(a) Find the angular speed of the wheel sprocket. (b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)

Question

The sprockets and chain of a bicycle are
shown in the figure. The pedal sprocket has a radius of 4 in.,
the wheel sprocket a radius of 2 in., and the wheel a radius of
13 in. The cyclist pedals at 40 rpm.

 

(a) Find the angular speed of the wheel sprocket.

 

(b) Find the speed of the bicycle. (Assume that the wheel
turns at the same rate as the wheel sprocket.)

Answer 1

The gear ratio depends on the circumferences of the two gears, so the 4 in pedal gives a 16pi inches circumference, vs the wheel sprocket having a 4pi circumference, for a multiplication of 4x.

 

With the pedal sprocket moving at 40rpm, the wheel sprocket will move at 160 rpm.

 

The wheel has a radius of 13 inches for a circumference of 531 inches.  At 160 rpm, that is 84,949 inches per minute,

or 7079 feet/min or 1.34 miles/min or 80.4 mph.