Rigid Body
Definition
In physics, a rigid body is an object that does not fracture or change its shape.
Explanation
In classical mechanics, one assumes a body is rigid when analyzing motion. Mathematically, a rigid body is defined by the requirement that the distance between any two points in the body does not change.
$$ | \mathbf{r}_{i} - \mathbf{r}_{j} | = \mathrm{constant}, \quad \forall i \ne j $$
Real objects deform to some degree when subjected to forces, so an ideal rigid body in the strict sense does not exist. However, when deformations have a negligible effect on the motion, treating the object as rigid greatly simplifies the analysis. Because the relative positions of all particles composing the rigid body are fixed, the motion of an object made of infinitely many particles can be described by only a few variables.
Degrees of freedom
A single free particle whose position is not constrained has one degree of freedom $3$. Although a rigid body consists of many particles, the constraints that keep the distances between pairs of points fixed greatly reduce the number of degrees of freedom. As a result, the number of degrees of freedom of a rigid body in three-dimensional space is $6$.
- The number of coordinates to specify the position of a point of the rigid body (usually the center of mass): $3$
- The number of coordinates to specify the orientation (rotation) of the rigid body about that point: $3$
Motion of a rigid body
That the degrees of freedom equal $6$ means any motion of a rigid body can be decomposed into the sum of translational motion and rotational motion.
- Translational motion: motion in which every point of the rigid body moves with the same velocity. It is described by the change of the center-of-mass position.
- Rotational motion: motion in which each point of the rigid body undergoes circular motion about some axis. It is described by changes in orientation.