2.5 Moments, levers and gears — Learn

Prerequisites: Section 2.1, especially forces and vectors; multiplying and dividing numbers; changing centimetres to metres.

By the end of this section you can: describe examples in which forces cause rotation; calculate the moment of a force; use balanced moments to find a force or a distance; and explain how levers and gears transmit the turning effect of a force.

1. Forces that cause rotation

A force can make an object turn about a fixed point. The fixed point is called the pivot. A door turns about its hinges, a spanner turns about a nut, and a see-saw turns about its centre.

2. The moment of a force

The turning effect of a force is called the moment of the force.

M = F d

where $M$ is the moment in newton-metres, $F$ is the force in newtons, and $d$ is the perpendicular distance from the pivot to the line of action of the force, in metres.

Perpendicular distance means the shortest distance from the pivot to the line along which the force acts. A larger force, or the same force further from the pivot, gives a larger moment.

Worked example 1. A force of 40 N acts at a perpendicular distance of 0.15 m from a pivot. Calculate the moment of the force.

\begin{aligned} M &= F d \\ &= 40 \times 0.15 \\ &= 6.0\,\text{N m} \end{aligned}

3. Balanced moments

An object that is balanced does not turn. The total clockwise moment about the pivot equals the total anticlockwise moment about the pivot.

Worked example 2. A weight of 300 N sits 2.0 m from the pivot of a balanced beam. A second weight sits 2.5 m from the pivot on the other side. Calculate the second weight.

The anticlockwise moment is $300 \times 2.0$ . The clockwise moment is $F \times 2.5$ . Because the beam is balanced, the two moments are equal.

\begin{aligned} 300 \times 2.0 &= F \times 2.5 \\ 600 &= F \times 2.5 \\ F &= \frac{600}{2.5} \\ &= 240\,\text{N} \end{aligned}

4. Levers

A lever is a bar that turns about a pivot. It transmits the turning effect of a force. If the effort is applied further from the pivot than the load, the lever multiplies the force: a small effort gives a large force on the load.

Worked example 3. A lever has an effort of 200 N at 0.80 m from the pivot and a load at 0.20 m from the pivot. Calculate the force on the load.

\begin{aligned} 200 \times 0.80 &= F \times 0.20 \\ 160 &= F \times 0.20 \\ F &= \frac{160}{0.20} \\ &= 800\,\text{N} \end{aligned}

The crowbar, the wheelbarrow and a spanner are all levers.

5. Gears

A gear is a wheel with teeth. When two gears mesh, the teeth of one turn the other. The force on the teeth transmits the turning effect from one gear to the other.

If a small gear drives a large gear, the large gear turns more slowly but with a larger moment. If a large gear drives a small gear, the small gear turns faster but with a smaller moment. The moments are in the same ratio as the numbers of teeth on the gears.

Worked example 4. A gear with 30 teeth drives a gear with 15 teeth. The driving gear has a moment of 8.0 N m. Calculate the moment of the driven gear.

The ratio of the teeth is $30 : 15$ , which is $2 : 1$ . The moment is multiplied by 2.

M = 8.0 \times 2 = 16\,\text{N m}

Watch out. Use the perpendicular distance from the pivot to the line of action of the force, not the distance to the point where the force is applied. If you push a spanner at an angle, the perpendicular distance is smaller than the length of the spanner, so the moment is smaller. Keep every distance in metres: $15\,\text{cm}$ is $0.15\,\text{m}$ . Re-check that your distances are less than about a metre for a hand tool.

Try it: balancing a lever

The figure shows a lever with a load on one side and an effort on the other. The reader sets the load and the two distances, and the figure works out the effort that balances the lever.

Predict first. Do not move the sliders yet. The load is 300 N at 1.0 m from the pivot. If the effort acts at 2.0 m from the pivot, what effort will balance the load? Write down your estimate.

Test. Set the values and check the balance. Then move the effort closer to the pivot and watch what happens to the effort needed.

Explain. Why does the effort needed get larger as the effort moves closer to the pivot? Why does a long spanner make a tight nut easier to loosen? Write your answer in two or three sentences.

6. Progress check

Return to the chapter problem. The hand pulls the brake lever with 60 N at 0.090 m from the pivot. Work out the moment. Then work out the force in the cable, which is 0.030 m from the pivot. Explain why the lever makes the brakes work better.

You should now be able to

describe examples in which forces cause rotation;
calculate the moment of a force, using $M = Fd$ ;
use balanced moments to find a force or a perpendicular distance;
explain how a lever transmits and can multiply the turning effect of a force;
explain how gears transmit the turning effect of a force, using the ratio of the numbers of teeth.