For a rigid body experiencing planar (two-dimensional) motion, the angular momentum is given by the following general scalar equation:

Where:

The figure below illustrates the general case where a rigid body is experiencing planar (two-dimensional) motion.

Note that

If the rigid body has a fixed point

The figure below illustrates the case where a rigid body is experiencing planar (two-dimensional) motion, and is rotating about a fixed point

For a rigid body experiencing planar (two-dimensional) motion, and rotating about a fixed point

Where:

For a rigid body experiencing planar (two-dimensional) motion, the sum of the moments about the center of mass

And if the rigid body is rotating about a fixed point

The left side of equations (3) and (4) represents the sum of the moments, either about point

Dropping the subscripts

Integrate this equation from time

The term on the left is defined as the external impulse acting on the rigid body (between

If no external moments act on the rigid body (either about point

Angular momentum is therefore conserved for the rigid body (between

For general three-dimensional motion, the angular momentum of a rigid body is given by the following general scalar equations:

Where:

The moment of inertia terms are given by

The product of inertia terms are given by

Note that we can orient the local

If

If the rigid body has a fixed point

For a rigid body experiencing general three-dimensional motion, the sum of the moments about the center of mass

Note that the moments are calculated with their components resolved along the local

From equation (8),

Integrate this equation from time

The term on the left is defined as the external impulse acting on the rigid body (between

Once again, this equation applies for both cases where the local

If no external moments act on the rigid body (either about point

Angular momentum is therefore conserved for the rigid body (between

For the case where we wish to determine the angular momentum of a rigid body about an arbitrary point, see problem # 9 in the momentum problems page.

The equations for planar (two-dimensional) motion follow naturally from the equations for general three-dimensional motion, so it is only necessary to derive the equations for three-dimensional motion.

To derive the three-dimensional equations for the angular momentum of a rigid body we must apply the equation for angular momentum of a small mass element in the rigid body, and then sum it over the entire rigid body. The problem is set up as shown in the figure below.

Note that, for the proof that follows the local

For a small mass element

Where:

Differentiate equation (10) with respect to time:

Now,

Therefore the first term on the right side of equation (11) is zero.

And from the equations for general motion of a rigid body:

Where:

Differentiate equation (12) with respect to time:

Where:

Substitute equation (13) into equation (11) and we get

By Newton’s Second Law,

where

Equation (14) becomes

Summing over all the particles

The second term on the right is zero because

To see why the above equation is true go to center of mass.

Thus,

By definition, the right side of this equation is the sum of the moments of the external forces acting on the rigid body, about point

The above equation tells us that the rate of change of angular momentum of the rigid body about point

Now, let’s look again at equation (10):

From the equations for general motion of a rigid body:

It remains to carry out the vector cross product multiplications in this equation and then sum

Now, if we were to start from the beginning and assume a fixed point

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