Question

a car is moving at a rate of 50mph and the diameter of its wheels is 2.5 feet. find the angular speed of the wheels in radians per minute

Answer 1

The car moves 50 miles in 1 hour (60 minutes), so it will have moved 50/60 (= 5/6) miles every minute. 1 land mile equals 5280 ft, so the car will travel 4400 ft every minute (5 * 5280 / 6 = 4400).

On the other hand, the perimeter of each wheel is π times its diameter. Since the diameter is 2.5 ft (= 5/2), the perimeter of each wheel is 5π/2 ft. Assuming no slip between the wheels and the road, each revolution will cause each wheel to advance one perimeter. Dividing the total length advanced per minute (4400 ft) into the length advanced per revolution (5π/2), we get 1760/π revolutions per minute. And given that one revolution equals 2π radians, we can multiply both to obtain 3520 radians per minute (1760 * 2 * π / π = 3520).

Of course, we could have skipped the second paragraph knowing that V = ωR and R = diameter/2 (4400 / 1.25 = 3520), but then we would not see why V = ωR.

Hope this helped!