2

OKAY I corrected a blunder and still have problems

I'm following this diagram from my book:

https://i.stack.imgur.com/urGXN.jpg

It asks that if the chainring is 150 mm in diameter, the sprocket is 80 mm in diameter, and (from information from a previous question) the wheel is 700 mm in diameter,

"How fast would one need to pedal, in revolutions per minute, in order to maintain a speed of 24 kilometers per hr?"

I'm assuming that 1,000,000 mm = 1 km, the chainring and the sprocket have the same linear velocity, the sprocket and the wheel have the same angular velocity, and that the question is technically asking for the speed of the chainring (that one is "pedaling.")

This is an odd number problem, so the answer was in the back of the book. I got double the actual speed. I will show my work and try to figure what went wrong.

The circumference of the wheel in kilometers is $2\pi \times 350mm$, or 2,199.11 mm. To find out how fast the wheel would have spin to go 40 km/hr, I converted 2,199.11 mm into kilometers and divided

$$\frac{24 km/hr}{2,199.11\times10^{-6}km} = 10914$$ Revolutions per hour.

Unit analysis.

$$\frac{10914 rev}{hr} \times \frac{1hr}{60min} = 182$$

revolutions per minute. Of the wheel, though. To find the speed of the chain ring, use the formula $v=r\omega$. Since the linear velocity of the sprocket and the chain are the same I can assume this

$$r_{1}\omega_{1}= r_{2}\omega_{2}$$ $$40mm \times 182rpm = 75mm \times \omega_{2}$$ $$\omega_{2} = \frac{40mm\times182rpm}{75mm}$$ $$\omega_{2} = 97.1rpm$$

But, according to my book the correct answer is 80.8rpm.

Gᴇᴏᴍᴇᴛᴇʀ
  • 915
  • 2
  • 14
  • 28

1 Answers1

1

Although there is a relation between the sizes of the chainrings and the freewheels and their tooth counts, it is more proper to talk of the latter. To count the teeth obviates the necessity to measure. What is of interest is the ratio between the tooth counts in both. In your example, the ratio is 150 to 80, so that for every revolution of the crank the wheel will make 1.875 revolutions. Multiply that by 700π to arrive at the distance traversed by one stroke, or 4123.340358 mm/stroke.

24 km/hour is equal to 24000000 mm/hour is equal to 400000 mm/minute. To get the strokes/minute required to make 24 km/hour, one divides 400000 mm/minute by 4123.340358 mm/stroke, giving 97.00872722 strokes/minute as the required cadence.

Senex Ægypti Parvi
  • 2,419