On the transformation of torques between the laboratory and center of mass reference frames

Abstract

It is commonly stated in Newtonian Mechanics that the torque with respect to the laboratory frame is equal to the torque with respect to the center of mass frame plus an $\mathbf{R}\times \mathbf{F}$ factor, with $\mathbf{R}$ being the position of the center of mass and $\mathbf{F}$ denoting the total external force. Although this assertion is true, there is a subtlety in the demonstration that is overlooked in the textbooks. In addition, it is necessary to point out that if the reference frame attached to the center of mass rotates with respect to certain inertial frame, the assertion is no longer true.