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A bicycle with  24
 -inch diameter wheels is traveling at  12
  mi/h.

 

Find the angular speed of the wheels in rad/min.

 

Enter the exact answer.

 

ω=
   

  rad/min

 

How many revolutions per minute do the wheels make?

 

Round your answer to three decimal places.

 

The wheels make  

  revolutions per minute.

 

To solve this, we'll break the problem into two parts: first, finding the angular speed (ω) in radians per minute, and then calculating the number of revolutions per minute.

### 1. **Find the angular speed (ω in rad/min):**

We are given the following information:
- Diameter of the wheel = 24 inches → Radius \( r = \frac{24}{2} = 12 \) inches.
- Speed of the bicycle = 12 miles per hour (mi/h).

We need to convert the speed to inches per minute, then use it to find the angular speed.

#### Step 1: Convert speed from mi/h to inches per minute.
1 mile = 5280 feet = 63,360 inches, and 1 hour = 60 minutes.

\[
12 \, \text{mi/h} = 12 \times 63,360 \, \text{inches/hour} = 760,320 \, \text{inches/hour}.
\]

Now, convert it to inches per minute:

\[
760,320 \, \text{inches/hour} \div 60 = 12,672 \, \text{inches/min}.
\]

#### Step 2: Relate linear speed to angular speed.
The formula that relates linear speed \( v \) to angular speed \( \omega \) is:

\[
v = \omega \times r
\]

Rearrange to solve for \( \omega \):

\[
\omega = \frac{v}{r}
\]

Substitute \( v = 12,672 \, \text{inches/min} \) and \( r = 12 \, \text{inches} \):

\[
\omega = \frac{12,672}{12} = 1,056 \, \text{rad/min}.
\]

So, the angular speed of the wheels is:

\[
\boxed{1,056 \, \text{rad/min}}.
\]

### 2. **Find the number of revolutions per minute:**

One full revolution corresponds to \( 2\pi \) radians. To find the number of revolutions per minute, divide the angular speed by \( 2\pi \):

\[
\text{Revolutions per minute} = \frac{\omega}{2\pi} = \frac{1,056}{2\pi}.
\]

Using \( \pi \approx 3.14159 \):

\[
\text{Revolutions per minute} = \frac{1,056}{6.28318} \approx 168.043.
\]

So, the number of revolutions per minute is:

\[
\boxed{168.043}.
\]

### Summary:
- Angular speed \( \omega = 1,056 \, \text{rad/min} \).
- The wheels make approximately 168.043 revolutions per minute.

---

Would you like any further details or clarifications? Here are some related questions:

1. How do angular speed and linear speed relate?
2. What is the difference between angular speed in radians and revolutions?
3. How can we convert from radians to degrees for angular speed?
4. What is the angular speed if the bicycle's speed was doubled?
5. How does the wheel's diameter affect its angular speed?

**Tip**: Remember that angular velocity is always in radians per time unit and is related directly to the linear speed and wheel radius.

A bicycle with 24-inch diameter wheels is traveling at 12 mi/h. Find the angular speed of the wheels in rad/min. Enter the exact answer. ω = rad/min. How many revolutions per minute do the wheels make? Round your answer to three decimal places.

Solve for the angular speed and revolutions per minute of a bicycle with 24-inch diameter wheels traveling at 12 mi/h. This problem involves angular velocity, linear velocity, and unit conversion. Suitable for high school students in Grades 10-12.

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