CALC The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct^3, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = p/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s^2. (c) What are θ and the angular velocity when the angular acceleration is 3.50 rad/s^2?

Video transcript

Welcome back everybody. We are given a sphere and we are told a couple of things about this. First and foremost, we are told that its angular position function of time is do this X. Which is some constant plus why another constant times time -7. Another constant times cubed. We're told that at times zero our sphere is going to have an angular position of I over three. We are told also at time zero that it has an angular velocity of 1.3 radiance per second. We're also told that at a time of 1.8 that has an angular acceleration of 0.9 radiance per second squared. Now we are tasked with finding When the angular acceleration is equal to 2.8 ingredients per second squared for some tea. What is the angular velocity evaluated at that? T. And the angular position evaluated at that. This is a lot to unpack. But here's what we're gonna do. We're gonna use these conditions that we are given to solve for X, Y and Z constants. Then we will need to find what time Makes the angular acceleration equal to 2.8 and then plug that time into these functions right here. So first and foremost, we can actually find these functions real quick and we'll sorry find the function for angular velocity and angular acceleration first. And that will make things a little bit easier. So let's let's tackle that first. Right. So the derivative of angular position with respect to time is equal to our angular velocity as a function of time. Let's take the derivative of this. Keeping in mind that X, Y and Z are constants of accents is constant is just zero. Plus that T. Is just gonna disappear on the Y. So we just have y minus this. Three is going to come down in front according to the powerful. I'm Z, I'm scared. Great. Now we're actually going to take the derivative again. So taking the derivative of our angular velocity with respect to time. If this our formula for angular acceleration as a function of time, this is equal to while the driven of wine is just zero because it's constant minus this two is going to come down in front again. So six C times our tea. Great. So now that we have these formulas, let's go ahead and plug in these conditions to solve for our constants of X, Y and Z. So our position angular position evaluated at zero is equal to pi over three. Well, let's go ahead and plug in T zero to this equation. We have X plus Y times zero minus e times zero cubed. This yields that pi over three is equal to X. So we found our X. Right. Now let's use this second condition right here that our angular velocity evaluated at time zero is equal to 1.3 radiance per second. Let's go ahead and plug in a time of zero into our angular velocity equation. We get y minus three Z times zero squared. So this yield that 1. radiance per second is in fact our why? Great. So now let's apply this last condition here. And it looks like we're probably gonna solve for Z with that. So Our income acceleration at 1. seconds to negative six times Z times one point eight. Great. Now dividing both sides, Sorry, this is equal to 0.9. So dividing Both sides by -6 times 1. eight. This yields that our Z is equal to negative 0.083 Ingredients per second. Cute! Great! So now we have our X, our Y and our Z. So now we can find this time right here and then plug it into our equations. Let's do that. So, our acceleration at this, given time that we're trying to solve for is equal to 2.82 point eight is equal to negative six times E, which is negative zero point yo Times our time. Fighting both sides by -6 times negative 0.08. Three of course, both sides over here too. This is going to yield that. The time that we are looking at is 5. seconds. Great. So now let's plug this time into our two equations to get our final answers. So, our angular velocity evaluated at 5.6 seconds is equal to our Y, which is 1.3 minus three times R Z of negative 0. times our times squared 5.6 squared. Which when you plug this into your calculator yet that are angular velocity at time, 5.6 is 9.13 radiance per second. Now our position evaluated at a time of 5. seconds is equal to our X. Of I over three plus Y. Which 1.3 times our time of 5. minus three times R. Z. Of negative 0.08 three Times 5.6 Cubed. Which when you plug into a calculator, you get an answer of 20.9 radiance. So now we have found our desired angular velocity, angular position or responding to answer choice B. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.