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. (a) Find a, b, and c, including their units.

Video transcript

Welcome back everybody. We are told that a toy ferris wheel has an angular position given as a function of time as the following X. Which is some constant plus Y. Which is another constant times, time minus the which is another constant times two cubed. We are told that a time at time zero the angular position equal to pi over two. We're also told that at time zero the angular velocity is 1.7 radiance per second. We are also told that the angular acceleration at Time 2.2 seconds is equal to 0.875 radiance per second squared. And we are asked to find what X, Y and Z are. So here's how we're going to do it first. We need to find the formulas for both angular velocity and angular acceleration. Once we have that, we will plug in these conditions to solve for our constants. Let's go and do that. The derivative of angular position with respect the time is going to give us our angular velocity as a function of time. Let's take the derivative of this function right here keeping in mind that X, y and z are constants term by term derivative, X is just zero. Plus through to the second term is going to be why that teal disappear minus. Using the power rule. This three is going to come down. So it'll be three times easy times. T squared. Now the derivative of angular velocity with respect to time is going to give us our angular acceleration as a function of time. So let's go ahead and take the derivative of this function now derivative of Y is zero as it's just a constant minus. Using the power rule again we're gonna have minus six C. E. Great. Now that we have our three formulas, we are ready to find X, Y and Z here. So we are told that at time equals zero. Our position angular position is equal to pi over two. Let's go ahead and plug in zero to this formula right here, X plus Y times zero minus Z times zero cubed. This yields that are X is equal to pi over two. Great. So now let's go ahead and plug in time zero into our angular velocity. We are told that at time zero it has angular velocity of 1. radiance per second. Which is equal to We're gonna plug in t to this formula right here. Why? Minus three times zero squared. This yields that are Y is equal to 1.7 radiance per second. Finally, at a time of 2.2, our angular acceleration is 0. radiance per second square. So let's go and plug it into this. Right here we have that. This is equal to negative six times our time of 2. squared. Actually gonna divide both sides by negative six times two point squared. Next time. Sorry, not squared, no squared there. Yes, Times 2.2. So -6 times 2.2. This is going to yield that our Z is equal to negative 0.0, Sorry, radiance six right there, ingredients second cubed. So now we have found our X, our Y and our Z corresponding to answer choice. A thank you all so much for watching. Hope this video helped. We will see you all in the next one.