# 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 systemWhen 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 pointPoint that an object rotates around. Sometimes called the fulcrum or rotational axis.
Both torque and work have SI units of $\text N \cdot \text m$, but they are not the same thing.
Work is a scalar that describes how much energy was transferred to a system and its units are equivalent to Joules. Torque describes a vector of how much twisting action a force puts on a body and is not equivalent to a Joule.

## Equations

EquationSymbol breakdownMeaning 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:
$\tau = rF\sin\theta$
Figure 1. Variables of the torque equation shown for a wrench and nut. The nut’s center is the pivot point.
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).
Figure 2. Components of the applied force $F$ aligned to the lever arm.The perpendicular component is $F_\perp$ and the parallel component is $F_\parallel$.

### 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).
Figure 3. Lever arm: these applied forces result in no torque on the wrench because of no lever arm $r$.
Figure 4. Direction of force: these applied forces result in no torque on the wrench because the applied force is parallel to the lever arm.

## 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.
Figure 5. The rotation of a clock’s hands is the reference for defining rotational direction. Counterclockwise is the positive rotation direction and clockwise is the negative direction.
For example, a torque that rotates an object counterclockwise is a positive torque (see figure 6 below).
Figure 6. An applied force that causes a positive counterclockwise torque.
A torque that rotates an object clockwise is a negative torque (see figure 7 below).
Figure 7. An applied force that causes a negative clockwise torque.