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. (b) What is the angular acceleration when θ = p/4 rad?

Video transcript

Welcome back everybody. We are looking at a toy ferris wheel whose angular position is given as a function of time by the following X. Which is some constant plus Y. Which is some other constant times t minus Z. Which is another constant times E cubed. And we are told a couple different things here, a couple different conditions. Right At time zero angular position is equal to I over five. Also at time zero, its angular velocity is equal to 1.6 radiance per second At a time of 1.4. Its angular acceleration equal to 1. radiance per second squared at some time that we do not know just yet. But we will have to solve for We have that our angular position is Pi over five and we have to find what the angular acceleration is at that same time. Well, let's look at our conditions here. And the first thing that I noticed is that well, when our angular position is pi over five our time is zero. So we really have to evaluate our angular acceleration at time equals zero. But we don't have a formula just yet for our angular acceleration as a function of time. So let's find that. First, this will be a couple of steps here. So, first, the derivative of angular position with respect to time is going to be our angular velocity with respect to time. This is equal to Well, let's take the derivative this guy keeping in mind that X, Y and Z. Or Constance. And just going to terminate time here derivative X is just zero because it's constant Plus why? That he will disappear minus using power. Will that three is going to come down? So three times C squared. Let's take the derivative again now. So the derivative of angular velocity is going to be our desired angular acceleration as a function of time. So then taking the derivative of this function derivative, Y is just going to be zero minus this. Two is going to come down again. Multiply with that three to become six times Z. T. Well, we have our formula for acceleration but we don't know what X. Y and Z are. We're actually gonna use these conditions right here to find our constants of X, Y and Z. So let's go ahead and do that. We're told that are angular position evaluated at zero is equal to pi over five. So let's go ahead and plug in zero to this equation Right here we have X plus Y times zero minus Z times zero cubed. This yields that our X is equal to i over five. Great. Also at time of zero, we were told that our angular velocity is equal to 1.6 radiance per second. So let's go ahead and plug in T. This equation right here equal to y minus three Z times zero squared. This yields that our Y is equal to 1.6 radiance per second. Great. Now our angular acceleration at time, 1.4 seconds equal to 1. radiance per second squared. So let's plug in 1.4 seconds to this formula right here we get that negative six times e times 1.4 is equal to 1.3 radiance per second. I'm gonna divide both sides by negative six times 1.4. Eight of six times one for this yields that are Z is equal to negative 0. ingredients per second, which is what we were looking for. So now that we have that, let's go ahead and evaluate our angular acceleration at time zero here, Right? Because that's what we are looking for, acceleration times zero is gonna be negative six times negative 0.155 ingredients per second cubed. I'm zero which is just going to be zero radiance per second squared. Giving us our final answer, answer choice B. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.