For a turntable were given the initial angular velocity, which is Omega and is 1.8 revolutions per second, which I convert to radiance per second by multiplying by two pies. Let's 11.3 radiance per second. The moment of inertia of the turntable I sub t is 0.2 kilograms meters squared. The party that we're gonna drop on the term table has a mass m of 200 grams or 2000.2 kilograms. So we convert everything two kilograms here because we want to use S I units and the distance away from the center that the party has dropped his 20.15 liters. So for part A, we are asked to find the initial kinetic energy of the turntable. So since it's all rotational, the initial kinetic energy of the turntable which we call Katie I is equal to 1/2 Iomega initial squares. This is a make, I swear. So we just simply have to plug those values in when we find that the initial kinetic energy of the turntable is equal to 1.2 a jewels so we can box set in as their solution. For part a part B says, What is the final rotational speed of the system with the lump of the putty now on the turntable? So to do that, we're gonna consider the fact that the end of their momentum is gonna be conserved to the final angular momentum. Elsa, death is equal. The initial angular momentum will The final angular momentum is going to be the total moment of inertia of the putty plus turntable system. So this is the moment of inertia of the turntable I 70 plus m r squared to get the moment of inertia of the putty. Uh, where m is the mass and R is the distance away from the center that the putty lies and then omega f is the final rotational velocity. What we're trying to find this is equal to the initial moment in the momentum of the turntable, which is I so tee times, omega chi. So we know everything here except for omega F. So we simply saw for Omega death we find that this is equal to I sub t times Omega chi divided by Isom t plus him are square. So simply plug these values into this expression and we find that the, uh there. Um, rotational speed in the end is 9.23 radiance per second. So we can box that in is our solution for B Heart C has asked us to find the final linear speed of the lump of putty. Okay, So defined the linear speed of the lump of putty. You simply have to use the definition that v final. The linear speed is equal to go make a final times the distance the putty is away from the centre are so just play these values in and we find that this is equal to 1.38 meters per second. Okay, so now for the next three questions D, e and F, what we're going to be doing is we're going to find the change in kinetic energy of these different parts of the system. So first, we're gonna find the change in kinetic energy of the turntable. Okay, so this is, um, parte de we'll call this Delta Katie for the change in kinetic energy of the table. So this is just gonna be Katie final minus Katie initial. But we already found we go up. We already found Katie. Initial right so we just need Katie final. So this is gonna be Katie Final is gonna be 1/2 Iomega final squared where I is the moment of inertia of the table. We could also write out the expression for Katie initial, but we already have a numerical value for Katie initial cause we found that in part a party. So we can just write this again as Katie I because we already have that numerical value. Otherwise, you could write it as 1/2 isom t omega initial squared. So plugging these values in, we find that Delta Katie is equal to negative 0.43 jewels. So it actually loses some energy, which makes sense because it's been slowed down due to the putty being dropped on it. Which is why we have that negative sign. Okay, now, party were asked to find the change in kinetic energy of the putty. We'll call this Don't decay, P. Okay, but this is equal to K p f minus k p i. But the party had no initial energy, so that goes to zero. So this is just KPs So K p f is 1/2 times the moment of inertia of the Putty, which is just m r squared times that and your speed of the putty in the final state Omega F Square. That's the same Omega value is the turntable since their rotating at the same speed. So playing this in, we find that this is equal to 0.19 jewels. So it actually gains kinetic energy, Which makes sense because it started out with none. And now it's rotating. So it has some. And then lastly for F, we're asked to do it for the entire system. So, um, so for the entire system, we'll just call this don't decay. This is gonna be K final minus k initial. Okay, so que final minus minus K initial. Well, what is K final minus K initial. So this is K for Final is que t final plus KP final minus Katie initial because there is no KP initial. So this comes out to be equal to negative 0.24 jewels so we can make a little statement about this so we can say that the the total kinetic energy there is a total change in kinetic energy, which we're going abbreviate with Katie in this system is equal to the sum of the kinetic energy changes in the individual constituents that make up the system. It's a little bit bigger and weaken box all of that in as our solution to Part F, which is the final part of the question.