The drive train of a bicycle has wheels 67.3 cm in diameter and pedal cranks 17.

Question

The drive train of a bicycle has wheels 67.3 cm in diameter and pedal cranks 17.5 cm long. The cyclist pedals at a steady angular rate of 72.0 rev/min. The chain engages with a front sprocket 15.2 cm in diameter and a rear sprocket 6.25 cm in diameter.

Calculate the speed of the bicycle relative to the road.

I know the speed of a link on the chain relative to the bike frame is 1.3195m/s, but I cannot figure out where to go from here. I've tried finding angular speed of the bike wheels by (v/r)=w where I used 1.3195 m/s for v and .3365 m for the radius of the wheel. Any help would be greatly appreciated.

Explanation / Answer

diameter of the wheel d = 67.3 cm length of the pedal cranks L = 17.5 cm angular speed of pedal = 72 rev/min                                       = (72*2/60) rad/s                                       = 7.536 rad/s diameter of the a front sprocket of chain d' = 15.2 cm radius of the a front sprocket of chain r = d'/2                                                             = 15.2 cm / 2                                                             = 7.6 cm                                                             = 0.076 m diameter of the a rear sprocket of chain d'' = 6.25 cm radius of the a rear sprocket of chain r' = d''/2                                                             = 6.25 cm / 2                                                             = 3.125 cm                                                             = 0.03125 m diameter of the a rear sprocket of chain d'' = 6.25 cm diameter of the a rear sprocket of chain d'' = 6.25 cm radius of the a rear sprocket of chain r' = d''/2                                                             = 6.25 cm / 2                                                             = 3.125 cm                                                             = 0.03125 m the speed of a link of the chain relative to the bicycle frame is           v = r             = (0.076 m)(7.536 rad/s)             = 0.5727 m/s angular speed of the bicycle wheels ,         ' = (v/r')             = [(0.5727 m/s) / (0.03125 m)]             = 18.32 rad/s .................................................................... radius of the wheel R = d/2                                  = (67.3 cm) /2                                    = 33.65 cm                                  = 0.3365 m the speed of the bicycle relative to the road is                  v' = 'R                      = (18.32 rad/s)(0.3365 m)                      = 6.16 m/s angular speed of the bicycle wheels ,         ' = (v/r')             = [(0.5727 m/s) / (0.03125 m)]             = 18.32 rad/s .................................................................... radius of the wheel R = d/2                                  = (67.3 cm) /2                                    = 33.65 cm                                  = 0.3365 m the speed of the bicycle relative to the road is                  v' = 'R                      = (18.32 rad/s)(0.3365 m)                      = 6.16 m/s

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