A wheel is rotating about an axis that is in the z-direction. The angular velocity ω_z is -6.00 rad/s at t = 0, increases linearly with time, and is +4.00 rad/s at t = 7.00 s. We have taken counterclockwise rotation to be positive. During what time interval is the speed of the wheel increasing? Decreasing?

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Welcome back, everybody. We are making observations about a toy helicopter and we are making observations about the rotation of its propeller here. So let me draw out its propeller and for a helicopter you're typically going to have for something like this, right? We're told that the counter clockwise direction is going to be positive and we are told, told a couple of other things as well at an initial time of zero seconds. So starting off, we have a initial velocity Angular velocity of negative 9.6 radiance per second and at a final time of 12 seconds, this angular velocity increases to 7. radiance per second. And we are tasked with finding two things. one, what is the time interval for which the velocity is increasing or positive? And b what is the time interval for which the angular velocity is decreasing or negative? Looking at all these terms here can dramatic equation comes to mind of this right here. Our final angular velocity is equal to our initial angular velocity plus our angular acceleration times our change in time here. So let's observe this just a little bit more here. We start out at a negative value and we go to a positive value. That means at some point which I'm just gonna label omega psi we know that our angular velocity is zero once again because it's going from negative to positive in the end. So what we need In order to differentiate between these two intervals is we need to find the time at which the angular velocity is zero. Therefore signifying going from negative to positive. So what else do we know? Well, we know an initial angular velocity of negative 9.6. I'm gonna label this are critical angular velocity of zero as our final angular velocity in terms of this equation right here. But we don't really know anything else. We don't know the time period in which it's solely increasing or decreasing. We don't know the change of angle or the entirety of this process. And we don't know the angular acceleration. Remember this delta T for either increasing or decreasing interval is what we are trying to find. So we need to then find one of these two values. And here's what I recommend. Let's go ahead and use this formula except with the values that we are already given in order to solve for our angular acceleration as our angular acceleration will be constant throughout this entire motion. So taking aside this equation here, I'm going to subtract our initial angular velocity from both sides. And then I will divide both sides by alpha you'll see that this alpha cancels out, this cancels out. And we are left with the fact that let's see our time is equal to our final angular velocity minus our initial angular velocity all over angular acceleration. We'll be able to use this equation when we find what our angular acceleration is, but we need to find our angular acceleration first. So what you can do is just switched these two values right here, We have that our angular acceleration is equal to our final angular velocity minus our initial angular velocity all divided by our time. And I will go ahead and use like I said, our initial values and just plug them in here. So we have a final angular velocity of positive 7. minus Our initial angular velocity of negative 9.6 divided by our total change in time. So will be our final minus our initial change or sorry, our final time minus our initial time, which is 12 seconds. When you plug all of this in your calculator, we get the angular acceleration of 1.43 radiance per second squared. Great. So now that we have that we now know that our angular acceleration is 1.43 ingredients per second squared. And then we can use these two values and plug it into this formula right here to find the time at which it changes from decreasing to increasing. So let's go ahead and do that. And that R T is equal to zero or are critical omega minus negative 9.6 divided by angular acceleration of 1.43. And this gives us a time of 6.71 seconds. Let's visualize this on a number line here. So here's our number line. We know that at the very beginning, we are going to have a time of zero seconds. And at the very end of our interval is 12 seconds. Somewhere in here just a little bit past the halfway point. We have our time of 6.71 seconds. And remember it started out as negative, it went positive. Therefore, this entire interval will be negative and this entire interval will be positive. We have now found the decreasing interval. Let me scroll down just a little bit. We have found the decreasing interval and the increasing interval corresponding to our final answer. Choice of C Thank you all so much for watching. Hope this for your help. We'll see you all in the next one.

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Key Concepts

Angular Velocity

Angular Acceleration

Direction of Rotation