CALC A fan blade rotates with angular velocity given by ω_z(t) = g - bt^2, where g = 5.00 rad/s and b = 0.800 rad/s^3. (b) Calculate the instantaneous angular acceleration α_z at t = 3.00 s and the average angular acceleration α_av-z for the time interval t = 0 to t = 3.00 s. How do these two quantities compare? If they are different, why?

Video transcript

Welcome back everybody. We are given the formula for angular velocity of some rotating particle as a function of time. And it is given as B, which is 3.8 minus D or 0.94 times I'm Where'd And we are asked to find both what the average angular acceleration is and the instantaneous Angular acceleration. And start out with this average angular acceleration. Well, this is going to be equal to our change in angular velocity over a given time interval. Now we are given this time interval. We are told that it goes from 0 to 2.5. Right, so really what we're looking at here is that we have that our average angular acceleration is going to Our final angular velocity. Our initial angular velocity over this interval of 2. seconds. What are our final and initial angular velocities? Well, we're going to actually plug in the starting beginning times into our formula right here to find that. So our Final angular velocity is just going to be our angular velocity evaluated at 2.5 Equal to 3.8 -0. times 2.5 Squared. Which when you plug into your calculator, you get that it is 1.45 radiance per second. Doing the same thing here, let's go ahead and find our initial angular velocity which is just us plugging in zero to our formula right here. So this is equal to 3.8 -0.94 times zero squared, giving us an initial angular velocity of 3.8 radiance per second. Great! Now that we have all of our turns, let's go ahead and plug these into our formula for average angular acceleration. This is equal to our final angular velocity of 1.5 minus our initial of 3.8. All over our time of 2.5 seconds. Giving us an average angular acceleration of negative 0.94 ingredients per cent squared. Great. So we found that. But what about the instantaneous acceleration at time? of 2.5 at that final moment here? Well outside of this formula, we also know that the derivative of our velocity are angular velocity with respect to time is going to give us a formula for acceleration. So if we just plug in that formula our desire time we'll find it. But first let's take the derivative of this equation right here And we'll just do it component by component here. So the derivative of 3.8 is just zero because it's constant and then you have minus. We're going to use the power rule here so that two comes down two times 0.94 times e. This is equal to negative 1.88 radiance per second. Dude, Times our time. Now let's go ahead and evaluate and find our acceleration at time 2.5 which will be our instantaneous acceleration at that time. Plugging in T here. We have negative two times 0. times 2.5. Giving us a final answer of negative 4. radiance per second squared. So now we found our average angular acceleration are instantaneous angular acceleration, and this corresponds to answer voice. Be Thank you all so much for watching. Hope this video helped. We will see you all in the next one.