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` (i) Sphere rolls without slipping with a constant velocity and no loss of energy.
(b) `h = R` (ii) Sphere spins clockwise, loses energy by friction.
(c ) `h = 3R//2` (iii) Sphere spins anti-clockwise, loses energy by friction.
(d) `h = 7R//5` (iv) Sphere has only a translational motion, looses energy by friction.

Explanation

Let the sphere of mass `m` and radius `R` be struck horizontally at a height `h` from the floor, as shown in Fig.
The sphere will roll without slipping when `omega = (v)/(R )`
Now, angular momentum of sphere, about c.m.
`mv(h - R) = I omega = ((2)/(5) mR^(2))((v)/(R)) or mv (h - R) = (2)/(5)mv R`
`h - R = (2)/(5)R or h = (7)/(5)R`
`:.` The sphere will roll without slipping with a constant velocity and no loss of energy when `h = (7)/(5)R`.
Therefore, `d rarr(i)`.
Torque due to applied force, about c.m., `tau = F (h - R)`
If `t = 0, h = R`, sphere will have only translational motion. If would lose energy by friction. Hence, `b rarr (iv)`.
The sphere will spin clockwise, when `tau` is positive, i.e.., `h gt R`.
Therefore, `c rarr (ii)`.
Again, the sphere will spin anticlockwise, when `tau` is negative, i.e. `h lt R`. Therefore, `a rarr(iii)`.