Classical mechanicsLesson 3 of 4 in the course Angular Momentum

Angular Momentum Conservation in Rotating Systems and Collisions

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A system's angular momentum changes when a net external torque acts on it. The rotational counterpart to Newton's second law is

∑τ⃗external=dL⃗dt\sum \vec{\tau}_{\text{external}} = \frac{d\vec{L}}{dt}

∑τ⃗external=0⇒dL⃗dt=0\sum \vec{\tau}_{\text{external}} = 0 \quad \Rightarrow \quad \frac{d\vec{L}}{dt}=0

L⃗=l⃗1+l⃗2+⋯+l⃗N=constant\vec{L}=\vec{l}_1+\vec{l}_2+\cdots+\vec{l}_N=\text{constant}

Definition: Angular momentum is conserved when the total angular momentum of a system remains constant because the net external torque about the chosen point is zero.

All flashcards

When is angular momentum conserved?
The total angular momentum about a chosen point in a fixed inertial frame is conserved when the net external torque about that point is zero.
What does total angular momentum conservation say about a system's parts?
Individual parts may exchange angular momentum and change their own angular momenta. The sum for the whole system remains constant if there is no net external torque.
Why does an ice skater spin faster after pulling in her arms?
Pulling the arms inward reduces moment of inertia. Since L=IωL=I\omega remains constant, angular velocity must increase.
What happens when flywheels with inertias I0I_0 and 3I03I_0 couple, if only the first initially rotates at ω0\omega_0?
With no net external torque, they share angular velocity ω0/4\omega_0/4 after coupling.
Why must the embedded bullet be included in a disk's final moment of inertia?
After impact, the bullet rotates with the disk, so both contribute to the total moment of inertia about the disk's axis.

Quiz questions

  1. What condition makes a system's total angular momentum constant about a chosen point?Answer: The net external torque about that point is zero.
  2. A flywheel with inertia I0I_0 rotates at ω0\omega_0 and couples to a stationary flywheel with inertia 3I03I_0. What is their shared angular velocity?Answer: ω0/4\omega_0/4
  3. In the flywheel coupling where the initial inertia is I0I_0 and the final inertia is 4I04I_0, what fraction of the initial rotational kinetic energy is lost?Answer: Three-quarters
  4. A gymnast's extended body length is approximated as a uniform rod. If tucking halves that length while angular momentum is conserved, how does angular velocity change?Answer: It increases by a factor of four.
  5. A bullet moves tangentially and strikes the edge of a stationary disk. What is the bullet's initial angular momentum magnitude about the disk's central axis?Answer: mvRmvR

Based on University Physics Volume 1 by William Moebs, Samuel J. Ling, Jeff Sanny (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

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