Chapter I
Rotational Dynamics and Equilibrium
Blinn College  Physics 2425  Terry Honan
The Vector Nature of Rotational Quantities
Torque about an Axis
Axis
r
”
r
¶
=
r
sin
q
F
F
¶
=
F
sin
q
q
q
We define torque as the rotational analog of force. Suppose you are trying to loosen a bolt. The axis of rotation is the center of the bolt. If
you are unable to give sufficient torque with your hand you grab a wrench. Take
r
”
as the vector from the axis to where the force
F
is applied.
Clearly the important part of the force is the component of the force perpendicular to the radial vector
r
”
. Moreover the larger
r
is the larger the
torque. This motivates the definition of torque
t =
r F
¶
If
q
is the angle between
r
”
and
F
then we can write
F
¶
=
F
sin
q
. Similarly we can
r
¶
=
r
sin
q
as the component of
r
”
perpendicular to
F
. This
gives us other ways of writing the torque.
t =
r F
¶
=
r F
sin
q =
r
¶
F
The sign of torque depends on the sign convention for kinematics. If a force tends to make something rotate in the positive direction then the
torque is positive and similarly negative torques tend to make things rotate in the negative direction.
Angular Velocity and Torque as Vectors
A rotation has a magnitude, the angle of rotation, and a direction, along the direction of the axis. Rotations are not vectors, though. Vector
addition is commutative but rotations are not. It turns out that infinitesimal rotations do commute and
are
vectors. We can write an infinitesimal
rotation as
„q
”
. Since angular velocity about an axis requires only an infinitesimal rotation,
w = „q
ê
„
t
, we can define the angular velocity vector
w
=
„q
”
„
t
.
There are two possible directions along an axis. We decide which direction by using the right hand rule. Wrap the fingers of your right hand in
the direction of rotation. The thumb points in the direction of the vector. If angular velocity is a vector then we can also make angular accelera
tion a vector.
a
=
„
„
t
w
Wit these considerations we can now make a vector out of the torque. We can assign its direction to the sense of rotation due to that torque.
r
”
and
F
are vectors; we will define the cross product so that the cross product of
r
”
and
F
is the torque
t
.
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t
=
r
”
ä
F
Interactive Figure
The Cross or Vector Product
The dot product, or scalar product, is a way of multiplying of two vectors that gives a scalar. The cross product, also known as the vector
product, is a multiplication that gives a vector.
A
ä
B
is a vector.
The magnitude of this vector is
A B
sin
q
. We will specify the direction with a unit vector
u
`
. The two vectors
A
and
B
define a plane; their cross
product is perpendicular to that plane. There are two unit vectors perpendicular to any plane; we use the right hand rule to find the correct one.
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 Summer '08
 Honan
 Physics, Angular Momentum, Force

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