217 Torque
A vector measure of a force’s turning effect that is used to describe how a force tends to change angular momentum about an origin, where an origin is the reference point about which the turning is measured.
Torque as the moment of a force. Torque is the cross product of the position of the point of application and the force. This principle is used to compute the turning effect of a force about a chosen origin.
The torque is
\[ \boldsymbol{\tau} = \mathbf{r}\times\mathbf{F} \]
where
- \(\boldsymbol{\tau}\) is the torque.
- \(\mathbf{r}\) is the position of the point of application relative to the origin.
- \(\mathbf{F}\) is the force.
Origin dependence. Torque depends on the choice of origin through the position vector. This principle is used to state which point the force is turning about.
The magnitude of torque. The magnitude of the torque is \(r F \sin\gamma\), where \(\gamma\) is the angle from \(\mathbf{r}\) to \(\mathbf{F}\). This principle is used to compute how much of the force is effective at turning.
The magnitude of the torque is
\[ \tau = r F \sin\gamma \]
where
- \(\tau\) is the magnitude of the torque.
- \(r\) is the distance from the origin to the point of application.
- \(F\) is the magnitude of the force.
- \(\gamma\) is the angle from \(\mathbf{r}\) to \(\mathbf{F}\).
The direction of torque. The direction of the torque is perpendicular to the plane of \(\mathbf{r}\) and \(\mathbf{F}\) by the right-hand rule. This principle is used to read the sense of the turning.
The torque-angular-momentum theorem. Torque about an origin equals the time rate of change of angular momentum about the same origin. This principle is used to relate a force’s turning effect to the change of rotational motion.
The torque-angular-momentum relation is
\[ \boldsymbol{\tau} = \dfrac{d\mathbf{L}}{dt} \]
where
- \(\boldsymbol{\tau}\) is the torque.
- \(\mathbf{L}\) is the angular momentum about the same origin.
- \(t\) is time.
Vanishing torque with conservation of angular momentum. If the net external torque vanishes, the total angular momentum is conserved. This principle is used to identify isolated rotational motion.
Vanishing net torque is
\[ \boldsymbol{\tau}_{\mathrm{ext}} = 0 \implies \mathbf{L} = \mathrm{constant} \]
where
- \(\boldsymbol{\tau}_{\mathrm{ext}}\) is the net external torque.
- \(\mathbf{L}\) is the total angular momentum.
Note: Also denoted \(\boldsymbol{\tau}\). Also called the moment of a force.