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Math Help - Word problem with revolutions, please help!

  1. #1
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    Word problem with revolutions, please help!

    I've been stuck on this problem forever, if someone could please show me step by step what the answer is and why that would be great!

    Mark has a bike with a 22 inch back wheel. The wheel sprocket is 3 inches in diameter and the petal sprocket is 4 inches in diameter. how fast in miles per hour is he going if he petals at a rate of 100 revolutions per minute?

    Thanks a lot!
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  2. #2
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    note the relationship between linear speed, v , of a point on the edge of a circle, and its angular speed, \omega ...

    v = r\omega

    the pedal sprocket and wheel sprocket move with the same linear speed as the chain that connects them ...

    \omega_p = 100 \, rpm

    r_w \omega_w = r_p \omega_p

    \omega_w = \frac{r_p \omega_p}{r_w}

    \omega_w = \frac{400}{3} \, rpm

    this angular speed of the wheel sprocket in rpm needs to be converted to radians per hour ...

    \left(\frac{400}{3} \, rpm\right) \cdot \left(\frac{2\pi \, rad}{rev}\right) \cdot \left(\frac{60 \, min}{hr}\right) = 16000\pi \, rad/hr

    the tire and the wheel sprocket spin at the same angular speed, therefore the linear speed of the back tire is ...

    v = (11 \, in) \cdot (16000\pi \, rad/hr) = 176000 \, in/hr

    I'll leave it to you to convert this linear speed into mph.
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