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Physics Chapter 4 : Topic 11
Conservation of Angular Momentum
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The law of conservation of angular momentum states that:
Reasoning: Angular momentum is conserved when the net external torque on a system is zero.
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Angular momentum of an isolated rotating system remains constant because:
Reasoning: External torque is required to change angular momentum.
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The mathematical expression for conservation of angular momentum is:
Reasoning: Angular momentum remains constant: $\mathbf{L}\mathbf{=}\mathbf{I}\mathbf{\omega}$
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Angular momentum of a system changes when:
Reasoning: External torque changes angular momentum.
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The SI unit of angular momentum is:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}$ $\mathbf{\Longrightarrow}$ Unit: $\frac{\mathbf{kg\ m}^{\mathbf{2}}}{\mathbf{s}}$
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If a rotating body decreases its moment of inertia, its angular velocity:
Reasoning: $\mathbf{I}_{\mathbf{1\ }}\mathbf{\omega}_{\mathbf{1}}\mathbf{=}\mathbf{I}_{\mathbf{2\ }}\mathbf{\omega}_{\mathbf{2}}$ ; Decrease in $\mathbf{I}$ causes increase in $\mathbf{\omega}$.
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A figure skater spins faster when pulling arms inward because:
Reasoning: Pulling arms inward reduces radius and moment of inertia.
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When a skater extends arms outward during spinning:
Reasoning: Increasing moment of inertia decreases angular velocity.
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A rotating person on a frictionless platform pulls weights closer to the body. The angular momentum:
Reasoning: No external torque acts, so angular momentum is conserved.
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A decrease in moment of inertia causes angular velocity to increase because:
Reasoning: The product $\mathbf{(I\omega)}$ remains constant.
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An ice skater spins faster by:
Reasoning: Pulling arms inward reduces moment of inertia.
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During a spin, an ice skater’s angular momentum remains constant because:
Reasoning: Ice provides very little frictional torque.
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A skater rotating with arms extended has:
Reasoning: Mass is distributed farther from the axis.
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If a skater reduces moment of inertia to half while spinning, angular velocity becomes:
Reasoning: The angular velocity of the skater becomes twice as large (doubles). If $\mathbf{(I)}$ becomes $\left( \frac{\mathbf{1}}{\mathbf{2}}\mathbf{I} \right)$, then $\mathbf{(\omega)}$ must become $\mathbf{(2}\mathbf{\omega)}$ to keep $\mathbf{(I\omega)}$ constant. $\mathbf{\Longrightarrow}$ $\mathbf{I}_{\mathbf{1\ }}\mathbf{\omega}_{\mathbf{1}}\mathbf{=}\mathbf{I}_{\mathbf{2\ }}\mathbf{\omega}_{\mathbf{2}}$
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The main principle involved in an ice skater’s spin is:
Reasoning: The skater changes rotation by changing moment of inertia.
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A flywheel is used to:
Reasoning: Flywheels store energy due to their rotation.
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A flywheel stores energy mainly because it has:
Reasoning: Large moment of inertia allows storage of large rotational kinetic energy.
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The rotational kinetic energy of a flywheel is:
Reasoning: $\mathbf{K\ .}\mathbf{E}_{\mathbf{\ rot}}\mathbf{=}$ $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{I}\mathbf{\omega}^{\mathbf{2}}$
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A flywheel is designed with mass concentrated away from its axis because:
Reasoning: Greater radius increases moment of inertia.
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A flywheel helps machines by:
Reasoning: Stored rotational energy helps maintain steady motion.
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A gyroscope works mainly on:
Reasoning: A spinning gyroscope resists changes in orientation due to angular momentum.
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A rapidly spinning gyroscope maintains its direction because:
Reasoning: Large angular momentum provides rotational stability.
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Gyroscopes are used in navigation systems because they:
Reasoning: Gyroscopes help determine orientation.
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A gyroscope with greater angular velocity has:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}$
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The resistance of a gyroscope to change in orientation is due to:
Reasoning: Angular momentum tends to remain constant.
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A rotating object has moment of inertia $\mathbf{4\ }\mathbf{kg\ m}^{\mathbf{2}}$ and angular velocity $\mathbf{5\ rad/s}$. Its angular momentum is:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}\mathbf{= 4(5) =}\mathbf{20\ kg\ }\mathbf{m}^{\mathbf{2}}\mathbf{/s}$
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A skater reduces moment of inertia from $\mathbf{8\ }\mathbf{kg\ m}^{\mathbf{2}}\mathbf{\ }$to $\mathbf{4\ }\mathbf{kg\ m}^{\mathbf{2}}$. If initialangular velocity is $\mathbf{3\ rad/s}$, final angular velocity is:
Reasoning: $\mathbf{I}_{\mathbf{1\ }}\mathbf{\omega}_{\mathbf{1}}\mathbf{=}\mathbf{I}_{\mathbf{2\ }}\mathbf{\omega}_{\mathbf{2}}$ ; Decrease in $\mathbf{I}$ causes increase in $\mathbf{\omega}$.$\mathbf{8(3) = \ 4}\mathbf{\omega}_{\mathbf{2}}$ $\mathbf{\Longrightarrow}$ $\mathbf{24 = \ 4}\mathbf{\omega}_{\mathbf{2}}$ $\mathbf{\Longrightarrow}$ $\mathbf{\omega}_{\mathbf{2}}\mathbf{=}\mathbf{6\ rad/s}$
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A wheel has angular momentum $\mathbf{200\ kg\ }\mathbf{m}^{\mathbf{2}}\mathbf{/}$s and moment of inertia $\mathbf{20\ kg\ }\mathbf{m}^{\mathbf{2}}$. Its angular velocity is:
Reasoning: $\mathbf{\omega}\mathbf{=}$ $\frac{\mathbf{L}}{\mathbf{I}}\mathbf{=}\frac{\mathbf{200}}{\mathbf{20}}\mathbf{=}\mathbf{10\ rad/s}$
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If angular velocity doubles while moment of inertia remains constant, angular momentum:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}$ ; Angular momentum doubles.This occurs because angular momentum is directly proportional to angular velocity when the moment of inertia does not change.
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A rotating system has constant angular momentum. If moment of inertia becomes four times, angular velocity becomes:
Reasoning: $\mathbf{I}_{\mathbf{1\ }}\mathbf{\omega}_{\mathbf{1}}\mathbf{=}\mathbf{I}_{\mathbf{2\ }}\mathbf{\omega}_{\mathbf{2}}$The angular velocity becomes one-fourth of its initial value, determined by the conservation of angular momentum and the inverse relation between inertia and speed.
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A satellite continues rotating in space because:
Reasoning: In absence of external torque, rotation continues.
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A rotating astronaut changes body shape. Which quantity remains constant?
Reasoning: Angular momentum is conserved when no external torque acts.
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Conservation of angular momentum applies when:
Reasoning: No external torque means no change in angular momentum.
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A spinning wheel becomes slower due to friction because:
Reasoning: Frictional torque changes angular momentum.
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The stability of a spinning top is due to:
Reasoning: Angular momentum resists changes in rotational axis.
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Which statement is correct?
Reasoning: Flywheels store energy through rotational motion.
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