Equations of Motion for Rigid Bodies

We are now ready to write down the general equations of motion for rigid bodies in terms of for the center of mass and for the rotation of the body about its center of mass.

As discussed above, it is useful to decompose the motion of a rigid body into

(1)

the linear velocity of its center of mass, and

(2)

its angular velocity about its center of mass.

The linear motion is governed by Newton's second law , where is the total mass, is the velocity of the center-of-mass, and is the sum of all external forces on the rigid body. (Equivalently, is the sum of the radial force components pointing toward or away from the center of mass.) Since this is so straightforward, essentially no harder than dealing with a point mass, we will not consider it further.

The angular motion is governed the rotational version of Newton's second law introduced in §B.4.19:

where is the vector torque defined in Eq.(B.27), is the angular momentum, is the mass moment of inertia tensor, and is the angular velocity of the rigid body about its center of mass. Note that if the center of mass is moving, we are in a moving coordinate system moving with the center of mass (see next section). We may call the intrinsic momentum of the rigid body, i.e., that in a coordinate system moving with the center of the mass. We will translate this to the non-moving coordinate system in §B.4.20 below.

The driving torque is given by the resultant moment of the external forces, using Eq.(B.27) for each external force to obtain its contribution to the total moment. In other words, the external moments (tangential forces times moment arms) sum up for the net torque just like the radial force components summed to produce the net driving force on the center of mass.

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Subsections

• Body-Fixed and Space-Fixed Frames of Reference
• Angular Motion in the Space-Fixed Frame
• Euler's Equations for Rotations in the Body-Fixed Frame
• Examples

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Previous: Newton's Second Law for Rotations
Next: Body-Fixed and Space-Fixed Frames of Reference

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.