Angular Momentum

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# Angular Momentum - PowerPoint PPT Presentation

Angular Momentum. Moment of Momentum . To continue the analysis of rotational motion, we must also extend the idea of momentum. Define an angular momentum L . Based on the momentum p = mv Includes a lever arm Follows the rules for torque. p. r. Momentum Cross Product .

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## PowerPoint Slideshow about 'Angular Momentum' - Mia_John

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Presentation Transcript

### Angular Momentum

Moment of Momentum
• To continue the analysis of rotational motion, we must also extend the idea of momentum.
• Define an angular momentum L.
• Based on the momentum p = mv
• Includes a lever arm
• Follows the rules for torque

p

r

Momentum Cross Product
• Angular momentum is a vector.
• Vector cross product of the lever arm and momentum.
• Direction follows the right-hand rule
• Magnitude from sine rule

L

p

r

Law of Angular Inertia
• The time derivative of angular momentum vector is the net torque vector.
• By the law of reaction, all internal torques come in canceling pairs.
Applying Torque
• An external torque changes angular momentum.
• dL/dt= t

L

L+rpsinq

w

w

p

A child of 180 N runs at 4.5 m/s and hops on the edge of a merry-go-round with radius 2.0 m.

If the motion is perpendicular to the radius, what is angular momentum?

The child starts with linear momentum.

W = mg, m = W/g = 18 kg.

p = mv = 81 kg m/s

At right angles, all the linear momentum contributes.

L = rp = 160 kg m2/s

The torque is due to the friction at contact.

Starting the Ride

m

r

Spinning Mass
• The moment of inertia is the analog of mass for rotational motion.
• The analog for angular momentum would be:

w

A child of 180 N runs at 3.0 m/s and hops on the edge of a merry-go-round with radius 2.0 m and mass of 160 kg.

What is the period of rotation?

The moment of inertia and the angular momentum for the child on the ride was found before.

I = Id + Ic =390 kg m2

L = rp = 160 kg m2/s

The period is related to the angular velocity.

L = Iw = I(2p/T)

T= 2p I/ L = 15 s

Initial Spin

m

M

r

Single Axis Rotation
• An axis of rotation that is fixed in direction gives a single axis rotation.
• Simplest case has the axis through the center of mass
• Angular momentum vector is parallel to the angular velocity

L

w

Limitations
• There are limitations to the relationship between angular momentum and angular velocity.
• Moving axis of rotation
• Asymmetric axis of rotation
• Angular momentum and angular velocity can have different directions.

p

L

r

Angular Momentum Vector
• The vector form of the law of rotational action is generalized to use angular momentum vectors.
• Correct for all axes
• Correct for changes in direction as well as angular velocity

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