# Torque

### Written by tutor Alexandra L.

Torque is often described as the analog of force in angular motion. More precisely, it is a quantity that

describes the ability of a force to cause twisting or turning of a body – that is, to produce angular motion.

Intuitively, we may think of applying a force of constant magnitude (such as a push) to a door at several

different distances away from its hinges – the farther away from the hinges we push, the more easily the door

swings shut. If we apply the force very close to the hinges, we must push harder in order to achieve the same

turning effect.

Algebraically, then, the turning effect is directly proportional to both the magnitude of the force, and

the perpendicular distance between the force vector and the hinge (more generally termed the **fulcrum**) around

which angular motion is to occur. This perpendicular distance is often called the **lever arm**, or the **moment
arm** of the force vector about the fulcrum.

In terms of vector algebra, the relation between torque, force, and lever arm is the cross product of the force vector

and the lever arm (whose direction points from the fulcrum to the point of force application):

In terms of the angle φ between the force vector and the lever arm vector, the magnitude of the torque is given by:

The direction of the resultant torque is given by the right hand rule – when the fingers of the right hand curl from the

direction of **r** to the direction of **F** (that is, in the direction of rotation produced by the torque), the thumb will point

in the direction of the torque vector **τ**. By convention, counterclockwise torques are positive, while clockwise torques are

defined as negative.

## Torque and angular acceleration

The angular acceleration of a rigid rotating body depends on the total amount of torque about the axis of rotation. This provides

us with a formula analogous to Newton’s second law for forces that describes the rotational motion of rigid objects:

Στ = *Iα*

The analog of mass in linear motion is *I*, the **moment of inertia** of the object.

## Torque Practice Quiz

True or False? If you grasp the handle of a spanner closer to the nut you are trying to tighten, you will need to apply less force to tighten it.

**A.**

True

**B.**

False

**B**.

True or False? Maximum torque is produced when the force vector is perpendicular to the lever arm connecting the fulcrum and the point of force application.

**A.**

True

**B.**

False

**A**.

Two children are sitting on a see-saw. The first child is twice as heavy as the second child, and is sitting a distance of 1.0 m away from the middle of the see-saw. How far from the middle of the see-saw (on the other side) must the second child sit in order to balance out the see-saw?

**A.**

1.0 m

**B.**

0.5 m

**C.**

2.0 m

**C**.