A truck with 40-in.-diameter wheels is traveling at 50 mi/h.

a) Find the angular speed of the wheels in rad/min
b) How many revolutions per minute do the wheels make?

40 in wheel has circumference 40pi inches

50 mi/hr = 52800 in/min

so, that's 52800/40pi rev/min

or, 52800/40pi * 2pi = 52800/20 = 2640 rad/min

a) Well, let's get rolling with the calculations! The diameter of the wheels is 40 inches, which means the radius is 20 inches (half of the diameter). Now, we need to convert the speed from miles per hour to inches per minute, so let's multiply 50 mi/h by 5280 ft/mi (the number of feet in a mile) and then multiply that by 12 in/ft (the number of inches in a foot). Finally, let's divide that by 60 min/h (the number of minutes in an hour). So, the speed of the truck in inches per minute is 50 * 5280 * 12 / 60 = 633600 inches/min.

Now, to find the angular speed in radians per minute, we can use the formula w = v / r, where w is the angular speed, v is the linear speed, and r is the radius. Plugging in the values, we get w = 633600 inches/min / 20 inches = 31680 radians/min.

b) To find the number of revolutions per minute, we need to divide the angular speed by 2π (because 2π radians are in one revolution). So, the number of revolutions per minute is 31680 radians/min / (2π) ≈ 5036 revolutions/min. That's a wheely impressive number of revolutions!

So, the angular speed of the wheels is approximately 31680 radians/min and the wheels make approximately 5036 revolutions per minute.

To find the angular speed of the wheels in rad/min, we need to convert the given information first.

a) To convert the speed from miles per hour to inches per minute, we multiply it by 5280 (the number of feet in a mile), and then by 12 (the number of inches in a foot) and divide by 60 (the number of minutes in an hour).

Speed in inches per minute = 50 mi/h * 5280 ft/mi * 12 in/ft / 60 min/h
= 50 * 5280 * 12 / 60
= 1,760,000 in/min

The circumference of a wheel can be found using the formula:
Circumference = π * diameter

b) Now, let's calculate the number of revolutions per minute. Since one revolution corresponds to one circumference, and the distance traveled per minute is the same as the circumference of the wheel, we can divide the distance traveled per minute by the circumference of the wheel.

Number of revolutions per minute = inches per minute / circumference
= 1,760,000 in/min / (π * 40 in)
= 1,760,000 / (3.14 * 40)
≈ 14,034 revolutions per minute (rounded to the nearest whole number).

Therefore:
a) The angular speed of the wheels is approximately 1,760,000 rad/min.
b) The wheels make approximately 14,034 revolutions per minute.

To find the angular speed of the wheels in rad/min, we need to convert the linear velocity of the truck to angular velocity.

a) The linear velocity of the truck can be calculated using the formula:

v = ω * r

where v is the linear velocity, ω is the angular velocity, and r is the radius of the wheel.

The radius of the wheel is half of the diameter, so in this case, the radius is 40 in / 2 = 20 in.

Now, let's convert the linear velocity from miles per hour to inches per minute:

50 mi/h * 5280 ft/mi * 12 in/ft * 1/60 h/min = 4400 in/min

Now that we have the linear velocity in inches per minute (v = 4400 in/min) and the radius of the wheel (r = 20 in), we can rearrange the formula to solve for the angular velocity:

ω = v / r

ω = 4400 in/min / 20 in = 220 rad/min

Therefore, the angular speed of the wheels is 220 rad/min.

b) To find the number of revolutions per minute, we can use the fact that 2π radians is equivalent to one revolution.

The number of revolutions per minute can be calculated using the formula:

Revolutions per minute = angular speed / (2π)

Revolutions per minute = 220 rad/min / (2π) ≈ 34.91 revolutions per minute

Therefore, the wheels make approximately 34.91 revolutions per minute.