Angular Momentum about a Single Axis
The circulation of linear momentum about the specified axis, being proportional to the component of momentum or each mass along a circle about the axis and the radius of the circle. Angular momentum is changed by external torques, and therefore is constant when these sum to zero. The angular momentum of a rigid body is proportional to its moment of inertia times its angular velocity.
Motivation for Concept
Imagine a satellite spinning in space. While there is a gravitational force on the satellite which causes its linear momentum to change, the satellite will continue to spin at the same speed about its axis of rotation. To change its rate of spin would require the application of an external force that did not act on the center of mass, but rather acted at some point off the axis of rotation. Moreover, the force would not change the spin rate if it acted directly toward or away from the axis - only if it had some component along or against the rotational velocity of the point where it was applied.
For a rigid body, angular momentum is analogous to the linear momentum, but is a more complicated topic than conservation of linear momentum because internal motions of the system can change the rate of rotation of a system whereas internal motions cannot change its total linear momentum. For example, if the satellite spread out solar panels, its rate of rotation would change, just as a twirling skater who pulls in his arms will rotate faster. Thus angular velocity can change even when angular momentum about some point is conserved due to the fact that there is no applied torque about that axis.
Utility of Single-Axis Case
The specification of a single axis of rotation allows a complete description of the angular momentum only in certain cases. To ensure that a problem can be completely described using angular momentum about a single axis, the following conditions must hold.
Movement in a Plane
Rotational motion in introductory physics will be concerned only with systems where the position of the center of mass and the linear momentum of each constituent is confined to a plane (by our convention, the xy plane).
Angular velocity and angular momentum Perpendicular to the Plane
Each rigid body in this system that is rotating must only rotate such that its angular velocity and angular acceleration are directed perpendicular to the xy plane containing the centers of mass, that is in the +z or -z direction.
Examples of Valid Systems
Systems that can be treated as having one-dimensional angular momentum include:
- Ball rolling from rest down an inclined plane.
- A yo-yo.
- Non-spinning baseball moving horizontally strikes bat pivoted horizontally at the handle with ball center of mass at the same height as bat center of mass.
Examples of Invalid Systems
Systems that require more than 1-D angular momentum:
- Styrofoam cup (wider on the top than the bottom) rolling down an inclined plane (the cup will begin to turn as it rolls).
- Baseball moving horizontally with spin along a horizontal axis strikes bat pivoted horizontally at the handle with the ball center of mass at same height as bat center of mass (ball and bat angular momenta perpendicular).
- Baseball moving horizontally with no spin strikes bat pivoted horizontally at the handle with ball center of mass above or below the bat center of mass (ball center of mass and bat center of mass move in different planes).
- [Gyroscopes] (the one invalid system that is generally covered in introductory mechanics).
Conditions on the Chosen Axis of Rotation
For a consistent definition of angular momentum, the axis of rotation must satisfy one of two possibilities:
- The axis is fixed (does not move) in an inertial reference frame.
- The axis intersects the xy plane at the point of the system's center of mass. (Note that the system's center of mass may be moving and even accelerating.)
Definition of Angular Momentum
Angular Momentum of a Point Particle
Beginning with Newton's 2nd Law for a point particle:
\begin
[ \sum \vec
= \frac{d(m\vec
)}
] \end
we take the cross product of the particle's position measured in the xy plane from the point of intersection with the chosen axis of rotation:
\begin
[ \sum \vec
\times\vec
= \vec
\times \frac{d(m\vec
)}
]\end
Assuming m is constant, we can pass it out of the derivative. We cannot assume that it is reasonable to pass r into the derivative, however, since it certainly has a time dependence (unless the velocity is zero). We can, however, prove that it is reasonable to pass r into the derivative. To see this, we evaluate:
\begin
[ \frac{d(\vec
\times\vec
)}
= \frac{d\vec
}
\times \vec
+ \vec
\times\frac{d\vec{v}}
= \vec
\times\vec
+ \vec
\times\frac{d\vec
}
]\end
Noting that the cross product of any vector with itself is zero, we then see that if the particle's mass is constant:
\begin
[ \sum \vec
\times\vec
= \frac{d(m\vec
\times\vec
)}
] \end
The left hand side is simply the net torque (single-axis) on the particle about the chosen axis. Thus, in the absence of net torque, the quantity:
\begin
[ L_
= m\vec
\times\vec
]\end
is constant in time. We therefore choose to define this quantity as the angular momentum of the point particle about the chosen axis. Note that L is a vector quantity, but it is one-dimensional (it must point perpendicular to the xy plane because of the cross product and our assumptions). We have therefore indicated the vector nature by a component subscript instead of a vector arrow.
