Chapter 8, Problem 1Q

A bicycle odometer (which counts revolutions and is calibrated to report distance traveled) is attached near the wheel axle and is calibrated for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

To determine

The result when an odometer calibrated for 27-inch wheels is used with 24-inch wheels.

Answer to Problem 1Q

Solution:

The odometer will end up showing a higher distance reading than the actual distance covered by the bicycle.

Explanation of Solution

Formula used:

1. $s=v\times t$
2. $v=\omega \times r$

Calculation: The odometer calculates the distance covered by multiplying the speed with time covered, i.e., $s=v\times t$ .

But the velocity it uses in this calculation depends on its calibration. It is the angular velocity that remains the same, which is counted by the odometer. Velocity is obtained by multiplying with the radius, i.e., $v=\omega \times r$ .

Therefore, if an odometer calibrated for a 27-inch wheel is used on a 24-inch wheel, the velocity the odometer uses will be greater than the actual velocity of the bicycle.

Since for the same time $s\propto v$ , the distance being calculated will also be greater than the actual distance covered by the bicycle.

Conclusion

Therefore in the given scenario, the odometer will end up showing a higher distance reading than the actual distance covered by the bicycle.