A 6.0-cm-diameter gear rotates with angular velocity ω = ( 20 ─ ½ t² ) rad/s where t is in seconds. At t = 4.0 s, what are: b. The tangential acceleration of a tooth on the gear?

Video transcript

Welcome back, everyone. We are making observations about a sprocket and we are told a couple of different properties here, we are told that it has a circumference of 56.55 centimeters or 0.5655 m. And we are told that it follows an angular velocity according to this angular velocity equation. Here, we are tasked with finding what is the tangential acceleration at a point on the circumference of the sprocket after a time of nine seconds. Well, for tangential acceleration, the way we convert here is we just multiply our angular acceleration times our radius. But we need to find both of these terms. First and foremost, let's just start out with angular acceleration. We know that angular acceleration is this the derivative of angular velocity with respect to time. So let's go ahead and take the derivative of our formula for angular velocity. Taking it turn by turn here, the derivative of 66 is constant. So it'll just be zero uh plus the derivative of 16 T squared by the power rule that two will come down multiply 16 to give us one third times T giving us that our angular velo uh sorry, our angular acceleration formula is just one third times t great. So now what we can do is we can go ahead and say that our angular acceleration is equal to one third times nine seconds, giving us an angular acceleration of three radiant per second squared after nine seconds. Great. So now we need to go ahead and find our radius here. And the way we're going to do that is we are going to use the circumference. We know that the circumference of a circle is two pi times our radius. And we were given the circumference was 20. m. Let me go ahead and divide both sides by two pi here. That'll cancel out the two pi on the left hand side and we get that our radius is equal to 20.5655 divided by two pi which gives us 0.9 m. So now we are ready to go ahead and find our tangential acceleration. This is just going to be equal to three times our radius of 0.9 giving us a final answer of 0. m per second squared, which corresponds to our final answer. Choice of D Thank you all so much for watching. Hope this video helped we will see you all in the next one.