# Torque

In this Physics tutorial, you will learn:

• What is torque?
• Where does torque change from the moment of force?
• How torque is calculated?
• What is the positive direction of torque?

## Introduction

In tutorial "Moment of Force. Conditions of Equilibrium", we discussed about moment of force or as it is otherwise known, the turning effect of force. However, the focus in that tutorial was setting up the equilibrium of a system, i.e. when the total clockwise moment is numerically equal to its opposite, i.e. the total anticlockwise moment of force.

In this tutorial we will focus on another aspect of turning effect, i.e. when forces "help" each other to make the turning effect easier. Obviosly, they must act in the same direction of rotation.

## The Meaning of Torque

By definition, "Torque is a non-zero resultant moment of force usually produced by a single force or a couple of forces that help each other when trying to rotate a system around a fixed point."

Torque can be either produced by a single force of by a couple of forces as long as they are able to create rotation in the system. As you see, the meaning of torque is very close to that of moment of force. The most important difference between these two concepts lies in the fact that torque is a movement effect whereas moment of force is a static effect. This means torque is used where there is rotation involved whereas moment of force is used where there is no rotation (when there is equilibrium).

### Examples of torque involving a couple of forces include:

1. Rotating the steering wheel of a car,
2. Rotating the handle bar of a bike,
3. Two men rotating a grinding mill, etc.

### Other examples of torque but which involve the use of a single force include:

1. Pushing a door as it swings on its hinges when pushed,
2. Turning the key,
3. Turning the dood knob, etc.

As you can see, all three examples shown above (in the second set of examples), involve the same process: opening a door. Thus, torque is applied three times in a row in this process: turning the key (action 1), turning the door knob (2) and pushing the door as it swings around its hinges (3). Look at the figure. On the other hand, in the first set of the abovementioned examples, there are two forces acting in the same direction or rotation on both sides of a system. Let's take the steering wheel for illustration.

Both forces applied by our hands help each other making the rotation of steering wheel easier. As a result, the car turns in the desired direction. In the figure below, forces F1 and F2 help each other rotate the steering wheel clockwise. As a result, the car turns right. In most cases (although it is not a must), the distance from the turning force is the same for each force applied when they form a couple. This helps the users balance their senses. In the specific case, this distance is equal to the radius of the steering wheel.

Both Moment of Force and Torque have the same unit i.e. N-m but The equation of torque for a single force applied is:

τ = r × F

where r is the distance from the turning point (usually the radius of a circle), and F is the perpendicular force to the bar (line of action) or the tangent force used at that specific point of the circle.

The scalar equivalent of the above equation is

|τ | = |r| × |F| × sin θ

where θ is the angle between the axis of rotation and the force applied. Look at the figure: In this figure there is a single force that produce torque. However, not all the force goes for the rotation of the system; some of it is wasted (the component of force in the direction of the bar). Therefore, since only the component of force perpendicular to the bar is able to produce rotation, we consider only it as a turning force.

### Example 1

A 60 N acts at the edge of a bar in the direction shown in the figure. Giving that cos 400 = 0.766 and sin 400 = 0.643, calculate the length of bar in cm to produce a 10 N × m torque.

### Solution 1

From the scalar equivalent of the equation of torque

|τ| = |r| × |F| × sin θ

we obtain after substitutions,

10 = |r| × 60 × 0.643
10 = |r| × 38.58
|r| = 10/38.58
= 0.2592 m
= 25.92 cm
≈ 26 cm

As for the torque of a couple, we obtain the following equation:

τ tot = r 1 × F 1 + r 2 × F 2

When the couple of forces is used to rotate a kind of wheel (such as a steering wheel), we have

r1
= r2
= r

### Example 2

A steering wheel has not been lubricated properly and a 12 N force is needed to overcome friction. What is the minimum force a driver must use on the steering wheel with either hand to make the car turn left at 24 N × m? The diameter of steering wheel is 36 cm. ### Solution 2

The steering wheel must rotate anticlockwise to make the car turn left. Also, F1 = F2 = F and r1 = r2 = r = d/2 = 36 cm / 2 = 18 cm = 0.18 m.

The torque given in the clues is the resultant (net) torque. It inludes the positive torque produced by the driver and the negative torque of the friction against motion. Therefore, we have

τ net = τ driver + τ friction = 24 N × m

For the negative torque produced by friction, we have

τ friction = -f × r

If we take the angle θ = 90° (as we need the minimum values of force), we obtain

τ friction = -12 N × 0.18 m
= -2.16 N × m

Therefore, the torque produced by the driver is

24 N × m = τ driver - 2.16 N × m
τ driver = 24 N × m + 2.16 N × m
= 26.16 N × m

This torque is produced by a couple of equal forces acting at r = 0.18 cm away from the turning point. Thus, we have

τ driver = r 1 × F 1 + r 2 × F 2
= r × F + r × F
= 2r × F
= 26.16 N × m

Thus,

2 × 0.18 × F = 26.16
0.36 × F = 26.16
F = 26.16/0.36
= 72.7 N

Therefore, the driver must apply a 72.7 N by either hand to make the system rotate as described.

Remark! As for the signs of the directions of rotation, anticlockwise is taken as positive [clockwise is negative].

## Summary

By definition, "Torque is a non-zero resultant moment of force usually produced by a single force or a couple of forces that help each other when trying to rotate a system around a fixed point."

Torque can be either produced by a single force of by a couple of forces as long as they are able to create rotation in the system. The meaning of torque is very close to that of moment of force. The most important difference between these two concepts lies in the fact that torque is a movement effect whereas moment of force is a static effect. This means torque is used where there is rotation involved whereas moment of force is used where there is no rotation (when there is equilibrium).

Examples of torque involving a couple of forces include:

1. Rotating the steering wheel of a car,
2. Rotating the handle bar of a bike,
3. Two men rotating a grinding mill, etc.

Other examples of torque but which involve the use of a single force include:

1. Pushing a door as it swings on its hinges when pushed,
2. Turning the key,
3. Turning the dood knob, etc.

Both Moment of Force and Torque have the same unit i.e. N-m but The equation of torque for a single force applied is:

τ = r × F

where r is the distance from the turning point (usually the radius of a circle), and F is the perpendicular force to the bar (line of action) or the tangent force used at that specific point of the circle.

The scalar equivalent of the above equation is

|τ| = |r| × |F| × sin θ

where θ is the angle between the axis of rotation and the force applied.

As for the torque of a couple, we obtain the following equation:

τ tot = r 1 × F 1 + r 2 × F 2

When the couple of forces is used to rotate a kind of wheel (such as a steering wheel), we have r1 = r2 = r

As for the signs of the directions of rotation, anticlockwise is taken as positive [clockwise is negative].

## Torque Revision Questions

1) Calculate the net torque for the system shown in the figure (take anticlockwise as positive). What is the net torque of the system if cos 370 = 0.8 and sin 370 = 0.6?

1. 110 N × m
2. 32 N × m
3. - 26 N × m
4. 26 N × m

2) Calculate the net torque of the system shown in the figure. (Take anticlockwise as positive) 1. - 90 N × m
2. 90 N × m
3. - 30 N × m
4. - 18 N × m

3) Rank the torques the four forces in the figure produce in the system (greatest first). 1. τ4 > τ1 > τ2 > τ3
2. τ4 > τ3 > τ1 = τ2
3. τ4 > τ3 > τ1 > τ2
4. τ4 > τ1 > τ3 = τ2