Wheels, turns and moving hands

One turn of a wheel

Paint a spot on the tyre of a bicycle and roll it along the road. When the spot touches the road again, the wheel has made one turn, which is also called one rotation or revolution. In that time the wheel has rolled exactly one circumference.

• In one turn, the wheel rolls $C = \pi d$.
• In $n$ turns, the wheel rolls $n \times \pi d$.
• To cover a distance $s$, the wheel needs $n = \dfrac{s}{\pi d}$ turns.

Before you use these, change every length to one unit. A wheel diameter is usually in centimetres and a road is in metres or kilometres.

Example 1. A wheel has a diameter of $70\ \text{cm}$. How far does it roll in $500$ turns? Take $\pi = \dfrac{22}{7}$.

1. One turn. $C = \dfrac{22}{7} \times 70 = 220\ \text{cm}$, which is $2.2\ \text{m}$.
2. 500 turns. $500 \times 2.2 = 1100\ \text{m}$, which is $1.1\ \text{km}$.

Example 2, the number of turns. A wheel has a radius of $28\ \text{cm}$. How many turns does it make on a road $1.32\ \text{km}$ long? Take $\pi = \dfrac{22}{7}$.

1. One turn. $C = 2 \times \dfrac{22}{7} \times 28 = 176\ \text{cm}$.
2. The road in centimetres. $1.32\ \text{km} = 1320\ \text{m} = 132\,000\ \text{cm}$.
3. The turns. $\dfrac{132\,000}{176} = 750$ turns.

Example 3, speed. A wheel of diameter $0.84\ \text{m}$ makes $250$ turns in one minute. Take $\pi = \dfrac{22}{7}$.

1. One turn. $C = \dfrac{22}{7} \times 0.84 = 2.64\ \text{m}$.
2. One minute. $250 \times 2.64 = 660\ \text{m}$ in a minute.
3. One hour. $660 \times 60 = 39\,600\ \text{m}$, which is $39.6\ \text{km}$. The speed is $39.6\ \text{km/h}$.

Two wheels on the same road. On the same road, a small wheel needs more turns than a big wheel. The number of turns is $\dfrac{s}{2\pi r}$, so it is inversely proportional to the radius. A wheel with half the radius makes twice as many turns.

A hand or a wiper that sweeps a fraction of a circle

The tip of the minute hand of a clock moves along a circle. The hand itself sweeps the inside of the circle, like the blade of a wiper that sweeps a half circle of glass.

If the hand turns through a fraction of a full turn, it sweeps the same fraction of the area of the circle, and its tip moves the same fraction of the circumference.

• A minute hand turns once in $60$ minutes. In $m$ minutes it turns $\dfrac{m}{60}$ of a circle.
• An hour hand turns once in $12$ hours, which is $720$ minutes. In $m$ minutes it turns $\dfrac{m}{720}$ of a circle.

Example 4. A minute hand is $21\ \text{cm}$ long. Take $\pi = \dfrac{22}{7}$. Find the area it sweeps in $20$ minutes, and the distance moved by its tip.

1. The fraction. $\dfrac{20}{60} = \dfrac{1}{3}$ of a circle.
2. The area. $\dfrac{1}{3} \times \dfrac{22}{7} \times 21^2 = \dfrac{1}{3} \times 1386 = 462\ \text{cm}^2$.
3. The tip. $\dfrac{1}{3} \times 2 \times \dfrac{22}{7} \times 21 = \dfrac{132}{3} = 44\ \text{cm}$.

Example 5, a wiper. A wiper blade is a straight arm of length $35\ \text{cm}$. It sweeps a third of a full circle. It cleans $\dfrac{1}{3} \times \dfrac{22}{7} \times 35^2 = \dfrac{3850}{3} = 1283\dfrac{1}{3}\ \text{cm}^2$. The next course gives the area for any angle, such as $115^\circ$. Here the angle is a simple fraction of a turn.

Practise with new numbers

Try it in Python

This program turns centimetres and kilometres into metres before it calculates. It uses $\pi = \dfrac{22}{7}$ because the question says so.

Common mistakes

• Mixing units. A diameter in centimetres and a road in kilometres need one unit before dividing.
• Using the diameter where the formula wants the radius. $2\pi r$ and $\pi d$ are the same, but $\pi r$ is not.
• Taking the hand sweep as a full circle. Work out the fraction of a turn first.
• Treating the hour hand like the minute hand. The hour hand turns $12$ times more slowly.

Remember these

• One turn of a wheel rolls one circumference, $\pi d$.
• Distance $= n \times \pi d$, and the number of turns is $\dfrac{s}{\pi d}$.
• Change all lengths to one unit. $1\ \text{km} = 1000\ \text{m} = 100\,000\ \text{cm}$.
• A hand that turns through a fraction of a circle sweeps that fraction of the area.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A wheel has a diameter of $2\ \text{m}$. Write the distance it rolls in one turn in terms of $\pi$.

2. A wheel with a circumference of $2\ \text{m}$ makes $300$ turns. How far does it roll, in kilometres?

3. A wheel has a radius of $21\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. How many turns does it make on a road of $66\ \text{m}$?

4. What fraction of a circle does the hour hand of a clock sweep in $3$ hours?

5. A minute hand is $14\ \text{cm}$ long. Take $\pi = \dfrac{22}{7}$. How far does its tip move in $15$ minutes?