Physics 53
Rotational Motion 3
Sir, I have found you an argument, but I am not obliged to find you an understanding.
— Samuel Johnson
Angular momentum
With respect to rotational motion of a body, moment of inertia plays the same role that
mass plays in the translational motion of a particle. It measures the intrinsic reluctance
of a body to have its state of rotation changed.
Torque plays the role in rotation that force plays in the translational motion of a particle.
It describes the external influence that causes changes in the state of rotation.
But what describes “the state of rotation” itself? For a particle, the state of translational
motion is described by the linear momentum,
p
=
m
v
.
The corresponding quantity for rotational motion is the
angular momentum
.
Like torque, angular momentum has meaning only with respect to some specified
reference point. Also like torque, its magnitude depends on the distance from that point.
We begin with the simplest system, a single particle. Later we will generalize to systems
of particles, with special interest in rigid bodies. The definition for a particle is:
Angular momentum of a particle
L
=
r
×
p
Here
r
is the position vector of the particle relative to the reference point, and
p
=
m
v
is
its linear momentum.
Some properties of
L
:
•
L
is a vector, perpendicular to the plane containing
r
and
p
, and thus perpendicular
to both
r
and
p
.
•
L
is zero if the particle moves along the line of
r
, i.e., directly toward or away from
the reference point.
PHY 53
1
Rotations 3
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•
The magnitude is given by
L
=
r
⊥
p
, where
r
⊥
is the
moment arm
, defined to be the
perpendicular distance from the reference point to the line along which the particle
moves
.
•
Alternatively,
L
=
rp
⊥
, where
p
⊥
is the component of
p
perpendicular to
r
.
•
L
is a maximum, equal to
rp
, if
p
is perpendicular to
r
. This is the case if the particle
moves (at least momentarily) in a circle about the reference point.
Like linear momentum and kinetic energy, angular momentum is an important aspect
of the state of motion of a particle, especially of orbital motion around some center of
force. It is also an important property of the behavior of a system of particles.
Torque as a vector
Here is the general definition of the torque of a force about a given reference point:
Torque
τ
=
r
×
F
Here
r
specifies the location, relative to the reference point, of the point at which the
force
F
is applied.
Some properties of
τ
:
•
Torque is a vector, perpendicular to the plane containing
r
and
F
, and thus
perpendicular to both
r
and
F
.
•
The torque is zero if
F
acts along the line of
r
, i.e., directly toward or away from the
reference point.
•
The magnitude is given by the two formulas introduced earlier:
τ
=
rF
⊥
=
r
⊥
F
.
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 Spring '07
 Mueller
 Physics, Angular Momentum, Force, Inertia, Mass, Momentum, Iω

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