Question

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 2 in., and the wheel a radius of130in. The cyclist pedals at 40rpm.
(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.)

Short answer

(a). $\mathbf{160}\mathit{\pi }\mathit{r}\mathit{a}\mathit{d}\mathbf{/}\mathit{m}\mathit{i}\mathit{n}$

(b). The bicycle's speed when the cyclist pedals at 40rpm ,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., is inch $\mathbf{2080}\mathit{\pi }$.

Step-by-step solution

a. Step 1. Given information.

Find the unknown angular speed of the sprocket by using known.
Then convert it inrad/min.
The given values are:
${\text{r}}_1=4$inch and ${\text{\hspace{0.17em}\hspace{0.17em}r}}_2=2$inch and angular speed ${\text{w}}_1=40\text{rpm}$.
To find: The angular speed of the second sprocket.

a. Step 2.Finding the value of w2.

Since,
the linear speed of both wheels sprockets is the same.

a. Step 3. Finding the value of w2.

As we know that
1 revolution$=2\pi$
So,
Angular speed in rad/min is $w_2=2\pi \left(80\right)$

$w_2=160\pi \text{rad}/\text{min}$

Thus, this is the angular speed of the wheel sprocket.

b. Step 1. Given the information

Use the angular speed of wheel from the part b.

The given values are:
The radius of the wheel $r=13\text{in}$.

To find:

The linear speed of the bicycle for the given condition.

b. Step 2. Formula Used

Assume the angular speed of the wheel sprocket as the angular speed of the wheel and use the formula of liner speed $v=rw$.

b. Step 3. Finding the value of v.

Assume,
The wheel's angular speed = wheel sprocket angular speed
As we have already got the angular speed wheel from part b is $w=160\pi$
The linear speed of the wheel $v=rw$

$\begin{array}{l}v=13\cdot \left(160\pi \right)\text{inch/min}\\ v=2080\pi \text{inch/min}\end{array}$

So,. this is the value of the speed of the wheel.