An electric fan is turned off, and its angular velocity decreases...

Question

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

Accepted Answer

Welcome back everybody. We are looking at the timeline of a motor here. So let me actually draw out, I'm line here and when I say timeline by a motor, I mean that when a motor is switched off, there's a certain time afterwards, we're still gonna keep going based off rotational inertia and kinetic energy. Right? So we're told a couple of different things here. Let's first look at this interval right At the time that the motor is switched off, we have that it has an angular velocity of rpm, which if we divide by 60, we're gonna have that. The initial angular velocity here is 10.8 revolutions per second. Alright, now we are then told at the end of this interval It slows down to 300 rpm, which when we divide by 60 we get that it is five revolutions per second. Now for the motor to slow down from 650 revolutions per minute to 300 revolutions per minute. It takes 3.5 seconds. Alright, now let's look at this second interval right here. This time, I'm going to say that our initial velocity is this? Well, that's correct because at the start of this interval it's just going to be what the velocity was at the end of the previous one. So that makes sense. And we are told that it slows all the way down. Now we are interested in finding the total time, delta t total. It takes for all of this to slow down considering that the acceleration is constant. That's going to be key here that our acceleration is constant. So what we are going to want to do is find our acceleration, then find this delta T. Of this interval. Then add the two delta teas together. So first and foremost let's go ahead and find our constant acceleration. Well, I'm gonna use this kid a magic formula and I'm going to look at this first interval right here. I'm gonna use the quadratic formula that says our final angular velocity is equal to our initial angular velocity plus our acceleration times time. Subtracting our initial angular velocity from both sides and dividing by time this yields that our acceleration is equal to our final angular velocity minus our initial angular velocity. All over delta T. We have all these values. So let's go ahead and plug that in here. Our angular acceleration is equal to our final which is five revolutions per second minus our initial of 10.8. Divided by the total time it takes in this first interval remember of 3.5. This gives us a constant acceleration of negative 1. revolutions per second. Great! Now that we have that I'm gonna go to this second interval with this acceleration and solve for this delta T. Right here, we're actually gonna use the same formula here. We are going to have that our final our final angular velocity is equal to our initial angular velocity plus our acceleration times time. But remember our initial and final have changed because we're looking at a different interval here, We're looking at the second interval. Right, so here's what I'm gonna do. I'm gonna do the same thing here. I want to try to isolate this time variable here. So I'm gonna subtract our initial angular velocity from both sides and divide by our angular acceleration which gives us that our time is equal to our final angular velocity minus our initial angular velocity all over our angular acceleration. So let's go ahead and plug in some values here. We have that our change in time for this second interval is equal to final which is zero minus our initial of five revolutions per second divided by our angular acceleration of negative 1.66 revolutions per second. Sorry, that should be squared revolutions per second squared which is equal to 3. seconds. So now let's go ahead and add our delta T. From the first angle plus our delta T. From the second angle to get our delta T total, which is what we are looking for. This is going to be 3.5 plus three .01. Giving us a final answer of 6.51 seconds corresponding to answer choice b. Thank you all so much for watching. Hope this video helped. We will see you all in the next one

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