Torque about an Axis

The idea of torque about a single point extends naturally to the more physically relevant idea of torque about a fixed axis, which is what actually governs a rigid body constrained to rotate about a line rather than free to spin about an arbitrary point.

Setup. Consider a rigid body free to rotate about a fixed axis $AB$, with a force $\vec F$ acting at some point $P$ on the body — note that $\vec F$ need not even lie in the plane containing the axis and $P$. The origin $O$ used to compute the ordinary torque $\vec\tau=\vec r\times\vec F$ can be taken to be any point lying on the axis $AB$ itself. The torque about the axis $AB$ is then defined as the component of $\vec\tau$ that lies along the axis — found by taking the angle $\phi$ between $\vec\tau$ and $AB$, giving a parallel (along-axis) component $rF\sin\theta\cos\phi$ and a perpendicular (across-axis) component $rF\sin\theta\sin\phi$.

Physical meaning of the split. The along-axis component is what actually rotates the body about the fixed axis. The perpendicular component instead tends to tilt the axis itself out of its fixed orientation — a phenomenon that becomes visible as precession, most familiarly in a spinning top that, as it slows down under the pull of gravity, traces out a slow conical wobble of its own spin axis even while continuing to spin rapidly about that (wobbling) axis. A detailed treatment of precession is beyond the scope of this course; instead, it is simply assumed that whatever mechanical constraints hold the axis fixed also automatically supply whatever additional force is needed to cancel the perpendicular torque component, so that only the along-axis component needs to be considered from here onward.

Practical simplifications used throughout the rest of the chapter, as a consequence:

• Only forces that lie in planes perpendicular to the axis, and do not themselves intersect the axis, are considered — a force parallel to the axis produces only a perpendicular (axis-tilting) torque, which can be ignored.
• Only position vectors perpendicular to the axis are considered — a position vector along the axis likewise contributes only a perpendicular torque.
• Any force whose line of action passes through (intersects) the axis contributes zero torque about it, since its own $r=0$ measured from the axis. …