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Chapter 8: Problem 31

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

Expert verified
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

01

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 \(C_2 = \pi(62) \mathrm{~cm}\).
02

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 \(C_2/C_1 = 62/72\).
03

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 \(((72/62)-1) \times 100 \% \).

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Most popular questions from this chapter

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