A painted tooth on a spinning gear has angular position θ = (6.0 rad/s⁴)t⁴. What is the tooth's angular acceleration at the end of 10 revolutions?

Video transcript

Welcome back, everyone. We are making observations about a rotating disk and we are told that the position of a hole engraved on that rotating disc obeys this equation for angular position. And we are tasked with finding what is the angular acceleration of the hole after a total number of revolutions. Well, we know that angular acceleration is just going to be the second derivative of angular position. So first, let's find that. But even before doing that, let's go over our answer choices, we have a 329 radiant per second squared. B 178 radiant per second squared, C 59.2 radiant per second squared or D 110 radiant per second squared. All right. Well, in order to get the double derivative of position, we, we have that our an angular position formula is just eight times T to the fourth. So we're really just gonna apply the chain rule twice. We have four times three times eight times T squared. And what this, what this gives us is 96 T squared. But what do we plug in for T in order to find our angular acceleration. Well, that's gonna be given by our initial condition right here. 15 revolutions multiplied by two pi radians. Since there's two pi radiant per revolution is going to be equal to our formula for our angular position. So this is going to be equal to eight T to the fourth. Now, if we just solve for T we will have the time to plug in to find our angular acceleration. Let's go ahead and divide both sides by eight. And you'll see that on the right hand side, the number eight cancels out. And then we just need to take the fourth route of both sides as well. And what this gives us is that T is equal to 1.85 seconds after a total number of 15 revolutions. So now that we have that we are ready to find our angular acceleration, we have that our angular acceleration is equal to 96 times 1. squared, giving us a final answer of 329 radiance per second squared, which corresponds to our final answer. Choice of a. Thank you all so much for watching. I hope this video helped. We will see you all in the next one.