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Rigid Body Features

We have seen that a rigid body has six degree of freedom. Three of these degrees correspond to the translational motion and other three to rotational motion.

For example, under the action of a net external force F, the CM of the body accelerates like a point particle of mass M located at CM co-ordinates. The three co-ordinates of CM represent three translational degrees of freedom of the body. In case of pure translation, the orientation of the body in space does not change, and all particles (or points) of the body move parallel to each other with equal velocities.

The rotational degrees of freedom are associated with the changes in orientation of the body in space. If any one point of the body is fixed (pivoted) in space, the body cannot translate. No movement of the body is now possible except a change in its orientation; hence, subsequent motion is pure rotation. We state, without proof, that

If one point of rigid body is fixed in space, then the most general displacement of body is equivalent to a (single) rotation about ‘some’ axis in space, passing through the fixed point. This is known as Euler’s theorem.

In general, a rigid body can both translate and rotate in space. Again we state, without proof, that

The general motion of a rigid body can be analyzed as the superposition of two independent motions: (i) a pure translation of any arbitrary reference point of the rigid body plus (ii) a pure rotation of the body about ‘some’ axis passing through that reference point (considering that point to be fixed in space). This is known as Chasles’ theorem.

If one point of the rigid body is fixed (pivoted) in space, then the above separation is trivial. There is no translation; the motion is pure rotation about the fixed point.

If the body also translates, then CM is generally taken as the reference point. the motion can be neatly separated into a pure translation of CM, described by

F = M a_CM

and a pure rotation about ‘some’ axis through CM, described by

The angular momentum also neatly divides into contributions from translation of CM and rotation about CM, as given by

L = L_CM + R_CM × M V_CM

As we shall see that similar division holds for the total kinetic energy too.

The angular velocity of rotation , at any instant, is same no matter what reference point we choose. By definition, rigid body rotation means that all points of the body are rotating together with same angular velocity about ‘some’ axis in space, at any instant; the axis is an ‘instantaneous axis’ of rotation.

If we choose CM as the reference point, instantaneous axis of rotation passes through CM; if we choose some other point of reference in the body, axis passes through there. However, both axes are parallel in space along the direction of vector .

To sum up: The key feature if rigid body dynamics is that it can be separated into a translation and a rotation.