Question

If in a system of forces acting on a rigid body $R.G \ne 0$ then the given system of forces is equivalent to
A) degenerate wrench
B) non-degenerate wrench
C) a couple
D) a single force

Answer 1

Hint: In a rigid body the deformation is zero or negligible. Hence the force or couple can not deform the body. There can be couples acting on the body which means a pair of equal and opposite forces acting on two different points away from the center of mass. A wrench is the collection of the forces and torques acting on the body.

Complete step by step answer:
By the problem $R.G \ne 0$, hence neither the couple nor the forces are zero.
Hence the collection of the force and the torques will have a distinct non-zero realization. Hence, the wrench is non-degenerate.

If in a system of forces acting on a rigid body then the given system of forces is equivalent to non-degenerate wrench. So, option (B) is correct.

Additional Information:
To get a mathematical formulation for the geometry of the central lines of the rigid body which is the axes of spatial movement and the lines of action of forces, Screw theory is used. This is the algebraic formulation of vectors i.e. forces and moments or angular and linear velocity in the kinematics and dynamics of rigid bodies.
A wrench is a screw consisting of the torque and the force of the system.
A degenerate wrench is such a wrench where, either the force or the couple is zero.
Similarly, if neither of the force or the couple is zero, the wrench is called a non-degenerate wrench.

Note:
The dot product is not equal to zero. This means that neither the values can be zero and the angle between the vectors is never $\dfrac{\pi }{2}$. Thus the couple can not be zero and the same for the resulting force. This means the wrench also can not be degenerated.