First slide

Dynamics of rotational motion about fixed axis

Question

Three objects, A: (a solid sphere), B: (a thin circular disk) and C: (a circular ring), each have the same mass M and radius R. They all spin with the same angular speed $ω$ about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation

Moderate

A

$W_{C} > W_{B} > W_{A}$
B

$W_{A} > W_{B} > W_{C}$
C

$W_{B} > W_{A} > W_{C}$
D

$W_{A} > W_{C} > W_{B}$

Solution

Work done required to bring a object to rest
$Δ W = Δ K E$
$Δ W = \frac{1}{2} I ω^{2}; where I = moment of inertia$
$For same ω, Δ W \propto I$
$For a solid sphere, I_{A} = \frac{2}{5} M R^{2}$
$For a thin circular disk, I_{B} = \frac{1}{2} M R^{2}$
$For a circular ring, I_{C} = M R^{2}$
$∴ I_{C} > I_{B} > I_{A}$
$∴ W_{C} > W_{B} > W_{A}$

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