We can break this problem into two parts:

1. Finding the Angular Speed in radians per minute (rad/min)
2. Finding the Revolutions Per Minute (rpm)

Given:

• Diameter of the wheel = 42 inches
• Speed of the truck = 60 miles per hour (mi/h)

Step 1: Convert the speed from miles per hour to inches per minute

1 mile = 63,360 inches (since 1 mile = 5280 feet and 1 foot = 12 inches).

Thus, the truck is moving at:

$60 \text{ miles/hour} \times 63,360 \text{ inches/mile} = 3,801,600 \text{ inches/hour}$

Now, convert this to inches per minute:

$\frac{3,801,600 \text{ inches/hour}}{60 \text{ minutes/hour}} = 63,360 \text{ inches/minute}$

Step 2: Find the wheel's circumference

The circumference $C$ of the wheel is given by the formula:

$C = \pi \times d$

where $d$ is the diameter of the wheel.

$C = \pi \times 42 \text{ inches} \approx 131.946 \text{ inches}$

Step 3: Find the number of revolutions per minute (rpm)

Now, calculate how many revolutions the wheel makes each minute by dividing the total distance traveled by the circumference of the wheel:

$\text{rpm} = \frac{\text{Distance traveled per minute}}{\text{Circumference of the wheel}} = \frac{63,360 \text{ inches/minute}}{131.946 \text{ inches}} \approx 480 \text{ rpm}$

Step 4: Find the Angular Speed in rad/min

The angular speed is related to the number of revolutions per minute by the formula:

$\text{Angular speed (rad/min)} = \text{rpm} \times 2\pi$

$\text{Angular speed} = 480 \times 2\pi \approx 3,015.93 \text{ rad/min}$

Final Answers:

• Angular speed: $\boxed{3015.9 \, \text{rad/min}}$ (rounded to 1 decimal place)
• Revolutions per minute (rpm): $\boxed{480 \, \text{rpm}}$

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Would you like further clarification on any part of the solution? Here are a few related questions you might find interesting:

1. How would the answers change if the wheel diameter was doubled?
2. How do you derive the relationship between angular speed and linear speed?
3. How do you convert angular velocity from radians per minute to degrees per second?
4. What would happen if the truck's speed were in kilometers per hour instead of miles per hour?
5. How can we relate the truck's speed to the time it takes for one revolution?

Tip: Always remember to convert units consistently when working with speed, distance, and angular quantities—this helps simplify the calculations.