Question

A uniform sphere of mass m and radius R is placed on a rough horizontal surface (Figure). The sphere is struck horizontally at a height h from the floor. Match the following:

Column I	Column II
(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.

Answer 1

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

Explanation:

Mass of the sphere = m

Radius = R

h = height from the floor

The sphere will roll without slipping when ω = V/R

Where v is linear velocity and to is the angular velocity of the sphere.

Now, angular momentum of the sphere is about centre of mass .....[We are applying conservation of angular momentum just before and after struck.]

Then by the law of conservation of angular momentum

`mv(h - R) = I_ω`

`mv(h - R) = 2/5 mR^2 v/R`

`h - R = 2/5 R`

`h = 2/5 R +R = 7/5 R`

Therefore, the sphere rolls without slipping with a constant velocity and no loss of energy. Thus (d) - (i)

Torque due to force `F = τ = (h - R) xx F`

If τ = 0, h – R = 0 and thus h = R

In this case, the sphere will only have a translation motion and slip against the force of friction. Thus (b) - (iv)

For clockwise rotation of the sphere τ > 0

`(h - R) xx F > 0`

Or `h > R`, thus (c) - (ii)

For anti-clockwise rotation `τ < 0`

`(h - R) xx F < 0`

`h < R`,Thus (a) - (iii)