A computer hard disk 8.0 cm in diameter is initially at rest. A small dot is painted on the edge of the disk. The disk accelerates at 600 rad/s² for ½s, then coasts at a steady angular velocity for another ½s. b. Through how many revolutions has the disk turned?

Video transcript

Welcome back, everyone. We are making observations about a circular disk attached to an electric motor. We are told that the diameter of the circular disk is 15 centimeters. So that means it has a radius of 7.5 centimeters and or 0.75. Now, we are told that the disc has a coupling screw located seven centimeters from the center of the disc. So 70.7 m away from the center of the disc, we're told that the disc is accelerated for or at a rate of 1400 radiance per second squared for a time of 1.5 seconds. And then it's allowed to spin at a uniform angular velocity for one second. Meaning during that second time interval, it is going to have an angular acceleration of zero. And we are tasked with finding the total number of rotations made by the disks. So here's how we are going to do this first. Let's find the number of rotations after the first interval. And then we will be able to find the number of rotations after the second interval. So let's use these kinematic formulas. We have that our final number of rotations is equal to our initial number of rotations plus our initial angular velocity times the change in time plus one half times our angular velocity times our time period squared. And we will also use the kinematic formula that our final angular velocity is equal to our initial angular velocity plus angular acceleration times our period of time. So let's go ahead and try to find first beta one, the number of rotations after 1.5 seconds. So using our formula, this is equal to beta knot plus omega knot times our first time period plus one half times our initial angular acceleration times our first time period squared. This is equal to well starts off from nothing. So that's gonna be zero also starts at rest. So that means our initial angular velocity is zero making the second term zero plus one half times 1400. And let me go ahead and move this down just a little bit. So I can finish up the rest of the term here one half times 1400 times 1.5 squared. What this gives us for our uh number of revolutions after the first time period is equal to 1575. Wonderful. All right. So now let's go ahead and find the number of revolutions after time period. Two, it's gonna be equal to the number of revolutions after time period. One plus our, let's see here, our velocity after time period, one times our second time period plus one half times our second angular acceleration times our delta T two squared. And let's go ahead and plug in all of our values we have that this is +1575 plus. Well, we need to figure out it looks like what our angular velocity was at the end of our first time interval. So let's go ahead and do that first before plugging in this term right here, we have that our velocity at the end of our first time period is equal to using this second formula right here. Our initial velocity plus our angular acceleration times delta T this just gives us zero plus 1400 times 1.5 giving us a angular velocity at the end of time period, one of 2100 radiance per second. Let me go ahead and scroll down here just a little bit. All right. So now we can take this number and plug it into this above formula. So we have, let's see here. We have 2100 times our second time period of one second plus one half times our second angular acceleration, which is just zero times our time of one squared. This just becomes zero. And what we get for the number of revolutions at the end of the second time period is 300 or sorry, 3675. In fact, this is not our number of revolutions and misspoke. This is the number of radiant that are completed, but we need to convert this to revolutions. So let's go ahead and do that. We have that. Our theta two is equal to, to 3675 radiant. And we know that in one revolution there is two pi radians, these units are going to cancel out, which leaves us with our final answer of 585 revolutions corresponding to our answer. choice of B Thank you all so much for watching. I hope this video helped. We will see you all in the next one.