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Chapter 1: Problem 58

Bicycle Velocity A five-speed bicycle with 26 -in.-diameter wheels has a chain ring with 44 teeth and five cogs with \(13,\) \(15,17,20,\) and 23 teeth. If a cyclist pedals at 1 revolution per second using any of these cogs, then what is the maximum velocity for this bicycle in \(\mathrm{mi} / \mathrm{hr} ?\)

Short Answer

Expert verified
The maximum velocity is approximately 15.7 mi/hr.

Step by step solution

01

- Determine Gear Ratios

Calculate the gear ratios for the bicycle using the formula: \(\text{Gear Ratio} = \frac{Teeth \, on \, Chain \, Ring}{Teeth \, on \, Cog}\). For each of the cogs: \(13\) teeth: \( \frac{44}{13} \approx 3.38 \) \(15\) teeth: \( \frac{44}{15} \approx 2.93 \) \(17\) teeth: \( \frac{44}{17} \approx 2.59 \) \(20\) teeth: \( \frac{44}{20} = 2.2 \) \(23\) teeth: \( \frac{44}{23} \approx 1.91 \)
02

- Find Maximum Gear Ratio

Identify the highest gear ratio from the calculated values: The highest gear ratio is \( \frac{44}{13} \approx 3.38 \).
03

- Determine Wheel Circumference

Calculate the circumference of the wheel using the formula: \(Circumference = \pi \times Diameter \). For a 26-inch wheel, \(Circumference = \pi \times 26 \approx 81.68 \) inches.
04

- Determine Distance Per Pedal Revolution

Calculate the distance traveled per pedal revolution by multiplying the gear ratio by the wheel circumference: \(Distance \, per \, Pedal \, Revolution = Gear \, Ratio \times Circumference \, of \, Wheel \). For the gear ratio \(3.38\): \(Distance = 3.38 \times 81.68 \approx 276.08 \) inches/revolution.
05

- Convert Inches to Miles

Convert the distance traveled per pedal revolution from inches to miles. There are 63,360 inches in a mile, so: \( \frac{276.08 \, inches}{63,360 \, inches/mile} \approx 0.00436 \) miles per revolution.
06

- Calculate Velocity in Miles per Hour

Given that the cyclist pedals at 1 revolution per second, convert this to revolutions per hour: \(1 \, revolution/second \times 3600 \, seconds/hour = 3600 \, revolutions/hour\). Then, multiply by the distance per revolution to get the velocity: \(3600 \times 0.00436 \approx 15.7 \, miles/hour\).
07

Conclusion

The maximum velocity for the bicycle, with the cyclist pedaling at 1 revolution per second using the cog with 13 teeth, is approximately \(15.7 \mathrm{mi}/\mathrm{hr}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gear Ratios
To calculate the velocity of a bicycle, we first need to understand gear ratios. A gear ratio is the relationship between the number of teeth on the chain ring and the number of teeth on the cog. This ratio determines how many times the wheel turns for each revolution of the pedal. For example, if the chain ring has 44 teeth and the cog has 13, the gear ratio is calculated as \(\text{Gear Ratio} = \frac{44}{13} \approx 3.38\). This means that for every pedal turn, the wheel will turn approximately 3.38 times. By computing this for each cog, we can find the impact of different gears on the bicycle鈥檚 performance.
Wheel Circumference
Next, we move to wheel circumference, which is essential for determining how far the bicycle travels with each wheel revolution. The circumference is calculated using the formula: \( \text{Circumference} = \pi \times \text{Diameter} \). For a bicycle with 26-inch diameter wheels, the wheel circumference is \(\text{Circumference} \approx 81.68 \) inches. The circumference helps us quantify the distance traveled per revolution of the wheel.
Distance per Revolution
To find how far the bicycle travels per pedal revolution, we multiply the gear ratio by the wheel circumference. Using the highest gear ratio of 3.38 we calculated earlier, the distance per pedal revolution is \( \text{Distance} = \text{Gear Ratio} \times \text{Circumference} \). This gives us \(\text{Distance} = 3.38 \times 81.68 \approx 276.08 \) inches per revolution. This calculation shows how gear ratios and wheel circumference combine to determine the travel distance per pedal turn.
Unit Conversion
It鈥檚 crucial to convert the distance from inches to miles when calculating bicycle velocity in miles per hour (mph). We know 1 mile is 63,360 inches. So, to convert 276.08 inches per pedal revolution to miles, we divide by 63,360: \( \frac{276.08}{63,360} \approx 0.00436 \) miles per revolution. Unit conversions help standardize our measurements, making it easier to understand and compare results.
Velocity Calculation
Finally, we calculate the bike鈥檚 velocity. If the cyclist pedals at 1 revolution per second, this converts to 3600 revolutions per hour (1 revolution/second \times 3600 seconds/hour). We then multiply the revolutions per hour by the distance per revolution (0.00436 miles/revolution) to find the velocity: \(3600 \times 0.00436 \approx 15.7 \) miles per hour. This shows how gear ratios and pedal speed work together to affect the bike鈥檚 speed, giving us a comprehensive understanding of bicycle velocity.

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