#### Question

# Pre-Calculus Help something with a bicycle pedal and wheel!?

The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket has a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 30 rpm.

a.) Find the rotations per minute of the wheel sprocket.

b.) Find the speed of the bicycle. Assume that the wheel turns at the same rate as the wheel sprocket. (Round your answer to 2 decimal places.)

#### Answer

The key here is that the linear velocity of the chain must be the same on both sprockets. 30 rpm = angular velocity of 60pi radians/min; linear velocity = radius* angular velocity = 240pi inches/min.

Working backward on the 2" sprocket, we get an angular velocity of 120pi radians/min which equals 60 rpm.

The bicycle tire is also rotating at 60 rpm which equals 120pi radians per minute which equals a linear velocity of 120pi*13 = 1560pi inches/min. If you want the speed in miles per hour

1560pi inches/min(60min/hour)(1ft/12inches)(1mi… = 4.64 mph.