A truck is traveling at 70mph and has wheels with a diameter of 26in. Find the angular velocity in radians per minute. Also find in revolutions per minute.
I've attempted this question and am completely stumped. I can't figure out how to make it work with the formula I'm given which is angular velocity=theta(angle)/time
one revolution covers 26π inches. Now let's start the conversion.
70mi/hr * 1hr/60min * 5280ft/mi * 12in/ft * 2πrad/26πin = 5686 rad/min
To find the angular velocity in radians per minute, we need to determine the angular displacement of the truck's wheels over time.
First, let's find the circumference of the truck's wheels. The circumference (C) of a circle is given by the formula: C = π * d, where d is the diameter.
In this case, the diameter (d) is given as 26 inches. So, the circumference of the truck's wheels is: C = π * 26 inches.
Next, we need to convert this circumference into the distance that the truck's wheels cover in 1 minute. Since the truck is traveling at 70 mph, which means it covers 70 miles in 1 hour, we can calculate the distance covered in 1 minute using the formula:
Distance = (Speed * Time) / 60 minutes.
In this case, the speed is given as 70 mph, and we want to find the distance covered in 1 minute. So, the distance covered in 1 minute is:
Distance = (70 * 1) / 60 miles.
Now, we need to convert this distance into inches, as the circumference of the wheels is also given in inches. Since 1 mile is equal to 5280 feet, and 1 foot is equal to 12 inches, we can convert 70 miles into inches using the following conversions:
70 miles = 70 * 5280 feet = (70 * 5280) * 12 inches.
So, the distance covered by the truck's wheels in 1 minute is:
Distance = [(70 * 5280) * 12) inches] / 60 minutes.
Now we can calculate the angular velocity in radians per minute. The angular displacement (θ) is given by the formula:
θ = Distance / Circumference.
In this case, the Distance is the distance covered by the truck's wheels in 1 minute, and the Circumference is the circumference of the wheels.
Using the values calculated above, we have:
θ = [(70 * 5280 * 12) inches / 60 minutes] / (π * 26 inches).
Simplifying this expression, we have:
θ = [(70 * 5280 * 12) inches / 60 minutes] * (1 / (π * 26)).
Now we can compute the angular velocity. The angular velocity (ω) is given by the formula:
ω = θ / Time.
In this case, the Time is 1 minute.
Using the values calculated above, we have:
ω = [(70 * 5280 * 12) inches / 60 minutes] * (1 / (π * 26)) / 1 minute.
This expression simplifies to:
ω = [(70 * 5280 * 12) inches] / (60 * (π * 26)) radians per minute.
Now you can calculate the final value for the angular velocity by evaluating this expression.