Question

A speedometer on the front wheel of a bicycle gives a reading that is directly proportional to the angular speed of the wheel. Suppose that such a speedometer is calibrated for a wheel of diameter \(72 \mathrm{~cm}\) but is mistakenly used on a wheel of diameter \(62 \mathrm{~cm} .\) Would the linear speed reading be wrong? If so, in what sense and by what fraction of the true speed?

Short answer

Yes, the linear speed reading would be wrong. The speedometer would be showing a speed that is approximately 16.13% higher than the true speed.

Step-by-step solution

Find the Circumference of Both Wheels

The first step is to find the circumference of both the wheels. The circumference \(C\) of a circle can be calculated by the formula \(C= \pi D\), where \(D\) is the diameter of the circle. Here the diameter of the first wheel is \(72 \mathrm{~cm}\) and the diameter of the second wheel is \(62 \mathrm{~cm}\). Thus, the circumference of the first wheel will be \(C_1= \pi(72) \mathrm{~cm}\) and of the second wheel will be \(C_2 = \pi(62) \mathrm{~cm}\).

Determine the Ratio of the Circumferences

By finding the ratio of the circumferences, we can examine the relationship between the angular speed (as measured by the speedometer) and the actual linear speed. The ratio of the circumferences is \(C_2/C_1 = 62/72\).

Analyze the Discrepancy

The ratio represents the discrepancy between the actual speed and the speed displayed by the speedometer. Therefore, the speedometer will show a reading that is \(72/62\) times the true speed when used on the smaller wheel. If we express this discrepancy as a percentage, it would be \(((72/62)-1) \times 100 \% \).

Key concepts

Linear Speed

Linear speed refers to how fast an object is moving along a path. In the context of a bicycle, it indicates how quickly the bike is traveling forward. Linear speed is often measured in units like meters per second or kilometers per hour.
It can be directly related to the rotational motion of the wheels. Essentially, linear speed (\(v\)) is related to angular speed (\(\omega\)) by the radius (\(r\)) of the wheel:

• \(v = r \cdot \omega\)

This equation shows that the larger the radius, the higher the linear speed for the same angular speed. Therefore, if the wheel diameter changes but the speedometer calibration remains the same, this affects the linear speed measurement. For the smaller 62 cm wheel, the actual linear speed will be underestimated by the speedometer calibrated for the 72 cm wheel.

Circumference

The circumference of a circle is the distance around it, which plays a crucial role in understanding the relationship between the wheel size and the speed reading. For any circle, you can find the circumference using the formula:

• \(C = \pi \cdot D\)

where \(\pi\) is a mathematical constant (approximately 3.14159), and \(D\) is the diameter of the wheel.
In this case, the circumference of the calibrated wheel (72 cm diameter) is \(C_1 = \pi \times 72\) cm, and for the smaller wheel (62 cm diameter), it is \(C_2 = \pi \times 62\) cm.
The key is that the linear speed is directly related to the circumference. So, if you swap to a smaller wheel but use the same speedometer, your correction factor becomes important. The ratio of the circumferences, \(\frac{62}{72}\), helps calculate how much the speedometer will over-register the speed on the wrong wheel.

Speedometer

A speedometer is a device that shows the speed of a vehicle, commonly found on bicycles and cars. For bicycles, this device is usually connected to the wheel to measure the rotational speed, or angular speed, of the wheel.
The speedometer translates this angular speed into linear speed using the wheel's circumference. If you mistakenly use a speedometer calibrated for a large wheel on a smaller wheel, the speed reading will be inaccurate because it still calculates speed using the larger wheel's circumference.
Calculating the expected discrepancy involves finding the ratio of the two circumferences. With a smaller circumference (\(C_2\) for the 62 cm wheel) being substituted into the equation for a speedometer calibrated for the 72 cm wheel, you obtain a reading that is \(\frac{72}{62}\) times the true speed. As a result, the speedometer displays higher speed than the bicycle actually travels, which can be expressed as a percentage error of \(((\frac{72}{62})-1) \times 100\%\). This serves as a warning about relying solely on speedometer readings when wheel sizes differ from calibration.