Question 733: Centre of Mass and Rotational Motion

Question

Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moment of inertia $I$ and $2 I$ respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2 \omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.

Accepted Answer

Loss in kinetic energy
= (K.E.)$$_{initial}$$ $$-$$ (K.E.)$$_{final}$$
$$ = \left[ {{1 \over 2}I{{(2\omega )}^2} + {1 \over 2}(2I){\omega ^2}} ight] - \left[ {{1 \over 2}(I + 2I){{\left( {{4 \over 3}\omega } ight)}^2}} ight]$$
$$ = [2I{\omega ^2} + I{\omega ^2}] - \left[ {{3 \over 2}I 	imes {4 \over 3} 	imes {4 \over 3}{\omega ^2}} ight]$$
$$ = 3I{\omega ^2} - {8 \over 3}I{\omega ^2} = {{I{\omega ^2}} \over 3}$$

Answer

$$\frac{I\omega^2}{2}$$

Answer

$$\frac{I\omega^2}{3}$$

Answer

$$\frac{I\omega^2}{4}$$

Answer

$$\frac{I\omega^2}{6}$$

Centre of Mass and Rotational Motion

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