Two ice skaters, both of mass 68 kg, approach on parallel path...

Question

Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia. Calculate the change in kinetic energy for this process.

Accepted Answer

Welcome back. Everyone in this problem. Two twin kids, each of mass 26 kg are in two different electric ride on cars. Each of mass 20 kg, they head toward each other moving in parallel paths separated by a distance of 1.1 m at a speed of 1.2 m per second. The kids extend their hands out the window of the cars toward each other and grab the other's hand as they pass, keeping the same 1.1 m distance from one another's trajectories as a consequence, both of the cars start rotating about the other afterward. Focusing on the transition from their approach to the moment they connect hands determine the change in kinetic energy, viewing the cars as point masses. For our answer choices A says that it's zero jewels, B 33 jewels, C 66 jewels and D 130 jewels. No, we're trying to determine the change in kinetic energy. OK? And for our energy notice that initially we had linear motion, OK? They were moving toward each other in parallel paths and after their hands uh connect then the cars start to rotate. So we're talking about rotational motion before we do that. Let's make a note of all the information that we have here. So, so far we know that the mass of each child is 26 kg and the mass of the electric ride on car is 20 kg. So the total mass for both objects is going to be equal to 20 plus 26 which is 46 kg. That's the, that's our point mass. OK. Together next, we also know that they are separated by a distance of 1.1 m. So if we think about it, when the cars start to rotate, the distance between both is 1.1 m. So the radius four or circular motion that develops is going to be a half of 1.1 m or 0.55 m. So we can say that R here R one and R two both equals 0.55 m. Next, we also know that they have a speed, a linear speed of 1.2 m per second. And now we're trying to use all of this to determine the change in kinetic energy. OK? No, let's ask ourselves, what do we know here? What type of energy uh are or what's happening with our energy? Well, solving for our kinetic energy or change in kinetic energy is going to be equal to the difference between the initial kinetic energy and the final kinetic energy. No, what we already know here. So far, OK. Is that initially there was linear motion and then uh in, in the final motion, it was changed to rotational motion. OK. So here, let's ask ourselves, what do we know about kinetic energy in both cases? Well, we know that when it comes to a translational or linear motion, the kinetic energy is going to be equal to a half of MV squared. And for our rotational energy, we can call that ke that's going to be equal to a half of I omega squared where I is our moment of inertia and omega is our angular velocity. So that means that would imply then that our changing kinetic energy OK is going to be equal to our initial linear motion. OK, which is going to be the sum of both kinetic energies of both twin kids and their ride on cars. So that's going to be a half. Well, remember both of them have the same speed and the same mass. So it's going to be two multiplied by a half of MV squared because they're both the same OK, minus our final kinetic energy, which is going to be equal to a half of our final moment of inertia multiplied by our final angular velocity. No two cancers are half. So really then the change in kinetic energy equals MV squared minus a half of if omega F squared. No, this is just the start because even though we know the mass and the velocity we don't know what our moment of inertia is. OK. And we don't know what our uh final angular velocity is. So we need to figure out what both of those are to then solve for our change in kinetic energy. So let's start with our moment of inertia. So now we're solving for if, now, what do we know about the moment of inertia? Well, recall, OK. Recall that a moment of inertia I is equal to Mr squared. OK. Where R is our radius and M is our mass. So with that in mind, then that means our final moment of inertia is going to be the sum of both the moments of inertia for both kids and their Charlie. So that's going to be equal to two multiplied by M multiplied by our radius squared, which is two Mr squared. So we have that so far. Next, let's try to find our angular velocity. OK. Now, let's think about it here. We're not given much in terms of what the angular velocity is. But let's ask ourselves in this situation, what do we know is going to happen? Well, we know, OK, we know that by the conservation of angular momentum, the total initial angular momentum of the system is going to be equal to the total final angular momentum. OK. So by applying that idea, then we know that L let L be our initial angular momentum of the system is going to be equal to the final angular momentum. If we think about it, both of them have angular momentum. OK. So L one I or initial angular momentum for our first and L two, I for our second child and, and, and ride on car is going to be equal to our final angular momentum. Now, if we take that a bit further, we know they both weigh the same and they, they have the same radius, they're the same distance apart from each other. OK. So that means then two multiplied by our moment of inertia multiplied by our an angular velocity is going to be equal to our final or this should be our initial here. It is going to be equal to our final moment of inertia multiplied by our final angular velocity. But we already know what our final moment of inertia is. OK. We already figured that out here. So we can use it, we can substitute it into our problem. OK? Because no, that means mm that two multiplied by our initial moment of inertia. So that's going to be mm multiplied by ri which we know it's the same mass and the same radar. So that's two multiplied by Mr squared multiplied by Omega I, OK. Which we know that Omega is defined as our velocity divided by our radius is going to be equal to if OK. Which oh we already know that we know it's two Mr squared. OK. Multiplied by Omega F. Now do you notice something here. Well, yes, notice that both sides, both sides have two M squared. So we can cancel both of them here to tell us that Omega f our final angular velocity is going to be equal to V divided by R. No. What does that mean? Well, we know V is equal to 1.2 m per second and we know our radius is 0.55 m. So when we divide both of those, we end up with the final angular velocity of 2.18 radiance per second. No, this is pretty good. This is pretty good because now we have our final angular velocity, we can substitute it into our formula for our change in kinetic energy to eventually solve. So let's do that because no, by applying that idea, this means that the change in kinetic energy is going to be equal to MV squared minus as we said earlier, if which we found to be two Mr squared, two Mr squared multiplied by our final angular velocity substituting all of these known values. Well, I think we can, we can even factor out M here. So that's M multiplied by V squared minus two R squared Omega F. So that's going to be 46 kg or mass multiplied by V squared as 1.2 m per second. All squared, all of that is squared. Yeah, multiplied by Omega F. Oops, sorry, my bad, my nose, sorry. Two R squared. So that's two multiplied by 0.55 m squared, multiplied by Omega F OK. So that notice here, Omega F squared. Sorry, I forgot my squared here. My apologies. This should be Omega F squared. So that's gonna be 2.18 radiance per second or squared. OK. So always remember to keep your science because that, that can get to you at the end of the day, right? So here we have our expression for our changing kinetic energy. And no, we can subtract no MV squared 46 multiplied by 1.2. Here we're going to get an initial kinetic energy of 66.2. For Jules likewise two multiplied by 0.55 squared, multiplied by 2.18 squared, including all of the correct units is also going to be 66.24 joules and 66.24 minus 66.24 equals zero. Thus, our change in kinetic energy is zero. In other words, there is no change in kinetic energy. If we look back at our answer choices, that means a is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Conservation of Angular Momentum

Rotational Kinetic Energy

Kinetic Energy Change

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