in vertical direction, so for the hoop,
`N+N^'=mg` (3)
From Eqs. (2) and (3),
`N=2mg-(mv^2)/(R)` (4)
As the hoop does not bonus, `Nge0` (5)
So from Eqs. (1), (4) and (5),
`(8gR-v_0^2)/(3R)ge0` or `8gRgev_0^2`
Hence `v_0lesqrt(8gR)`