FIGURE EX4.23 shows the angular-velocity-versus-time graph for a particle moving in a circle. How many revolutions does the object make during the first 4 s?

Welcome back, everybody. We are making observations about a cylinder that is fixed on the top of a rotating platform were given this graph of angular velocity versus time of the cylinder. And we are tasked with finding the total number of rotations made between the times of zero seconds and six seconds. Let's think about this. Conceptually. First, we know that angular velocity is just equal to the change in our angular position divided by the change in time. I'm gonna multiply both sides by our change in time here. And we get that our change in angular position is equal to our angular velocity D T. Now let's integrate both sides here. The bounds for this left hand side is just going to be the initial angular position to the final angular position. And for the right side, it's the initial angular velocity to the final angular velocity. What this gives us is our desired total change in angular position is equal to the integral from the initial to the final angular velocity of omega D T. Now, this may seem a little complicated here, but let's look at the geometric interpretation of this integral right here. It's as simple as this, all we have to do since we're given a graph of time versus angular velocity is just find the area underneath this curve right here. And we can do it by finding the area of these four regions which I have labeled as A B C and D. So let's go ahead and do that area of A is just the area of a triangle. So this will be one half times the base of two times the height of 10 giving us 10 radiance area of B is just going to be the base of three times. The height of 10 giving us 30 radiance area of C is going to be base of one times height of 10 giving us 10 radiance and the area of D is one half times The height of 20 times the base of one giving us 10 radiance. And when you add all of these up, we get a total change of 60 radiance. But as you see, we need this in terms of revolutions or rotations. So let's convert real quick. We know that there are two pi radiance in one revolution. Let's cancel out this unit on top of bottom, giving us a final answer of 9. revolutions corresponding to our answer. Choice of a. Thank you all so much for watching. Hope this video helped we will see you all in the next one.