# Math Help - bicycle wheel

1. ## bicycle 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 a radius of 2in., and the wheel a radius of 13in. the cyclist pedals at 40rpm

a.) find the angular velocity 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, aeroflix!

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

(a) Find the angular velocity of the wheel sprocket

(b) Find the speed of the bicycle.

The pedal sprocket is turned at $40\text{ rpm} \:=\:40(2\pi) \:=\:80\pi\text{ rad/min}$

The pedal sprocket has a circumference of $2\pi r \,=\,8\pi$ inches.

Hence, the chain is moving at: . $(80\pi)(8\pi) \:=\:640\pi^2\text{ in/min}$

The wheel sprocket has a circumference of $2\pi(2) \:=\:4\pi\text{ inches.}$

It is turning at: . $\frac{640\pi^2}{4\pi} \:=\:160\pi\text{ rad/min }\;\;(a)$

The wheel has a circumference of: $2\pi(13) \:=\:26\pi\text{ inches.}$

It moves along the ground at: $(26\pi)(160\pi) \:=\:4160\pi^2\text{ in/min}$

The bicycle's speed is: . $\begin{Bmatrix}
41,\!057.5543 & \text{in/min} \\
3,\!421.462859 & \text{ft/min} \\
205,\!287.7715 & \text{ft/hr} \\
38.88025976 & \text{mi/hr}
\end{Bmatrix} \;\;\approx\;\; 39\text{ mph }\;(b)$