Angular Momentum of a Rigid Body in Pure Rotation
By dividing a rigid body into N tiny mass elements Δmi (i ranging from 1 to N), we can express its angular momentum about a specific axis as:
\begin
[ L_
= \sum_
^
\vec
_
\times\vec
_
(\Delta m_
) ]\end
The assumption of pure rotation, however, means that the velocity is completely tangential (perpendicular to r) and equal in magnitude to the magnitude of rω where ω is the angular velocity associated with the rotation. We can therefore write:
\begin
[ L_
= \sum_
^
r_
^
\omega \Delta m_
= \omega \sum_
^
r_
^
\Delta m_
]\end
where we have removed ω from the sum since a body in pure rotation has a uniform angular velocity.
In the limit of a continuous body, the sum goes over to an integral:
\begin
[ \sum r_
^
\Delta m_
\rightarrow \int r^
dm ]\end
which is simply the definition of the body's moment of inertia about the chosen axis.
Thus, for a rigid body:
\begin
[ L_
= I_
\omega]\end
where Ia is the body's moment of inertia about the chosen axis.
Angular Momentum of a Rigid Body both Translating and Rotating
A rigid body's center of mass can be translating with respect to the axis chosen. If we again imagine dividing the body into N mass elements Δmi, we note that these mass elements must execute pure rotation about the rigid body's center of mass.
If the mass elements of the rigid body could change their radial separation from the center of mass, then the relative distance between points in the body would change. Such a change would violate the definition of rigid body.
Because of this property of rigid bodies, it makes sense to restate the sum:
\begin
[ L_
= \sum_
^
\vec
_
\times\vec
_
(\Delta m_
) ]\end
by defining body coordinates r' such that:
\begin
[ \vec
= \vec
\:' + \vec
_
]\end
where rcm is the position of the body's center of mass measured from the axis of rotation. Taking a time derivative causes us to similarly define:
\begin
[ \vec
= \vec
\:' + \vec
_
]\end
We can now use the constraint of the rigid body (pure rotation about the center of mass) to define an ωcm (the angular velocity of the body about the center of mass) which allows us to write:
\begin
[ \vec
\times\vec
= \omega_
^
+ \vec
_
\times\vec
\:' + \vec
\:'\times\vec
_
+\vec
_
\times\vec
_
]\end
Inserting this expression into the sum for the body's angular momentum gives:
\begin
[ L_
= \sum_
^
(\omega_
_
^
+ \vec
_
\times\vec
_
\:' + \vec
_
\:'\times\vec
_
+ \vec
_
\times\vec
_
) (\Delta m_
) ]\end
In the primed coordinates, however, the center of mass velocity and the center of mass position are by definition equal to zero. Thus:
\begin
[ \sum_
^
\vec
_
\:' \:\Delta m_
=\sum_
^
\vec
_
\:'\:\Delta m_
= 0 ]\end
Therefore, we have:
\begin
[ L_
= \left(\omega_
\sum_
^
_
^
\Delta m_
\right) + \left(\vec
_
\times\vec
_
\right) \left(\sum_
^
\Delta m_
\right) = I_
\omega_
+ M_
\vec
_
\times\vec
_
]\end
where Mtot is the total mass of the rigid body.
Comparing this expression to those found in the previous sections, we see that the angular momentum of a rigid body that is both translating and rotating with respect to the chosen axis of rotation can be thought of as the sum of the angular momentum of the center of mass treated as a point particle with the mass of the entire rigid body and the angular momentum contributed by the body's rotation about the center of mass.
Advanced Section: Valid Choices for the Axis of Rotation
Under construction.