Physics · Ch 5 — Motion of System of Particles and Rigid Bodies

Definition of Torque

5.2.1

Definition of Torque

Torque is defined as the moment of an externally applied force about a chosen point or axis of rotation. If a force $\vec F$ acts at a point whose position vector, relative to the reference point, is $\vec r$ , the torque is the vector (cross) product

\vec\tau=\vec r\times\vec F.

Since the cross product of two vectors always produces a third vector perpendicular to both of the originals, torque is itself a genuine vector quantity, perpendicular to the plane containing $\vec r$ and $\vec F$ . Its SI unit is the newton metre (N m).

Magnitude. Writing $\theta$ for the angle between $\vec r$ and $\vec F$ , the magnitude of the torque is

\tau=rF\sin\theta.

This can be computed in two entirely equivalent ways, by associating the $\sin\theta$ factor with either vector: $\tau=r(F\sin\theta)=rF_\perp$ , using $F_\perp$ , the component of the force perpendicular to $\vec r$ ; or $\tau=(r\sin\theta)F=r_\perp F$ , using $r_\perp$ , the component of the position vector perpendicular to $\vec F$ (equivalently, the perpendicular distance from the reference point to the force's line of action).

Direction — the right-hand rule. If the fingers of the right hand are curled from the direction of $\vec r$ towards the direction of $\vec F$ , the extended thumb points along the direction of $\vec\tau$ . This direction has a direct physical meaning: torque pointing out of the page produces an anticlockwise rotation; torque pointing into the page produces a clockwise rotation. …

Figure 5.4Torque on a rigid body

What this figure shows. A force F is applied at a point on a rigid body located by a position vector r drawn from a reference point O, with the angle theta marked between r and F; the torque about O is the cross product of these two vectors, a vector perpendicular to the plane that contains …

Figure 5.5Direction of torque by the right-hand rule

What this figure shows. The right-hand rule for torque is illustrated: if the fingers of the right hand are curled from the direction of the position vector r towards the direction of the force F, the extended thumb points along the direction of the resulting torque vector, which is always perpendicular to the plane containing r and F …

Figure 5.6Torque direction and the sense of rotation

What this figure shows. Two panels show a force F applied at a distance r from a pivot: in panel (a) the torque vector points out of the page and the resulting rotation is anticlockwise; in panel (b) the torque vector points into the page and the resulting rotation is clockwise, establishing the rule that torque out of the page means anticlockwise turning and torque into the page means clockw …

Figure 5.7Two equivalent ways of computing torque magnitude

What this figure shows. Two panels show the same force F applied at position r from an origin O, at angle theta between them: panel (a) resolves the force into a component F sin(theta) perpendicular to r, giving torque = r times (F sin theta); panel (b) instead resolves the position vector into a component r sin(theta) perpendicular to F, giving the same torque as (r sin theta) times F, showing both decompositions yield an identical torq …

Table 5.1The value of torque for different relative orientations of r and F

Case	Angle between r and F	Torque
r and F perpendicular	90 degrees	tau_max = rF
r and F parallel	0 degrees	tau = 0
r and F antiparallel	180 degrees	tau = 0

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