The angular velocity of a process control motor is ω = ( 20 ─ ½ t² ) rad/s, where t is in seconds. b. Through what angle does the motor turn between t = 0 s and the instant at which it reverses direction?

Video transcript

Welcome back, everyone. We are making observations about a shaft and we are told that it has an angular velocity described by this equation up top right here. And we are tasked with finding what is the total number of revolutions for the shaft between a time period or a time of zero seconds. And when the direction of the spinning shaft is reversed, let's go over our answer choices here. We have a zero radiance B 115 radiant C 83. radiance or D negative 16.7 radiant here. Well, first and foremost, we know that in order to get revolutions, that's just going to be the integral between our two times of our angular velocity function with respect to time. However, we need to know what this top bound or time T is. But we are told that it's when the angular staff starts rotating in the opposite direction. So for a for a quick second, it'll be when the angular velocity is zero. And luckily, we're given a formula which is 25 minus T squared. So we just solve for this to get our upper bound of T. I'm gonna go ahead and add squared to both sides of our equation. And what this gives us is that T squared is equal to 25. If we take the square root of both sides, we will find that our T is equal to five seconds when the shaft starts rotating in the opposite direction. Great. So now we are ready to go ahead and perform our integration here. We have that our total revolutions is equal to the integral from 0 to 5 of 25 minus T squared. All with respect to T, what this gives us is 25 T minus T cubed over three, evaluated from 0 to 5. And this gives us 25 times five minus 1, 25/ minus 25 times zero minus zero cubed over three. All of this just becomes zero. So when we plug in this first term into our calculator, what we get is a final answer of 83.3 radiance which corresponds to our answer. Choice of C Thank you all so much for watching. I hope this video helped. We will see you all in the next one.