The problem involves a truck with 38-inch diameter wheels traveling at a speed of 55 miles per hour (mi/h). We need to find the angular speed in radians per minute (rad/min) and the number of revolutions per minute (rpm) the wheels make.

Step 1: Find the Circumference of the Wheel

The circumference of the wheel, which is the distance it covers in one complete revolution, is given by the formula:

$C = \pi \times d$

where $d$ is the diameter of the wheel. In this case, the diameter is 38 inches, so:

$C = \pi \times 38 \text{ inches}$

$C \approx 119.38 \text{ inches}$

Step 2: Convert the Truck's Speed to Inches per Minute

The truck is traveling at 55 miles per hour. To find how many inches it travels per minute, we need to convert miles per hour to inches per minute.

1 mile = 5280 feet, and 1 foot = 12 inches. Therefore:

$55 \text{ miles per hour} = 55 \times 5280 \times 12 \text{ inches per hour}$

Now, convert hours to minutes (1 hour = 60 minutes):

$55 \times 5280 \times 12 \div 60 \text{ inches per minute}$

Step 3: Find the Number of Revolutions per Minute (rpm)

To find the revolutions per minute, we divide the truck's speed in inches per minute by the circumference of the wheel:

$\text{rpm} = \frac{\text{Truck's speed in inches per minute}}{\text{Circumference of the wheel}}$

Step 4: Find the Angular Speed in radians per minute

Since each revolution corresponds to $2\pi$ radians, the angular speed in radians per minute is:

$\text{Angular speed} = \text{rpm} \times 2\pi$

Let's calculate the values for revolutions per minute and angular speed. math

Given values

diameter = 38 # in inches speed_mph = 55 # in miles per hour

Step 1: Calculate the circumference of the wheel

circumference = math.pi * diameter # in inches

Step 2: Convert truck speed to inches per minute

speed_ipm = speed_mph * 5280 * 12 / 60 # inches per minute

Step 3: Calculate revolutions per minute (rpm)

rpm = speed_ipm / circumference

Step 4: Calculate the angular speed in radians per minute

angular_speed_rad_per_min = rpm * 2 * math.pi

circumference, speed_ipm, rpm, angular_speed_rad_per_minHere are the results:

1. The circumference of the wheel is approximately 119.38 inches.
2. The truck's speed is 58,080 inches per minute.
3. The wheels make approximately 486.51 revolutions per minute (rpm).
4. The angular speed of the wheels is approximately 3056.84 radians per minute (rad/min).

Would you like further details or explanations? Let me know if you have any questions.

Here are some related questions for further exploration:

1. How do changes in wheel size affect the angular speed of a vehicle?
2. How would the calculations change if the truck's speed were different (e.g., 60 mph)?
3. What is the relationship between linear speed and angular velocity in rotational motion?
4. How would you calculate angular speed for a non-circular wheel?
5. How would you convert angular speed from radians per minute to revolutions per second?

Tip: Always ensure to convert units consistently when solving physics problems to avoid errors in calculations.