The independent coordinates of a rigid body

Consider a rigid body with three non-collinear points p, q, and r. Since each particle has three degrees of freedom, a total of nine are needed. But the interparticle spacing is set because of the presence of limitations. Consequently, there will be three equations of constraints between them.

Hence the total number of the degrees of freedom is reduced to 9-3 =6. The position of each further particle, say P, requires 3 coordinates but there will be 3 equations of constraints for this particle (because distance of P from p,q and r is fixed).

We can also show alternatively that 6 generalized coordinates are required to fix the configuration of the body. For assigning generalized coordinates, the fixed point in the body, which registers its translation and may with advantage be taken coincident with C.M of the body is given three Cartesian coordinates relative to the coordinate axes of the external space. The remaining three coordinates then specify the orientation of the body relative to the external frame or reference. Orientation of the body can be described elegantly by locating a Cartesian set of coordinates fixed in the rigid body (denoted by primed axes) called body set of axes in fig 2.

Therefore, if the location and orientation of a Cartesian coordinate system fixed in the body (body set of axes x′, y′, and z′) are known, the configuration of a rigid body with respect to some Cartesian coordinates system in space (space set of axes x, y, and z) can be determined completely. Three coordinates that also track body translation can be simply assigned in order to fix the origin of the coordinate system. The direction cosines of the body set of axes (also known as the primed system) can be used to determine the body set of axes' orientation (unprimed system).

Let α, β, γ be the dc's and i,j,k and i′, j′, k′ be the unit vectors along the axes x, y,z fixed in space and x′, y′, z′ axes fixed in the body. Then

The direct coordinates (dc) also provide relationships between a given point's coordinates in one system and coordinates in the other system. The components of the position vector r, along the system's axes, are what make up a point's coordinates in a particular reference frame. In terms of x, y, and z, the x′ coordinate is then given by

This note is taken from Classical Mechanics, MSC physics, Nepal.

This note is a part of the Physics Repository.

Next topic: Euler's theorem

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