Final answer:
To calculate the truck's wheel angular speed in rad/min, convert the speed from mi/h to in/min and the wheel's diameter to radius in feet. Then, divide the linear speed by the radius to find that the angular speed is approximately 146.667 rad/min.
Step-by-step explanation:
To find the angular speed of the wheels in radians per minute, we first need to know the wheels' linear speed and the relationship between linear and angular speed. The linear speed (v) is already given as 45 miles per hour. To calculate the angular speed (ω), we use the formula ω = v / r, where r is the radius of the wheel.
First, convert 45 miles per hour to inches per minute since the wheel's diameter is given in inches:
- 45 mi/h = 45 * 5280 ft/mi * 12 in/ft * 1/60 min/h = 3960 inches per minute.
Now, calculate the radius of the wheel in inches:
- The diameter of the wheel is 54 inches, the radius (r) = diameter / 2 = 54 in / 2 = 27 inches.
Convert the radius to feet to match the linear speed units:
- 27 in * 1 ft / 12 in = 2.25 feet.
Next, calculate the angular speed:
- ω = linear speed / radius = 3960 in/min / (2.25 ft * 12 in/ft) ≈ 146.667 rad/min.
Notice the conversion factor of 2π radians for one revolution to get from linear speed to angular speed.
So, the angular speed of the truck's wheels is approximately 146.667 radians per minute.