Angular Momentum

Angular momentum is a measure of the momentum of an object around an axis.

Linear momentum (p) is defined as the mass (m) of an object multiplied by the velocity (v) of that object:

p = m*v.


With a bit of a simplification, angular momentum (L) is defined as the distance of the object from a rotation axis multiplied by the linear momentum:

L = r*p or L = mvr.
This equation works for a single particle moving around a central point, for example a planet orbiting around the Sun or a rock tied onto a string that is swung in a circle.  The mass is that of the planet, the velocity is the planet's orbital velocity, the distance can be taken as the semi-major axis of the orbit. (More accurately, the velocity and distance from the Sun both change as the planet moves in an elliptical orbit, but the product of the velocity times the distance stays constant, so we can pick any one point in the orbit and calculate L).  For an object in orbit, we know that the velocity can be written as
v = 2*pi*R/Torb
where Torb is the orbital period of revolution, in seconds.  Thus, we can rewrite our equation for
L as
Lorb = M*(2*pi*R/Torb)*R= MR2 * 2*pi/Torb,
For a rigid body rotating on an axis (e.g., the Earth spinning), the angular momentum is the product of the moment of inertia (I) and the angular velocity (w):
Lrot = I*w,
and for a rigid, spherical body,
I = 0.4MR2,
where M is the mass and R is the radius of the sphere.  The angular velocity w is measured in radians per second (the distance around the circumference of a circle is 2*pi radians), and can be calculated from
w = 2*pi / Trot,
where T is the period of rotation of the sphere.  Thus,
Lrot = 0.4 MR2 * 2*pi / Trot = 0.8*pi * MR2 / Trot.
Example 1: The rotational angular momentum of the Earth is
Lrot =  0.8*pi * MR2 / Trot
Lrot =  0.8 * 3.14 * (5.98 x 1024 kg)(6.37 x 106m)2 / (8.64 x 104 s)
Lrot =  7.1 x 1033 kg m2 s-1
Example 2: The orbital angular momentum of the Earth is
Lorb = MR2 * 2*pi/Torb
Lorb = 2 * 3.14 *(5.98 x 1024 kg)(1.496 x 1011m)2/(3.15 x 107 s)
Lorb = 2.7 x 1040 kg m2 s-1
Clearly, most of the Earth's angular momentum is in its orbit, not its rotation.