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7.4 Conservation of Angular Momentum

#angularmomentum

#torque

#nettorque

#inertia

โฑ๏ธย ย 3 min read

written by

Peter Apps

peter apps

June 8, 2020


Enduring Understanding 5.E

The angular momentum of a system is conserved.

Essential Knowledge 5.E.1

If the net external torque exerted on the system is zero, the angular momentum of the system does not change.

Angular Momentum - similar linear momentum - is conserved when there are no external torques on the object(s) in the system.

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Image courtesy of ScienceABC.

Using the skater example from above, the skater has a constant angular momentum throughout the images. Since pulling their hands in is an internal force (and therefore an internal torque), angular momentum is conserved. In terms of I and ๐œ”, the first image has a large I and small ๐œ”, while the second image has a smaller I and larger ๐œ”.

Other common situations with the conservation of angular momentum involve collisions and planetary motion.

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As shown in the image above, the radius of the orbit changes, resulting in different velocities for the planet. You may have covered this idea earlier in terms of potential and kinetic energies, now we can also explain it in terms of conservation of momentum. This idea also leads to Keplerโ€™s 2nd Law (The law of areas).

EXAMPLE: (AP Classroom)

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The left end of a rod of length d and rotational inertia I is attached to a frictionless horizontal surface by a frictionless pivot, as shown above. Point C marks the center (midpoint) of the rod. The rod is initially motionless but is free to rotate around the pivot. A student will slide a disk of mass:

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toward the rod with velocity v0 perpendicular to the rod, and the disk will stick to the rod a distance x from the pivot. The student wants the rod-disk system to end up with as much angular speed as possible.

a. Immediately before colliding with the rod, the diskโ€™s rotational inertia about the pivot is:

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and its angular momentum with respect to the pivot is:

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Derive an equation for the post-collision angular speed ฯ‰ of the rod. Express your answer in terms of d, mdisk, I, x, v0, and physical constants, as appropriate.

STEP 1: Identify applicable equations -

  • The question mentions angular momentum and angular speed. We may also need conservation of momentum because a collision is mentioned

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STEP 2: Plug in knowns & solve for ฯ‰

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Your Answer should be:

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b. Consider the collision for which your equation in part (a) was derived, except now suppose the disk bounces backward off the rod instead of sticking to the rod. Is the post-collision angular speed of the rod when the disk bounces off it greater than, less than, or equal to the post-collision angular speed of the rod when the disk sticks to it?

_____Greater than _____Less than _____Equal to

Briefly explain your reasoning.

CORRECT ANSWER: Greater than

REASONING: When the disk hits the rod, it transfers angular momentum to the rod. This is an internal collison so angular momentum is conserved before and after the collision. When the disk bounces off the rod, its angular momentum changes from positive to negative. This is a larger change in its momentum than when it was brought to rest by sticking to the rod. Since the total momentum is constant, the rod must gain more angular momentum to balance out the now negative momentum of the disk.

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