Work (physics) - Work of Forces Acting On A Rigid Body

Work of Forces Acting On A Rigid Body

The work of forces acting at various points on a single rigid body can be calculated from the work of a resultant force and torque. To see this, let the forces F₁, F₂ ... F_n act on the points X₁, X₂ ... X_n in a rigid body.

The trajectories of X_i, i=1,...,n are defined by the movement of the rigid body. This movement is given by the set of rotations and the trajectory d(t) of a reference point in the body. Let the coordinates x_i i=1,...,n define these points in the moving rigid body's reference frame M, so that the trajectories traced in the fixed frame F are given by

The velocity of the points X_i along their trajectories are

where ω is the angular velocity vector obtained from the skew symmetric matrix

known as the angular velocity matrix.

The small amount of work by the forces over the small displacements δr_i can be determined by approximating the displacement by δr=vδt so

or

This formula can be rewritten to obtain

where F and T are the resultant force and torque applied at the reference point d of the moving frame M in the rigid body.