angular momentum

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Angular momentum

Physics 7C lecture 14

Thursday November 14, 8:00 AM – 9:20 AMEngineering Hall 1200

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Torque as a vector

• Torque can be expressed as a vector using the vector product.

• How to find the direction of torque using a right hand rule.

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Rigid body rotation about a moving axis• The motion of a rigid body is a

combination of translational motion of the center of mass and rotation about the center of mass

• The kinetic energy of a rotating and translating rigid body is K = 1/2 Mvcm

2 + 1/2 Icm2.

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Rolling without slipping• The condition for rolling without slipping is vcm = R.

• Figure 10.13 shows the combined motion of points on a rolling wheel.

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Work and power in rotational motion • The total work done on a body by the torque is equal to the

change in rotational kinetic energy of the body and the power due to a torque is P = zz.

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Angular momentum • What’s the analogue of momentum P for angular motion?

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Angular momentum • Momentum : p = m v

• angular momentum: τ = r × p = m r × v

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Angular momentum • The angular momentum of a rigid body rotating about a

symmetry axis is parallel to the angular velocity and is given by L = I.

• Right hand rule!

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Angular momentum • For any system of particles = dL/dt.

• For a rigid body rotating about the z-axis z = Iz.

• It makes sense!

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Conservation of angular momentum

• When the net external torque acting on a system is zero, the total angular momentum of the system is constant (conserved).

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Conservation of angular momentum

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A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K?

A. L and K both increase.

B. L stays the same; K increases.

C. L increases; K stays the same.

D. L and K both stay the same.

Q10.11

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A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K?

A. L and K both increase.

B. L stays the same; K increases.

C. L increases; K stays the same.

D. L and K both stay the same.

A10.11

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Conservation of angular momentum

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A rotational “collision”

• Find ω. Where is the energy lost?

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Angular momentum in a crime bust

• A bullet hits a door causing it to swing. Find ω.

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Angular momentum in a crime bust

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Gyroscopes and precession

• For a gyroscope, the axis of rotation changes direction. The motion of this axis is called precession.

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Non-rotating gyro

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A rotating flywheel• For a spinning flywheel, the magnitude of the angular

momentum stays the same, but its direction changes continuously.

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A rotating flywheel• precession angular frequency: Ω = τ / L.

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A precessing gyroscopic• What is the precession direction? CW or CCW?