Physics 204A Class Notes
111
Section 11 – Dynamics of Rotational Motion
What do objects do and why do they do it?
Objects rotate as well as translate.
We can describe this
motion with rotational kinematics and explain it in terms of torques and rotational kinetic energy.
However, the concept of torque can be expanded by treating it as a vector. Just as force is the cause of a
change in linear momentum, torque is the cause of changes in angular momentum.
Using the analogy to
the Law of Conservation of Linear Momentum, we will introduce a new law, the Conservation of
Angular Momentum.
Section Outline
1.
The Vector Cross Product
2.
Torque as a Vector
3.
Angular Momentum and Torque
4.
The Law of Conservation of Angular Momentum
5.
The Gyroscope
1.
The Vector Cross Product
There are two types of vector multiplication.
One produces a scalar quantity and is called the dot
product.
The second type produces a vector and is called the “Cross” or “Vector” product.
We need to
review the cross product to get a better understanding of torque.
Cross Product:
r
A
"
r
B
#
AB
sin
$
ˆ
n
=
A
y
B
z
%
A
z
B
y
( )
ˆ
i
+
A
z
B
x
%
A
x
B
z
( )
ˆ
j
+
A
x
B
y
%
A
y
B
x
( )
ˆ
k
The cross product is perpendicular to both vectors involved and it is often written as,
r
A
!
r
B
=
ˆ
i
ˆ
j
ˆ
k
A
x
A
y
A
z
B
x
B
y
B
z
Example 11.1: For the two vectors
r
A
=
10.0
ˆ
+
20.0
ˆ
j
and
r
B
=
"
30.0
ˆ
+
20.0
ˆ
find (a)their cross product and (b)the angle
between them.
Given:
r
A
=
10.0
ˆ
i
+
20.0
ˆ
j
and
r
B
=
"
30.0
ˆ
i
+
20.0
ˆ
j
.
Find:
r
A
"
r
B
and
θ
.
(a)Using the equation for the cross product,
r
A
"
r
B
=
ˆ
ˆ
ˆ
A
x
A
y
A
z
B
x
B
y
B
z
=
ˆ
ˆ
ˆ
10
20
0
#
30
20
0
=
0
ˆ
+
0
ˆ
j
+
(10)(20)
#
(20)(
#
30)
[ ]
ˆ
$
r
A
"
r
B
=
800
ˆ
.
The cross product points in the zdirection as it should.
x
y
r
A
r
B
θ
r
ˆ
n
r
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112
(b)Using the definition of the cross product,
r
A
"
r
B
=
AB
sin
#
$
=
arcsin
r
A
"
r
=
arcsin
r
A
"
r
x
2
+
A
y
2
x
2
+
B
y
2
Plugging in the numbers,
"
=
arcsin
800
10
2
+
20
2
(
#
30)
2
+
20
2
$
=
82.9˚
2.
Torque as a Vector
In the last chapter we found the torque by taking the part of the force
that is perpendicular to the radius vector and multiplying by r or by taking the
part of the radius vector that is perpendicular to the line of action of the force
(the lever arm) and multiplying by all of the force,
=
F
#
r
=
F
sin
$
( )
r
.
This looks so much like a cross product that we defined torque as,
r
!
"
r
r
#
r
F
,
which means, the torque points out of the paper or along the zaxis in this case.
This may be bothersome at first, but when you think about it, which way would
you point a vector that is describing a rotation in the xy plane.
If it pointed in the
xy plane, it would have to be moving.
This is problematic if the torque vector is
constant.
The only direction it could point is perpendicular to the xy plane, along
the zdirection.
The decision as to whether it points in the positive or negative zdirection is
determined by the convention called the “right hand rule.”
The Right Hand Rule:
 point your fingers along
r
r
.
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 Fall '09
 Kagan
 Physics, Angular Momentum, Law of Conservation of Angular Momentum

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