# Torque and equilibrium review

Review the concept of torque and how it is affected by the applied force and lever arm.

## Key terms

Term (symbol) | Meaning |
---|---|

Torque ($\tau$) | Measure of the twisting action caused by a force that can cause an object to rotate about an axis. Vector quantity with SI units of $\text N \cdot \text m$. |

Net torque ($\Sigma \tau$) | Sum of all the torques on a system |

Balanced system | When the net torque on a system is zero |

Lever arm | Perpendicular distance from the axis of rotation to where the force is applied. Vector quantity with SI units of $\text m$. |

Pivot point | Point that an object rotates around. Sometimes called the fulcrum or rotational axis. |

## Equations

Equation | Symbol breakdown | Meaning in words |
---|---|---|

$\tau = rF\sin\theta = r_\perp F$ | $\tau$ is torque, $F$ is applied force, $r$ is the radius from the axis of rotation to the location where the force is exerted, and $\theta$ is the angle between $F$ and $r$ when these vectors are placed tail to tail. | Torque is proportional to both the lever arm and the force component perpendicular to the lever arm. |

## How to visualize the torque equation

A wrench produces a torque on a nut if a force is applied to it correctly (see Figure 1). The equation for torque is:

To produce a torque, the force $F$ must be applied at some distance $r$ away from the pivot point. Since only the perpendicular component $F_\perp$ produces torque, the equation includes $\sin \theta$ (see Figure 2 below).

### The magnitude of the torque depends on:

- Applied force $F$: Larger forces increase torque.
- Radius $r$: Increasing the radius increases the torque.
- Angle between the force and lever arm $\theta$: Directing a force perpendicular to the lever arm increases the torque.

An applied force can result in zero torque if there is no lever arm or the applied force is parallel to the lever arm (see Figure 3 and 4 below).

## How to determine the direction of torque

The direction of rotation can be clockwise (cw) or counterclockwise (ccw). These terms refer to the movement of hands on a clock (see Figure 5). In physics, the counterclockwise direction is defined as positive and clockwise is negative for rotational variables.

For example, a torque that rotates an object counterclockwise is a positive torque (see figure 6 below).

A torque that rotates an object clockwise is a negative torque (see figure 7 below).

## Learn more

For deeper explanations of torque, see our video about torque and equilibrium.

To check your understanding and work toward mastering these concepts, check out our exercises: