Transcript Document

Angular Momentum
•
•
•
Angular momentum of rigid bodies
Newton’s 2nd Law for rotational motion
Torques and angular momentum in 3-D
Text sections 11.1 - 11.6
Physics 1D03 - Lecture 30
1
“Angular momentum” is the rotational analogue
of linear momentum.
Recall linear momentum: for a particle, p = mv .
Newton’s 2nd Law: The net external force
on a particle is equal to the rate of change
of its momentum.
Fexternal
dp

dt
To get the corresponding angular relations for a rigid body, replace:
m
v
F
p
I
w
t
L (“angular momentum”)
Physics 1D03 - Lecture 30
2
Angular momentum of a rotating rigid body:
Angular momentum, L, is the product of the
moment of inertia and the angular velocity.
L = Iw
Units: kg m2/s (no special name). Note similarity to: p=mv
Newton’s 2nd Law for rotation: the torque due to
external forces is equal to the rate of change of L.
For a rigid body (constant I ),
So, sometimes
t external
dL

dt
dL d ( Iw )
dw

I
 I
dt
dt
dt
t external  I (but not always).
Physics 1D03 - Lecture 30
3
Conservation of Angular momentum
There are three great conservation laws in classical mechanics:
1) Conservation of Energy
2) Conservation of linear momentum
3) and now, Conservation of Angular momentum:
In an isolated system (no external torques),
the total angular momentum is constant.
Physics 1D03 - Lecture 30
4
Angular Momentum Vector
w
For a symmetrical, rotating, rigid body,
the vector L will be along the axis of
rotation, parallel to the vector w, and
L
L = I w.
(In general L is not parallel to w, but Iw is still equal
to the component of L along the rotation axis.)
Physics 1D03 - Lecture 30
5
Angular momentum of a particle
L  r  p  r  (mv)
z
L
This is the real definition of L.
O
y
r
v
• L is a vector.
x
m
f
• Like torque, it depends on the
choice of origin (or “pivot”).
• If the particle motion is all in the x-y
plane, L is parallel to the z axis..
Physics 1D03 - Lecture 30
6
Angular momentum of a particle (2-D):
|L| = mrvt
= mvr sin θ
r
v
m
For a particle travelling in a circle (constant |r|, θ=90),
vt = rw, so:
L = mrvt = mr2w = Iw
Physics 1D03 - Lecture 30
7
Quiz
As a car travels forwards, the angular momentum
vector L of one of its wheels points:
A) forwards
B) backwards
C) up
D) down
E) left
F) right
Physics 1D03 - Lecture 30
8
Quiz
A physicist is spinning at the center of a frictionless
turntable, holding a heavy physics book in each hand with
his arms outstretched. As he brings his arms in, what
happens to the angular momentum?
A) increases
B) decreases
C) remains constant
What happens to the angular velocity?
Physics 1D03 - Lecture 30
9
Example:
A student sits on a rotating chair, holding two weights each of
mass 3.0kg. When his arms are extended to 1.0m from the
axis of rotation his angular speed is 0.75 rad/s. The students
then pulls the weights horizontally inward to 0.3m from the
axis of rotation.
Given that I = 3.0 kg m2 for the student and chair, what is the
new angular speed of the student ?
Physics 1D03 - Lecture 30
10
Example
Angular momentum provides a neat approach to
Atwood’s Machine. We will find the accelerations of
the masses using “external torque = rate of change
of L”.
O w
R
v
m1
m2
Physics 1D03 - Lecture 30
v
11
Atwoods Machine, frictionless (at pivot), massive pulley
For m1 : L1 = |r1 x p1|= Rp1
so L1 = m1vR
L2 = m2vR
Lpulley= Iw = Iv/R
O w
R
R
Thus L = (m1 + m2 + I/R2)v R
r
so dL/dt = (m1 + m2 + I/R2)a R
Torque, t = m1gR - m2gR
= (m1 - m2 )gR
v
p1
m1
m2
v
p1
Write t = dL/dt, and complete the calculation to solve for a.
Note that we only consider the external torques on the entire system.
Physics 1D03 - Lecture 30
12
Solution
Physics 1D03 - Lecture 30
13
Summary
Particle:
Any collection of particles:
L  r  p  r  (mv)
L   ri  p i
i
Rotating rigid body:
Newton’s 2nd Law for rotation:
L = I w.
t external
dL

dt
Angular momentum is conserved if there is no external torque.
Physics 1D03 - Lecture 30
14