First slide

General Planar motion

Question

In the figure mass of both spherical body and block is m. Moment of inertia of the spherical body about center of mass is 2mR². The spherical body rolls on the horizontal surface. There is no slipping at any surfaces in contact. The ratio of kinetic energy of the spherical body to that of block is

Difficult

A

$\frac{3}{4}$
B

$\frac{1}{3}$
C

$\frac{2}{3}$
D

$\frac{1}{2}$

Solution

Let v be the linear velocity of center of mass of the spherical body and $ω$ its angular velocity about centre of mass. Then
$As there is no slipping at B, v = ω (2 R) \Rightarrow ω = \frac{v}{2 R}$
K.E. of spherical body,
$\begin{array}{l} K_{1} = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} \\ = \frac{1}{2} m v^{2} + \frac{1}{2} (\frac{1}{2} m {(2 R)}^{2}) (\frac{v^{2}}{4 R^{2}}) \\ = \frac{3}{4} m v^{2} . . . . . . . . . . . . . . . (i) \end{array}$
Let say speed of the block is $v'$ .
$There is no slipping at A, v + R ω = v' \Rightarrow R 2 ω + R ω = v'$
$\begin{array}{l} \Rightarrow v' = (ω) (3 R) = 3 R ω = (3 R) (\frac{v}{2 R}) = \frac{3}{2} v \\ ∴ K.E. of block K_{2} = \frac{1}{2} m v^{' 2} = \frac{1}{2} m {(\frac{3}{2} v)}^{2} = \frac{9}{8} m v^{2} \dots . . . . . . . . (ii) \\ From equations (i) and (ii), \frac{K_{1}}{K_{2}} = \frac{2}{3} \end{array}$

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