A 25 g steel ball is attached to the top of a 24-cm-diameter vertical wheel. Starting from rest, the wheel accelerates at 470 rad/s². The ball is released after ¾ of a revolution. How high does it go above the center of the wheel?

Video transcript

Welcome back, everyone. We are making observations about a circular disk that spins in a vertical plane and is used to launch masses along a vertical tangent. You're told that the diameter of the disc is 0.5 m and that we are going to be launching a marble which has is a mass of 14.5 g. Now, we are told that the disc starts initially off at rest. So it'll have an initial angular velocity of zero. We are told that it is accelerated at a rate of 750 radiant per second squared. Now, we are tasked with finding after a total of five revolutions, we need to calculate the maximum height obtained above the launch point here. All right. Well, in order to figure out the launch point, let's take a look at this kinematic formula right here. This kinematic formula states that our final velocity squared is equal to our initial velocity squared plus two times our acceleration times our delta Y subtract our initial velocity squared from both sides. And then I'm going to divide both sides by two A as well. On the right hand side, this cancels out the acceleration of velocity terms. And what we get is that our delta Y is equal to our final velocity squared, vertical velocity squared minus our initial velocity squared all divided by two A. Now we know that our A is going to be the acceleration due to gravity which is negative 9.8 m per second squared. And we know that our final velocity is going to be zero because that's when projectile motion reaches a maximum height. But now we need to figure out what this initial velocity is going to be. Luckily, we have a formula for this. We have that our initial velocity converting between angular and vertical velocity. We're gonna have our radius times our final angular velocity after five revolutions. So let's go ahead and calculate that. And the way we're going to do that is we are going to do it with this formula right here. That says our final angular velocity is equal to squared is our equal to our initial angular velocity plus two times our angular acceleration times the number of revolutions. All I got to do is take the square root of both sides and you'll see that the exponents and the radical cancel out. And we get that our final angular velocity is equal to this quantity right here. So let's go ahead and plug in our values. We have zero squared plus two times 7 times the number of revolutions. But we need this in radiance So I'm gonna multiply our number of revolutions by two pi and what we get is 10 pi radiant. So I'm gonna plug in 10 pi right there, extend the radical a bit. And what we get is that our final angular velocity is 68. radiance per second. Great. So now let's go ahead and find our initial vertical velocity. This is going to be equal to our diameter divided by two to get the radius times 68.6 which gives us 17.15 m per second. Great. So now we are ready to go ahead and use our formula for the maximum height. So we have that delta Y is equal to final velocity squared, zero squared minus 17.15 squared all divided by two times negative 9.8 giving us a final answer of m which corresponds to our answer choice of D. Thank you all so much for watching. I hope this video helped. We will see you all in the next one.