# Off-axis torque

If one adds pure torque to a disc rotating about a fixed axis in such a way that the application of the torque is off the axis of rotation with the torque vector parallel to the axis of rotation, does the added torque provide angular acceleration to the disc's rotation about the fixed axis regardless of point of application? By pure torque I mean a couple with which no net force is exert on the disc. For example, if a rotating disc had a magnetic dipole attached to its outer edge with the magnetic moment facing the direction of rotation and then a uniform magnetic field was applied to the system such that the dipole's moment were not aligned with the applied field, at that moment in time a torque would be applied to the disc at the point where the dipole was attached.

• You need to clarify some terms: what is "pure torque." Every torque arises from some force. And by accelerate, do you mean linear acceleration, angular acceleration, or either one? – Bill N Apr 25 '19 at 2:33
• Is “pure torque” your term for a “couple”? – Farcher Apr 25 '19 at 4:51
• I hope my edited query clears those issues up. – William Prince Apr 26 '19 at 5:27
• "the application of the torque is off the axis of rotation" there is no way to do this physically or mathematically. Torque is not applied at a location. Forces only have locations since they only exist along lines in space (and hence have a location). Torques are free vectors. – John Alexiou Apr 26 '19 at 21:14

No, because you cannot add torque, you can just add a force. And then the torque generated depends on the point of application:

$$T=F·r$$ (A vector product)

If you could add pure torque it wouldn't depend on the point of application. But you add a force and then a torque appears.

• What about a force couple, which results in a net torque, but no net force? – John Alexiou Apr 26 '19 at 2:24
• They're still forces. T=2*F*D in that case or the distance between the forces instead of D. – user228687 Apr 27 '19 at 0:27
• What about a magnetic dipole in a uniform magnetic field when the moment of the dipole is not aligned with the field? If we consider the Gilbertian pole model to be obsolete, then we have real torque applied to the dipole's mass but no net force. – William Prince Apr 27 '19 at 2:45
• Remember that the force on a magnetic dipole in a magnetic field is determined by the gradient of the field and the gradient of a uniform field is zero. – William Prince Apr 27 '19 at 2:58
• Thank you, I didn't know it. – user228687 Apr 28 '19 at 12:54

A pure torque does not have a location where it is applied. It is felt uniformly by the entire rigid body. As a result, the body is going to rotate about the center of mass. If there are is zero net force, then the center of mass will continue moving with constant velocity (or be at rest).

The equations of motion state that net force affects the motion of the center of mass only, and net torque the motion (rotation) about the center of mass. This is a direct result of the definitions of momentum an angular momentum, and Newton's 2nd law of motion.