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Physics Chapter 4 : Topic 9
Moment of Inertia and Angular Momentum
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Moment of inertia is the rotational equivalent of:
Reasoning: Mass resists changes in linear motion, while moment of inertia resists changes in rotational motion.
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Moment of inertia of a body depends on:
Reasoning: Moment of inertia depends on both mass and how far the mass is distributed from the axis.
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The formula for moment of inertia of point masses is:
Reasoning: $\mathbf{I}\mathbf{=}\mathbf{m}\mathbf{r}^{\mathbf{2}}$ ; where $\mathbf{r}$ is distance from axis.
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The SI unit of moment of inertia is:
Reasoning: From $\mathbf{I}\mathbf{=}\mathbf{mr}^{\mathbf{2}}\mathbf{=}\mathbf{kg \times}\mathbf{m}^{\mathbf{2}}\mathbf{= kg}\mathbf{m}^{\mathbf{2}}$
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Increasing the distance of mass from the axis of rotation:
Reasoning: $\mathbf{I}\mathbf{=}\mathbf{m}\mathbf{r}^{\mathbf{2}}$ ; Moment of inertia depends on square of distance.
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A rotating object is easier to accelerate when:
Reasoning: Smaller rotational inertia requires less torque.
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Two objects have equal masses. The object with larger radius of rotation has:
Reasoning: Greater distance from axis increases rotational resistance.
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A body with mass concentrated near the axis has:
Reasoning: Smaller radius reduces $\mathbf{(}\mathbf{I)}$.
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Moment of inertia increases if:
Reasoning: Moment of inertia depends on distance squared.
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Which object has greater moment of inertia?
Reasoning: A ring has more mass farther from the axis.
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A $\mathbf{2\ kg}$ point mass rotates at a distance of $\mathbf{3\ m}$ from an axis. Its moment of inertia is:
Reasoning: $\mathbf{I}\mathbf{=}\mathbf{m}\mathbf{r}^{\mathbf{2}}$ $\mathbf{= 2}\mathbf{(3)}^{\mathbf{2}}\mathbf{= 18}\mathbf{\ kgm}^{\mathbf{2}}$
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A $\mathbf{5\ kg}$ mass is located $\mathbf{4\ m}$ from an axis. Moment of inertia is:
Reasoning: $\mathbf{I}\mathbf{= 5}\mathbf{(4)}^{\mathbf{2}}$ $\mathbf{= 80\ }\mathbf{kgm}^{\mathbf{2}}$
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If the distance of mass from axis is doubled, moment of inertia becomes:
Reasoning: $\mathbf{I}\mathbf{\propto}\mathbf{r}^{\mathbf{2}}$ ; Doubling radius gives $\mathbf{2}^{\mathbf{2}}\mathbf{= 4}$ times.
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A body has moment of inertia $\mathbf{50\ kg}\mathbf{m}^{\mathbf{2}}$. If its mass is $\mathbf{10\ kg}$, radius of gyration is:
Reasoning: $\mathbf{I}\mathbf{=}\mathbf{m}\mathbf{k}^{\mathbf{2}}$ $\mathbf{\Longrightarrow}$ $\mathbf{50 = 10}\mathbf{k}^{\mathbf{2}}$ $\mathbf{\Longrightarrow}$ $\mathbf{k}\mathbf{=}\sqrt{\mathbf{5}}\mathbf{\approx 2.2\ m}$
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Radius of gyration represents:
Reasoning: Radius of gyration gives equivalent distribution of mass.
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Angular momentum is the rotational equivalent of:
Reasoning: Linear momentum $\mathbf{p}\mathbf{=}\mathbf{mv}$, angular momentum $\mathbf{L}\mathbf{=}\mathbf{I\omega}$.
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The formula for angular momentum of a rotating body is:
Reasoning: Angular momentum depends on moment of inertia and angular velocity.
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The SI unit of angular momentum is:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}$ $\mathbf{\Longrightarrow}$ $\mathbf{kg\ m}^{\mathbf{2}}\mathbf{/s}$
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Angular momentum increases when:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}$
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A rotating body has zero angular momentum when:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}$ ; If $\mathbf{\omega}\mathbf{= 0}$, angular momentum is zero.
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A body has moment of inertia $\mathbf{4\ }\mathbf{kg\ m}^{\mathbf{2}}$ and angular velocity $\mathbf{5\ rad/s}$. Angular momentum is:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}\mathbf{= 4(5) =}\mathbf{20\ }\mathbf{kg\ m}^{\mathbf{2}}\mathbf{/s}$
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A wheel has angular momentum $\mathbf{100\ }\mathbf{kg\ m}^{\mathbf{2}}\mathbf{/s}$ and moment of inertia $\mathbf{20\ }\mathbf{kg\ m}^{\mathbf{2}}$. Its angular velocity is:
Reasoning: $\mathbf{\omega}\mathbf{=}$ $\frac{\mathbf{L}}{\mathbf{I}}\mathbf{=}\frac{\mathbf{100}}{\mathbf{20}}\mathbf{=}\mathbf{5\ rad/s}$
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If moment of inertia doubles while angular velocity remains constant, angular momentum:
Reasoning: $\mathbf{L}\mathbf{=}\mathbf{I\omega}$ ; Angular momentum is directly proportional to $\mathbf{I}$.
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A skater pulls arms inward while spinning. The angular velocity increases because:
Reasoning: Conservation of angular momentum: $\mathbf{I}_{\mathbf{1\ }}\mathbf{\omega}_{\mathbf{1}}\mathbf{=}\mathbf{I}_{\mathbf{2\ }}\mathbf{\omega}_{\mathbf{2}}$Decrease in $\mathbf{I}$ causes increase in $\mathbf{\omega}$.
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When a spinning skater extends arms outward:
Reasoning: Increasing $\mathbf{(}\mathbf{I}\mathbf{)}$ reduces $\mathbf{\omega}$ to conserve angular momentum.
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Angular momentum remains constant when:
Reasoning: Angular momentum is conserved when no external torque acts.
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A rotating astronaut changes position in space. Angular momentum remains constant because:
Reasoning: Conservation occurs in absence of external torque.
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A dancer spins faster by pulling arms inward because:
Reasoning: Decreasing $\mathbf{(}\mathbf{I}\mathbf{)}$ increases angular velocity.
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If external torque acts on a rotating body:
Reasoning: Torque changes angular momentum.
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The relationship between torque and angular momentum is:
Reasoning: Torque is the rate of change of angular momentum.
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A flywheel has a large moment of inertia because:
Reasoning: Large radius distribution increases rotational inertia.
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Two wheels have same mass and radius. The ring has greater moment of inertia because:
Reasoning: Mass farther from axis contributes more to $\mathbf{(}\mathbf{I)}$.
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A body with large moment of inertia requires:
Reasoning: $\mathbf{\tau}\mathbf{=}\mathbf{I\alpha}$
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Angular momentum depends on:
Reasoning: $\mathbf{L = I\omega}$
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A rotating object slows down if:
Reasoning: Lower angular momentum means reduced rotational motion.
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The correct statement is:
Reasoning: Angular momentum is the rotational form of linear momentum.
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