A rigid body, by definition, is a system of particles where internal forces of interaction are so strong that the inter-particle distances remain practically constant during the motion of the body.
A rigid body has six degrees of freedom
A system of n particles, in general, has 3n degrees of freedom, i.e. 3n number of co-ordinates are required to describe its configuration at any instant. However, in a rigid body, the constraint that inter-particle distances are fixed reduces its total degrees of freedom to just six. Let us see below?
A single particle has 3 degrees of freedom. Two unconstrained particles have 6 degrees of freedom. If the distance of 2nd particle from 1st is fixed, then we need only two co-ordinates to fix 2nd particle once the position of 1st particle is known. Hence, system now has only 5 degrees of freedom. One constraint reduces the degrees of freedom by one.
Three constrained particles have 9 degrees of freedom. If we impose the constraint that inter-particle distances are fixed, then we have 3 equations of constraint, viz. r12 = c1, r23 = c2, and r13 = c3 where, c1, c2 and c3 are the constants. System now possesses 9 – 3 = 6 degrees of freedom, 3 co-ordinates are required to fix the 2nd particle. Now, infinite planes pass through the line joining points (or particles) 1 and 2. Particle 3 lies on one of these planes. One (angular) co-ordinate is required to choose this plane. Once the plane is chosen, position of particle 3 is known because its distances from particles 1 and 2 are fixed. Hence, a total of 3 + 2 + 1 = 6 co-ordinates completely describe three arbitrary rigidly fixed points.
Now, let us consider a rigid system of n particles. We have seen above that, with respect to a given co-ordinate system in space, only 6 co-ordinates are required to fix any three arbitrary particles in the rigid body. Once these three particles are fixed, the position of the 4th particle is known because the distances of 4th particle from original three are fixed and given. This is equivalent to saying that three rods (of fixed lengths) which are required to rigidly attach the 4th particle to original three are given. Hence, 4th particle has no degree of freedom. That means, the rigid system of four particles along has got the same 6 degrees of freedom. Similarly, positions of remaining 5th to nth particles are also automatically fixed. Thus, a rigid body, in general, possesses six degrees of freedom.
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