Angular momentum
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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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Moment of momentum l.jpg
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 l.jpg
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 l.jpg
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.

    • Add external torques


Applying torque l.jpg
Applying Torque

  • An external torque changes angular momentum.

    • dL/dt= t

L

L+rpsinq

w

w

p


Starting the ride l.jpg

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 l.jpg
Spinning Mass merry-go-round with radius 2.0 m.

  • The moment of inertia is the analog of mass for rotational motion.

  • The analog for angular momentum would be:

w


Initial spin l.jpg

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 l.jpg
Single Axis Rotation merry-go-round with radius 2.0 m and mass of 160 kg.

  • 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 l.jpg
Limitations merry-go-round with radius 2.0 m and mass of 160 kg.

  • 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 l.jpg
Angular Momentum Vector merry-go-round with radius 2.0 m and mass of 160 kg.

  • 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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