a bicycle with 20-inch diameter wheels is traveling at 13 mi/h. find the angular speed of the wheels in rad/min. give exact answer

To find the angular speed of the bicycle's wheels in radians per minute, we can follow these steps:

### Step 1: Convert the linear speed from miles per hour to inches per minute.
1. The bicycle travels at 13 miles per hour. We need to convert this to inches per minute:
   - 1 mile = 5280 feet, and 1 foot = 12 inches, so 1 mile = \(5280 \times 12 = 63,360\) inches.
   - There are 60 minutes in an hour.
   
   Therefore, the speed in inches per minute is:
   \[
   \text{Speed in inches per minute} = 13 \, \text{mi/h} \times \frac{63,360 \, \text{in}}{1 \, \text{mi}} \times \frac{1}{60 \, \text{min}} = \frac{13 \times 63,360}{60} = 13,728 \, \text{in/min}.
   \]

### Step 2: Calculate the circumference of the wheel.
The diameter of the wheels is 20 inches, so the radius \(r\) is:
\[
r = \frac{20}{2} = 10 \, \text{in}.
\]
The circumference \(C\) of a circle (the distance traveled in one full revolution of the wheel) is given by:
\[
C = 2\pi r = 2\pi \times 10 = 20\pi \, \text{in}.
\]

### Step 3: Calculate the number of revolutions per minute.
To find the number of wheel revolutions per minute, we divide the linear speed (in inches per minute) by the circumference:
\[
\text{Revolutions per minute} = \frac{\text{Speed in inches per minute}}{\text{Circumference}} = \frac{13,728}{20\pi} = \frac{13,728}{62.8319} \approx 218.53 \, \text{rev/min}.
\]

### Step 4: Convert revolutions per minute to radians per minute.
Each revolution corresponds to \(2\pi\) radians. Therefore, the angular speed in radians per minute is:
\[
\text{Angular speed} = 218.53 \, \text{rev/min} \times 2\pi \, \text{rad/rev} = 218.53 \times 2\pi \approx 1373.1 \, \text{rad/min}.
\]

Thus, the exact angular speed of the wheels is:
\[
\boxed{\frac{13,728}{10} \, \text{rad/min}} = 1372.8 \, \text{rad/min}.
\]

Would you like more details or clarifications on any step? 

Here are 5 follow-up questions:
1. How do you convert from linear speed to angular speed?
2. What is the relationship between radius, circumference, and revolutions?
3. How would the angular speed change if the wheel size increased?
4. Can you convert the angular speed to radians per second?
5. How does the formula for angular speed apply to other circular objects, such as gears?

**Tip:** Remember that angular speed is related to both the speed of travel and the size of the wheel. Larger wheels need fewer revolutions to maintain the same linear speed.

A bicycle with 20-inch diameter wheels is traveling at 13 mi/h. Find the angular speed of the wheels in rad/min. Give exact answer.

Find the exact angular speed of a bicycle with 20-inch diameter wheels traveling at 13 miles per hour. This step-by-step solution covers key concepts like unit conversion, circular motion, and angular speed calculation. Suitable for high school students in Grades 10-12.

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