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 Column I with Column II.

	Column I		Column II
(A)		(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)		(r)	Sphere spins anti – clockwise loses energy by friction.
(D)		(s)	Sphere has only a translational motion. Loses energy by friction.

• 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 1

Answer: (A)

A - r, B - s, C - q, D – p

Answer 2

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 angular momentum of sphere about its centre of mass is. $\operatorname { Mv } ( \mathrm { h } - \mathrm { R } ) = \mathrm { I } \omega = \left( \frac { 2 } { 5 } \mathrm { MR } ^ { 2 } \right) \left( \frac { \mathrm { v } } { \mathrm { R } } \right)$

($\therefore$ For a sphere , $\mathrm { I } = \frac { 2 } { 5 } \mathrm { MR } ^ { 2 }$)

$\operatorname { Mv } ( \mathrm { h } - \mathrm { R } ) = \frac { 2 } { 5 } \operatorname { MvR }$

$h - R = \frac { 2 } { 5 } R$ or $h = \frac { 7 } { 5 } R$

$\therefore$ The sphere will roll without slipping with a constant velocity and no loss energy when,

Torque due to applied force about centre of mass, $\tau = \mathrm { F } ( \mathrm { h } - \mathrm { R } )$

If, sphere will have only translational motion. It would lose energy by friction.

$\therefore \mathrm { B } - \mathrm { s }$

The sphere will spin clockwise, when $\tau$ is positive, i.e., h > R

Again the sphere will spin anticlockwise, when $\tau$ is negative, i.e, , h < R

$\therefore \mathrm { A } - \mathrm { r }$