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Physics 170 - Mechanics
Lecture 27
Angular Momentum
Angular Momentum
For a particle of mass m
moving in a circle of radius r:
More generally:
L=rxp
Angular Momentum
For more general motion,
Sign of Angular Momentum
L>0
L>0
The angular momentum L is taken to be positive if the
angular position is increasing with time, i.e., if the motion
associated with L is a counterclockwise rotation.
Rotation as a Vector
We can treat rotation and other angular motion
quantities as vectors by using the right-hand rule: if the
fingers of your right hand follow the rotation direction,
then your thumb points along the rotation axis in the
vector direction of the angular velocity ω.
An alternative definition is that if a right-hand
threaded screw is rotated, then ω points in the direction
in which the screw advances.
The Vector Nature of Rotational Motion
Similar application
of the right-hand rule
gives the vector
direction of the
torque τ.
The Vector Nature of Rotational Motion
The vector
directions of the
angular velocity vector
ω and the angular
momentum vector L
are along the axis of
rotation. Applying the
right-hand rule gives
the direction.
Angular Momentum and
Angular Velocity as Vectors
Example: Angular Momentum
About the Origin
Find the angular momentum about the origin for the
following situations:
(a) A car of mass 1200 kg moves in a counterclockwise
circle in the xy plane of radius 20 m with a speed of 15 m/s;
(b) A uniform disk in the xy plane of radius 20 m and mass
1200 kg rotates at 0.75 rad/s along its axis, which is the z
axis.
Example: Angular Momentum
(a) What is the angular momentum of a 0.13 kg Frisbee, considered
to be a uniform disk of radius 7.5 cm, spinning with ω = 11.5 rad/s?
(b) What is the angular momentum of a 95 kg person running with a
speed of 5.1 m/s around a circular track of radius 25 m?
Example: The Spin Angular
Momentum of the Earth
What is the angular momentum of the Earth as it
rotates on its axis? (Assume a uniform sphere.)
Example: The Orbital Angular
Momentum of the Earth
What is the angular momentum of the
Earth as it orbits the Sun?
Changing Angular Momentum
Looking at the rate at which angular momentum
changes,
Therefore, if τ = 0, then L is constant with time.
If the net external torque on a system is zero, the
angular momentum is conserved.
Example: A Windmill
In a light wind, a windmill experiences a constant torque of 255 N m.
If the windmill is initially at rest, what is its angular momentum
after 2.00 s?
Notice that you do not need to know the moment of
inertia of the windmill to do this calculation.
Example: Jumping On
Running with a speed of 4.10 m/s, a
21.2 kg child heads toward the rim of a
merry-go-round of radius 2.00 m, as
shown.
What is the child’s angular
momentum L with respect to the
center of the merry-go-round?
(L = r x p)
Conservation of Angular Momentum
If the net external torque on a system is
zero, the angular momentum is conserved.
The most interesting consequences occur in
systems that are able to change shape:
3.74 rad/s
5.33 kg m2
1.60 kg m2
Example: A Stellar Performance
A star of radius Ri = 2.3 x 108 m rotates initially with
an angular speed of ωi = 2.4 x 10-6 rad/s.
If the star collapses to a neutron star of radius Rf =
20.0 km, what will be its final angular speed ωf ?