A car with a 44-inch diameter wheel is traveling at 45 mph. (Notes: round the answer to three decimal places.) Find the angular speed of the wheels in rad/min

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

### Step 1: Convert the diameter of the wheel to its radius
The diameter of the wheel is 44 inches, so the radius \( r \) is:
\[
r = \frac{44 \text{ inches}}{2} = 22 \text{ inches}
\]

### Step 2: Convert the car's speed from miles per hour to inches per minute
First, convert the speed from miles per hour to inches per hour:
\[
45 \text{ miles/hour} \times 5280 \text{ feet/mile} \times 12 \text{ inches/foot} = 2,851,200 \text{ inches/hour}
\]
Next, convert the speed to inches per minute:
\[
2,851,200 \text{ inches/hour} \div 60 \text{ minutes/hour} = 47,520 \text{ inches/minute}
\]

### Step 3: Find the number of wheel rotations per minute
The circumference \( C \) of the wheel is:
\[
C = 2\pi r = 2\pi \times 22 = 44\pi \text{ inches}
\]
The number of rotations per minute \( N \) is:
\[
N = \frac{\text{Linear speed in inches/minute}}{\text{Circumference of the wheel}} = \frac{47,520 \text{ inches/minute}}{44\pi \text{ inches}} = \frac{47,520}{44\pi}
\]

### Step 4: Calculate the angular speed in radians per minute
The angular speed \( \omega \) in radians per minute is given by:
\[
\omega = N \times 2\pi = \frac{47,520}{44\pi} \times 2\pi = \frac{47,520 \times 2}{44} = \frac{95,040}{44}
\]
\[
\omega \approx 2,160 \text{ radians per minute}
\]

Thus, the angular speed of the wheels is approximately **2,160 radians per minute**.

Do you want more details or have any questions? Here are 5 related questions for further exploration:

1. How would the angular speed change if the wheel diameter was different?
2. What is the relationship between linear and angular speed?
3. How does angular speed change if the car accelerates?
4. What if the car was traveling at a different speed—how would the angular speed change?
5. Can you convert angular speed from radians per minute to degrees per minute?

**Tip:** Remember, the circumference of a wheel directly affects the number of rotations it makes for a given linear distance.

Learn how to calculate the angular speed of a car's wheels traveling at 45 mph using mathematical formulas and conversions. This problem involves geometry, trigonometry, and physics concepts, including the circumference of a circle and linear speed conversion. Suitable for advanced high school students.

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