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. The figure on the right shows a door as viewed from above, and a force $\vec F$ being applied to the door. The pivot is the point that the door will rotate around. The vector $\vec r$ from the pivot to the location of the force is called the lever arm.
Aside
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.
Figure (a) shows the location where you should push a door to open it; towards the end of the door opposite that of the pivot. When the force is perpendicular to the lever arm, then the torque exerted by the force is
$$\tau = rF \hbox{ when $\vec r\perp \vec F$}$$
In (b), the force is applied closer to the pivot, and so the torque will be smaller. (You commonly see toddlers trying to open a door by pushing it close to the hinge, with little success.)
In figure (c), the force is applied to the edge of the door, directly towards the hinge. Common sense should tell you that this isn't going to make the door spin no matter how hard you push on it. When the force points in the same direction as the lever arm, or in the opposite direction, then there is no torque on the door.
$$\tau = 0 \hbox{ when $\vec r\,\Vert\,\vec F$}$$
Apparently, the torque must depend not only on how far the force is from the pivot, but also the angle of the force.
If \(\theta\) is the the angle between the force and its lever arm, then the torque \(\tau\) of $\vec F$ around that pivot is
$$\tau=rF\sin\theta$$
Remember that the Sine Function is a measure of perpendicularity, so the torque is largest when $\vec r$ and $\vec F$ are perpendicular. The units of torque are the length of the lever arm times the magnitude of the force; thus Newton-meters (Nm).
