6.4 Conservation of Angular Momentum

Learning objectives

6.4A—Describe the behavior of a system using conservation of angular momentumDescribe the behavior of a system using conservation of angular momentum.• The total angular momentum of a system about a rotational axis is the sum of the angular momenta of the system’s constituent parts about that axis.• Any change to a system’s angular momentum must be due to an interaction between the system and its surroundings.- i. The angular impulse exerted by one object or system on a second object or system is equal and opposite to the angular impulse exerted by the second object or system on the first. This is a direct result of Newton’s third law.- ii. A system may be selected so that the total angular momentum of that system is constant.- iii. The angular speed of a nonrigid system may change without the angular momentum of the system changing if the system changes shape by moving mass closer to or further from the rotational axis.- iv. If the total angular momentum of a system changes, that change will be equivalent to the angular impulse exerted on the system.6.4B—Describe how the selection of a system determines whether the angular momentum of that system changes. AP…Describe how the selection of a system determines whether the angular momentum of that system changes. AP Physics 1: Algebra-Based Course and Exam Description Energy and Momentum of Rotating Systems UNIT 6 TOPIC 6.5 Rolling• Angular momentum is conserved in all interactions.• If the net external torque exerted on a selected object or rigid system is zero, the total angular momentum of that system is constant.• If the net external torque exerted on a selected object or rigid system is nonzero, angular momentum is transferred between the system and the environment.

Add the parts about one axis

For a chosen rotational axis, a system's total angular momentum is the signed sum of the angular momenta of all its parts:

$L_{\mathrm{total}}=L_1+L_2+\cdots$ .

Check external angular impulse

$J_{\mathrm{ext}}=0\quad\Longrightarrow\quad L_{\mathrm{total},i}=L_{\mathrm{total},f}$

Calculate a nonrigid shape change

Shape change: a rotating person has $I_i=3.0\,\text{kg}\,\text{m}^2$ and $\omega_i=2.0\,\text{rad}\,\text{s}^{-1}$ . Pulling mass inward changes the inertia to $I_f=1.5\,\text{kg}\,\text{m}^2$ , with negligible external angular impulse.

$L_i=I_i\omega_i=(3.0)(2.0)=6.0\,\text{kg}\,\text{m}^2\,\text{s}^{-1}$

$\omega_f=\dfrac{L_i}{I_f}=\dfrac{6.0}{1.5}=4.0\,\text{rad}\,\text{s}^{-1}$ .

Track internal exchanges

Parts of the system can exchange angular momentum. Their interaction gives equal-and-opposite angular impulses, so those internal changes cancel in the system total even though each part's angular momentum may change.

Know when the total changes

Conservation does not mean angular momentum can never change. If the surroundings exert a net angular impulse, then $\Delta L_{\mathrm{system}}=J_{\mathrm{ext}}$ . Always name the system and axis before applying conservation.

Classify torque using the boundary

A torque is external only when the object exerting it lies outside the selected system. The same interaction can therefore be internal for one system choice and external for another.

Change only the selected system

Selected system	Torque from the interaction	Angular-momentum result
Both interacting objects together	Internal	Total $L$ is constant if other external torques are zero
Only one interacting object	External	That object's $L$ changes and momentum crosses the boundary

Apply both choices to coupled disks

Two disks couple on a low-friction axle. For the system containing both disks, their mutual frictional torques are internal and the total angular momentum is constant when the axle exerts negligible external torque. For disk A alone, disk B's frictional torque is external, so $L_A$ changes.

Account for boundary transfer

$\Delta L_{\mathrm{selected\ system}}=J_{\mathrm{external}}$

Do not conserve each object separately

“Angular momentum is conserved in interactions” refers to a sufficiently inclusive system. It does not mean every individual object keeps the same angular momentum. State the boundary, then test net external torque or angular impulse.

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