To solve this problem, we need to find both the angular velocity of the wheel in radians per minute and the wheel's spinning speed in revolutions per minute (RPM). Let's break it down into steps.

Step 1: Convert the car's speed to inches per minute

Given:

• Car speed = 55 miles per hour
• Radius of the wheel = 15 inches

First, we need to convert the car's speed from miles per hour to inches per minute.

1 mile = 63,360 inches, and 1 hour = 60 minutes.

So,

$\text{Speed in inches per minute} = 55 \times 63,360 \text{ inches per hour} \times \frac{1}{60} \text{ minutes per hour}$

$\text{Speed in inches per minute} = 55 \times 1,056 \text{ inches per minute}$

$\text{Speed in inches per minute} = 58,080 \text{ inches per minute}$

Step 2: Calculate the wheel's angular velocity in radians per minute

The formula for angular velocity $\omega$ is:

$\omega = \frac{v}{r}$

where:

• $v$ is the linear speed (in inches per minute),
• $r$ is the radius of the wheel.

So,

$\omega = \frac{58,080 \text{ inches per minute}}{15 \text{ inches}}$

$\omega = 3,872 \text{ radians per minute}$

Step 3: Calculate the wheel's spinning speed in revolutions per minute (RPM)

One revolution is $2\pi$ radians. So, to find the number of revolutions per minute (RPM):

$\text{RPM} = \frac{\omega}{2\pi}$

$\text{RPM} = \frac{3,872 \text{ radians per minute}}{2\pi}$

$\text{RPM} = \frac{3,872}{6.2832}$

$\text{RPM} \approx 616.14 \text{ revolutions per minute}$

Final Answers:

• Angular velocity: The angular velocity of the wheel is approximately 3,872 radians per minute.
• Revolutions per minute (RPM): The wheel spins at approximately 616.14 RPM.

Would you like more details or have any questions?

Here are 5 relative questions for you:

1. How would the angular velocity change if the car's speed increased to 70 miles per hour?
2. What is the relationship between linear speed and angular velocity?
3. How would you calculate the distance traveled by the car in 10 minutes using the wheel's RPM?
4. How does the size of the wheel's radius affect its angular velocity?
5. Can you derive the formula for linear velocity given the RPM and radius of the wheel?

Tip: When dealing with problems that involve units, always perform unit conversions at the start to avoid mistakes later on.