A uniform sphere of mass `m` and radius `R` is placed on a rough horizontal surface [Fig.] The sphere is struck horizontally at a hight `h` from the floor. Match the following :
(A) `h = R//2` (p) Sphere rolls without slipping with a constant velocity and no loss of energy.
(B) `h = R` (q) Sphere spins clockwise, loses energy by friction.
(C ) `h = 3R//2` (r) Sphere spins anti-clockwise, loses energy by friction.
(D) `h = 7R//5` (s) Sphere has only a translational motion, looses energy by friction.

Options

A. A -r, B - s, C - q, D - p
B. A - s, b - p, C - r, D - q
C. A - q, B - r, C - p, D - s
D. A - p, B - q, C - s, D - r

Answer

A

Explanation

Let the sphere of mass M and radius R be struck horizontally at a height h from the floor, as shown in the figure.

The sphere will roll without slipping when `omega=v/R`
Angulmar momentum of sphere about its center of mass,
`Mv(h-R)= Iomega= (2/5MR^(2))(v/R)`
`mv(h-R) = 2/5MvR`
`h-R=2/5R` or `h=7/5R`
`therefore` The sphere will roll without slipping with a consant velocity and no loss energy when `h=7/5R`
`therefore` The sphere will roll without slipping with a constant velocity and no less energy when `h=7/5R`
`therefore` D-p
Torque due to applied force about center of mass,
`tau = F(h-R)`
If `tau=0, h=R`, sphere will have only translational motion. It would lose energy by friction.
`therefore B-s`
The sphere will spin clockwise, when `tau` is positive, i.e., `h gt R`.
`therefore C-q`
Again, the sphere will spin anticlockwise, when `tau` is negative, i.e., `h lt R`
`therefore A-r`