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). At what values of v_(0) will the hoop move without bouncing?

A small body A is fixed to the inside of a thin rigid hoop of radius R

1.	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). At what values of v_(0) will the hoop move without bouncing?
Answer» Solution :One the hoop, the grond is giving a normal force `R`. Again the attached mass `A` is also giving a normal force `N`. Now value of `R` is changing as the attached body is changing its position. Value of `R` is minimum when the attached body is at the highest position in its circular MOTION. but this minimum value of `R` should also the greater than zero to PREVENT the hoop from bouncing. Now CONSIDER `FBD` of the attached body `N+mg=(mv^(2))/R`.i [where `v=`velocity of cenntre of mass of the hoop at tis moment. Again velocity of the body at its moment with RESPECT to the hooop centre of mass `=2v-v=v`] For the hoop `N+R=mg`.....ii From i and ii `(mv^(2))/R-mg+R=mgimpliesR=2mg=(mv^(2))/R`..........iii `Rge0, implies v^(2)le2ghr`........iv Now CONSERVING energying the reference rame of the ground: `mv_(0)^(2)1/2 mv_(0)^(2)+1/2MR^(2)omega_(0)^(2)` `=1/2mv^(2)+1/2(mR^(2))omega^(2)+1/2m(2v)^(2)+2mgR` `implies` Putting `v_(0)=Romega_(0)` and `v=Romega:` `v_(0)=sqrt(8gR)`