Transcript Angular momentum
Physics 7C lecture 14
Angular momentum
Thursday November 14, 8:00 AM – 9:20 AM Engineering 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
Mv
cm 2 + 1/2
I
cm 2 .
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• •
Rolling without slipping
The condition for rolling without slipping is
v
cm =
R
.
Figure 10.13 shows the combined motion of points on a rolling wheel. Copyright © 2012 Pearson Education Inc.
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 =
z
z
.
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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 Copyright © 2012 Pearson Education Inc.
•
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 = I
z
.
• 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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Q10.11
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.
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A10.11
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.
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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. Copyright © 2012 Pearson Education Inc.
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?
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