OpenStax College Physics for AP® Courses, Chapter 23, Problem 29 (Problems & Exercises)

This is College Physics Answers with Shaun Dychko. A generator with 500 turns has a peak voltage of 480 volts and the diameter of the coil, or the armature inside, is 8.00 centimeters and the magnetic field inside of it is 0.250 Tesla and the question is what angular speed does the generator need in order to create this peak voltage of 480 volts? And so we have a formula for peak voltage: it's number of turns times the area of the coil times the magnetic field strength times angular speed and we will replace area with the formula πr squared but we don't know what the radius is— we are given the diameter— and so we will substitute diameter divided by 2 in place of r and we end up with πd squared over 4 and then we will substitute that in for area. These coils are circular we are assuming... it might even say that actually... well it says diameter and so when it says this coil has a diameter that implies that it's a circle. So now let's solve this for ω and we are going to multiply both sides by 4 over Nπd squaredB. So we get the angular velocity then is 4 times the peak voltage divided by the number of turns times π times diameter squared times magnetic field strength. So that's 4 times 480 volts divided by 500 turns times π times 8.00 times 10 to the minus 2 meters squared times 0.250 Tesla which is 763.944 radians per second. The question asks us to express our answer in revolutions per minute so we multiply this by 60 seconds for every minute so the seconds cancel and we multiply by 1 revolution for every 2πrad— 2πrad is the number of radians in a full circle or one revolution, in other words— so we are left with revolutions per minute for our units but for some reason, there's a convention which is that this fraction—revolutions per minute—is abbreviated r p m for the words revolutions per minute. So that's 7.30 times 10 to the 3 r p m is our answer.