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Chapter 11
Angular Momentum;
General Rotation
Introduction
• Recap from Chapter 10
– Used torque with axis fixed in an inertial frame
– Used equivalent of Newton’s Second Law
– Rotational kinetic energy as mechanical
energy
• What’s ahead
– General rotation can be quite complicated
– Our goal is understanding several important
new aspects and our limitations
– Awareness of limits to what is known; e.g.
spin and its technological uses
Our approach
• Angular momentum provides insight
• New precision and math: the meaning and
use of the cross (vector) product
• Torque and angular momentum for a
particle, a system of particles, rigid bodies
• Collision of rotating objects (lab)
• Combining translation and rotation
• Comments on more complex systems and
rotating frames of reference
Angular Momentum of Objects
Rotating about a Fixed Axis
• Angular momentum as analogue of linear
momentum (What could we conclude?)
• Scalar expressions for angular
momentum, the relation of torque and
angular momentum, and conservation of
angular momentum
• Examples (demo, sports, weather)
• Question, Question
Angular Momentum of Objects
Rotating about a Fixed Axis…
• Direction of angular momentum
– Right-hand rule
– Vectors and pseudovectors (axial vectors)
– Directions of angular momentum and angular
velocity for a symmetrical object rotating
about a symmetry axis
– What remains true otherwise?
• Symmetry of nature corresponding to
conservation of angular momentum
Vector Cross Product
• Math for more precision, generality
• Torque as a vector
• The cross product
– Definition
– Examples
– Question, Question
• The torque vector
– Question
• Examples
Angular momentum of a particle
• Considering p and L in a simple system of
particles and their constancy if Force and
Torque are zero
• A more precise definition of angular
momentum
• Analogue of Newton’s 2nd Law
– Derivation question
– Limitations
– Question
Angular momentum & torque
for a system of particles
• Relation between angular momentum and
torque
– Definitions of total angular momentum, net
torque, and external torque
– Relationship
– Picture
– Limitations?
Angular momentum & torque
for a rigid object
• Relation between component of angular
momentum along axis of rotation and
angular velocity about that axis
– Derivation in book
• Relation between angular momentum and
angular velocity when rotation axis is a
symmetry axis through CM
– Discussion in book
Angular momentum & torque
for a rigid object (cont’d)
• General relation between angular
momentum and torque when calculation is
done about either (1) the origin of an
inertial frame, or (2) CM of system
– Previous result since a rigid body is a special
case of a system of particles
Angular momentum & torque for
a rigid object (cont’d…)
• From the previous, the relations among
the external torque along the rotation axis,
the angular momentum along the rotation
axis, the moment of inertia about the
rotation axis, the angular velocity about
the rotation axis, and the angular
acceleration about the rotation axis
• Examples of rotation only (Precession)
• Examples with translation (Pulleys,
Physlet 10.12, more)
More…
• More complicated examples
– Kepler’s Second Law (in text)
– Zero torque analogue (in Physlets)
• Rotating frames of reference
– Resulting fictitious forces
the end
What three general conditions could make
A x B = 0?
(Write examples out of actual vectors as
specifically as possible on the whiteboard
provided.)
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What is r x F, where
r = (1m, .5m, -2m)
F = (4N, 2N, -1N)
(Use the whiteboard provided.)
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A particle moves with a constant speed in a
straight line. How does its angular
momentum, calculated about any point not
in its path, change in time?
• Hint: A physics argument will show that
the net force on the particle is zero, so the
net torque must be zero about any point.
• Make a mathematical argument (on the
whiteboard)
• Make a diagrammatic argument (on the
whiteboard)
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Describe how astronauts floating in space
can move their arms to turn upside down?
Or turn around to face the opposite
direction?
(Demonstrate this.)
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Start with L = r x p
How is the net torque related to dL/dt?
Assume L is about a point at rest in an
inertial frame.
use whiteboards provided
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Conservation of angular momentum is
useful for understanding the motion of one
object, whereas conservation of linear
momentum is not and is useful for a
system of more than one object.
Why (in general)?
Give an example of its use for one object.
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We learned previously that v = rω, which is
true for the diagram given.
Which of the following is/are true:
1. v = rω
2. v = ωr
3. v = r x ω
4. v = ω x r
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