### 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.

#### Body-Fixed and Space-Fixed Frames of Reference

Rotation is always about some (instantaneous) axis of rotation that is
free to change over time. It is convenient to express rotations in a
coordinate system having its origin (
) located at the
center-of-mass of the rigid body (§B.4.1), and its coordinate axes
aligned along the principal directions for the body (§B.4.16).
This *body-fixed frame* then moves within a stationary
*space-fixed frame* (or ``star frame'').

In Eq.(B.29) above, we wrote down Newton's second law for angular
motion in the *body-fixed frame*, *i.e.*, the coordinate system
having its origin at the center of mass. Furthermore, it is simplest
(
is diagonal) when its axes lie along principal directions
(§B.4.16).

As an example of a local body-fixed coordinate system, consider a
spinning top. In the body-fixed frame, the ``vertical'' axis
coincides with the top's axis of rotation (spin). As the top loses
rotational kinetic energy due to friction, the top's rotation-axis
*precesses* around a circle, as observed in the space-fixed
frame. The other two body-fixed axes can be chosen as any two
mutually orthogonal axes intersecting each other (and the spin axis)
at the center of mass, and lying in the plane orthogonal to the spin
axis. The space-fixed frame is of course that of the outside
observer's inertial frame^{B.28}in which the top is spinning.

#### Angular Motion in the Space-Fixed Frame

Let's now consider angular motion in the presence of linear motion of the center of mass. In general, we have [270]

#### Euler's Equations for Rotations in the Body-Fixed Frame

Suppose now that the body-fixed frame is rotating in the space-fixed frame with angular velocity . Then the total torque on the rigid body becomes [270]

Similarly, the total external forces on the center of mass become

*cf.*Eq.(B.15))

Substituting this result into Eq.(B.30), we obtain the following equations of angular motion for an object rotating in the body-fixed frame defined by its three principal axes of rotation:

These are call *Euler's
equations:*^{B.29}Since these equations are in the body-fixed frame, is the mass
moment of inertia about principal axis , and is the
angular velocity about principal axis .

#### Examples

For a uniform sphere, the cross-terms disappear and the moments of inertia are all the same, leaving , for . Since any three orthogonal vectors can serve as eigenvectors of the moment of inertia tensor, we have that, for a uniform sphere, any three orthogonal axes can be chosen as principal axes.

For a cylinder that is not spinning about its axis, we similarly
obtain two uncoupled equations
, for , given
(no spin). Note, however, that if we replace the
circular cross-section of the cylinder by an *ellipse*, then
and there is a coupling term that drives
(unless happens to cancel it).

**Next Section:**

Young's Modulus

**Previous Section:**

Newton's Second Law for Rotations