# Math Help - Finding angular speed

1. ## Finding angular speed

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.)

2. Hello jasonlewiz
Originally Posted by jasonlewiz
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.)
Notice:

• The circumference of the pedal sprocket is twice the circumference of the wheel sprocket, because its radius is twice as big.

• As the pedal sprocket turns through 1 complete revolution, the chain moves through a distance equal to the circumference of the pedal sprocket.

Therefore the chain must make the wheel sprocket turn through 2 complete revolutions for every 1 revolution of the pedal sprocket. So:

(a) The angular speed of the wheel sprocket is $40\times2 = 80$ rpm.

(b) The wheel is also turning at 80 rpm. So in one minute, the bicycle moves a distance equal to 80 times the circumference of the wheel $= 80\times2\pi\times13$ inches.

Can you work it out from here?

(I make the answer 9.08 ft per sec, or 6.19 mph.)