Angular Momentum Angular Momentum – Concepts & Definition

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Angular momentum – concepts & definition

- Linear momentum: p = mv

- Angular (Rotational) momentum: L = moment of inertia x angular velocity = I

linear rotational inertia m I speed v  rigid body linear p=mv L=I angular momentum momentum

1 Angular momentum of a bowling ball  6.1. A bowling ball is rotating as shown about its mass center axis. Find it’s angular momentum about that axis, in kg.m2/s

A) 4 B) ½ C) 7 D) 2 E) ¼

 = 4 rad/s M = 5 kg R = 0.5 m I = 2/5 MR2 L  I

Angular momentum of a point particle 2 L I mr  mvTpTp r  mvrsin   mvr = p r  pr

vT   r O  tangential r r p vT    v

radial Note: L = 0 if v is parallel to r (radially in or out)

||L I mvTp r  mvr sin  mvr

= rpTp rp sin r p

rFTp rFsin  r F

2 Angular momentum for car

 5.2. A car of mass 1000 kg moves with a speed of 50 m/s on a circular track of radius 100 m. What is its angular momentum (in kg • m2/s) relative to the center of the race track (point “P”) ?

A) -5.0  102 A

6 B) -5.0  10 B

4 C) -2.5  10 P D) 5.0  106 E) 5.0  102

 5.3. What would the angular momentum about point “P” be if the car leaves the track at “A” and ends up at point “B” with the same velocity ? A) Same as above B) Different from above ||L I mvTp r  mvr sin  mvr C) Not Enough Information = rpTp rp sin r p

Net angular momentum

LLLLnet 123...

3 Example: calculating angular momentum for particles

PP10602-23*: Two objects are moving as shown in the figure . What is their total angular momentum about point O?

||L I mvTp r  mvr sin  mvr

= rpTp rp sin r p

 L II  Lt net  net tt

Conservation of angular momentum :

Angular momentum, L is consreved if  net  0 Conservation of net angular momentum for a system

Net angular momentum, Lnet is consreved if  net,ext  0

4 A puck on a frictionless air hockey table has a mass of 5.0 g and is attached to a cord passing through a hole in the surface as in the figure. The puck is revolving at a distance 2.0 m from the hole with an angular velocity of 3.0 rad/s. The cord is then pulled from below, shortening the radius to 1.0 m. The new angular velocity (in rad/s) is ______.

Tethered Astronauts

 6.3. Two astronauts each having mass m are connected by a mass-less rope of length d. They are isolated in space, orbiting their center of mass at identical speeds v. By pulling on the rope, one of them shortens the distance between them to d/2. What are the new net angular momentum L’ and speed v’ of the astronaut? A) L’ = mvd/2, v’= v/2 B) L’ = mvd, v’ = 2v C) L’ = 2mvd, v’ = v D) L’ = 2mv’d, v’ = v/2 E) L’ = mvd, v’= v/2

L  I

L  mv Tr

5 Demonstration: Spinning Professor: Web link

Isolated System

 net, ext 0 L constant

Iii  I f f

Moment of inertia changes Another link

Example A professor on a freely rotating platform extends his arms horizontally, holding a 10-kg mass in each hand.

He is rotating about a vertical axis with an angular velocity of 2 rad/s. If he pulls the mass closer to his body, what will the final angular velocity be if his body’s moment of inertia (excluding the two masses, of course) remains approximately constant at 10 kg m2, the moment of inertia of the platform is 5 kg m2, and the distance of the masses from the axis changes from 1 m to 0.2 m?

6 Example: A merry-go-round problem

A 40-kg child running at 4.0 m/s jumps tangentially onto a stationary circular merry-go-round platform whose radius is 2.0 m and whose moment of inertia is 20 kg-m2.

Find the angular velocity of the platform after the child has jumped on.