rigid body

A rigid body is a particular kind of physical object that occurs frequently in the study of dynamics. Specifically, an extended body that is rigid does not deform in any way, but rather always maintains its shape. A rigourous definition of this property is that any two points fixed within the body are always the same distance apart.

Rigid body dynamics is the first step beyond basic point particle dynamics, as a rigid body is the simplest object that nevertheless has more complicated motion than a point particle. While point particles can only undergo translational motion through three-dimensional space (i.e. they have three degrees of freedom), rigid bodies additionally have the possibility of rotational motion. It can be shown that the motion of a rigid body can be described completely by translational motion of the centre of mass and rotation around an axis passing through the centre of mass.

The dynamical properties of a rigid body are described by two quantities: the total mass and the moment of inertia. The mass describes the response of the body to forces through Newton's second law F = ma. The moment of inertia describes the response of the system to torques through the rotational analogue of Newton's second law, τ = Iα. (the expert reader will note that the moment of inertia is not a scalar but a second-rank tensor (matrix) and thus the torque is not necessarily in the same direction as the angular acceleration)

Unfortunately, no real body can be truly rigid. This is not a mere practical problem but is a fundamental fact. To see this, we must leave the safe confines of classical mechanics and consider special relativity. One of the fundamental principles of special relativity is that no influence of any sort can travel faster than the speed of light. To see how this disallows the possibility of a rigid body, it is illustrative to consider an example.

Consider a rigid rod 1 A.U. (yes, 1 A.U.) in length, or about 8.5 light-minutes. This rod is rotating at a constant rate around its midpoint, tracing out a circle with a diameter of half of the earth's orbit with its endpoints. Now, consider the situation where one end of the bar is brought to a halt. If the rod is truly rigid, this situation will not cause any deformation of the bar.

Now, no influence can travel faster than the speed of light, so the influence of this action can take no less than eight and a half minutes to reach the far end of the rod. In that time, the far end has continued its motion around the circle, as it is physically impossible for it to tell that the other end has stopped rotating. Now when the influence finally reaches the other end of the rod it is no longer diametrically opposite the first end, so the two ends are clearly no longer 1 A.U. apart. Therefore, the rod cannot be rigid.

Although this example is rather far-fetched, the principle is the same for ordinary-scale objects. For a 1 meter rod the time interval is on the order of a few nanoseconds but this still prevents it from being truly rigid. (It is possible to create a small-scale object that is rigid for all practical purposes, but it can never be perfectly rigid) This proof is not only applicable to rods but can be seen to be valid for an arbitrarily-shaped body, from which a rod-shaped subset can be taken and the proof applied to it.

Despite the relativistic flaw, rigid body dynamics is a useful idealisation for the classical calculations that are used for most real-world engineering applications. The rigid body model is sufficiently simpler than a model that supports nonrigid bodies that the error caused by approximating to a rigid body is worth avoiding the more involved calculations involving deformation.