Torque

Torque

$\tau$

Units:N m

Sometimes it's not enough for the forces to balance. For example, suppose two equal forces were applied on the box as shown here. If you tried this yourself, you would see that the box will rotate clockwise. The tendency of a force to cause an object to rotate around a particular axis or pivot is called torque.

Torque depends not only on how strong the force is, but also where and how the force is applied relative to the pivot. For example, in this figure

the torque due to $\vec F_B$ is larger than the torque due to $\vec F_C$, because it is easier to open a door when you push on the side that is opposite the hinges.
On the other hand, $\vec F_A$ exerts no torque at all, because if you push on the edge of the door directly towards the hinges then you won't make it spin at all, no matter how hard you push.

To define the torque as a variable, we first need to define the lever arm vector $\vec r$ of a force around a pivot, which is a vector from the pivot to the place where the force is being applied. We also need to specify the angle $\theta$ between the force and its lever arm vector. Given those two quantities, we can define the torque $\tau$ of a force F around a pivot to be

$$\tau=rF\sin\theta$$

In Trigonometric Functions we learned that the sine of the angle between two vectors tells you how perpendicular the two vectors are. When the force points directly towards or directly away from the pivot, then $\sin\theta=0$ and the torque is zero. When the force is perpendicular to the lever arm, on the other hand then the torque is simply $\tau=rF$, from which we can see that the units of torque are Newton-meters (Nm).

Note that we use the lowercase Greek letter $\tau$ for torque; be sure to distinguish it from a lower-case $t$ or upper-case $T$. See The Greek Alphabet for a suggestion of how to write the letter distinctly.

There are two other ways we can write this equation, if the two vectors are not perpendicular. First, we can write $$\tau=rF_\perp$$ where $F_\perp$ is the component of the force which is perpendicular to the lever arm. A little trig will show us that $F_\perp=F\sin\theta$, which is where this equation comes from. We can also write $$\tau=r_\perp F$$ where $r_\perp$ is the component of the lever arm that is perpendicular to the force. This quantity $r_\perp$ is sometimes called the "moment arm"; we will avoid this term to avoid confusion with the lever arm $\vec r$.