Two balls are connected by a 150-cm-long massless rod. The center of mass is 35 cm from a 75 g ball on one end. What is the mass attached to the other end?

Video transcript

Hey, everyone. Welcome back in this problem. We have a rigid bar with negligible mass of length, 80 centimeters that connects a large sphere with a mass M one of 25 g to a small sphere with an unknown mass M two. And the large sphere is located at one end of the bar with the center of mass of the bar located 22 centimeters away from this sphere. Let's stop there for a minute. We're gonna draw a diagram of what we have and we're gonna come back and figure out what this question is asking us to find. So we have a bar with negligible mass. It is 80 centimeters long. Yeah. And it's gonna connect two spheres. So on the left, we have a large sphere and, and it has a mass M one of 25 g. On the other end, we have a smaller sphere with an mass M two and we don't know the value of M two. Now, we're told that the large sphere is 22 centimeters away from the location of the center of mass. And so we have some position X C M along this bar that is 22 centimeters from the left end. All right. Now, getting back to the problem, it is asking us to determine the mass of the sphere in grams at the other end of the bar. OK. So that's this mass M two. We're given four answer choices all in grams. Option A six, option B 12, option C nine and option D 18. Now, we're given information about one mass. We have information about the center of mass. So let's go ahead and write out our equation for the center of mass. We know that the position of the center of mass X C M, it's gonna be equal to M one X one plus M two, X two divided by M one plus M two. And you can recall this equation for a center of mass with two objects. No M one again is the mass of that big sphere. M two is the mass we're looking for of that smaller sphere. X one is gonna be the distance of the big sphere from whichever point we're going to use as a reference point X two is the same for that mass M two. Now, we know that the center of mass is 22 centimeters from the left end. OK. So we're gonna draw a red point on that left end and we're gonna use that as a reference point. So we know that the center of mass is at centimeters. OK. This is gonna be equal to the mass M 1 g multiplied by its position relative to a reference point. Now, our reference point is on the left end of this bar, which is exactly where that large sphere is located. So the distance there is actually gonna be zero centimeters, then we're gonna add the mass and two, which is what we're looking for, multiplied by its position or its distance from this reference point. That's gonna be the entire length of the bar since it's on the opposite end. And so that's gonna be 80 centimeters. And all of this is gonna be divided by the sum of the masses, 25 g plus M two. So we're gonna multiply this denominator on both sides. OK? So that we can get rid of it and work just in the numerator, we have 22 centimeters multiplied by 25 g plus M two. And that's going to equal to M two multiplied by centimeters. OK. The first term in the new radar on the right hand side just goes to zero because we're multiplying by zero centimeters. Now, we have M two term on both sides and we want to expand and rearrange this so that all of our M two terms are together. So expanding on the left hand side, we have 22 centimeters multiplied by 25 g, which gives us 550 centimeter grams. Then we have 22 centimeters multiplied by this mass M two. So we're adding that 22 centimeters supplied by M two and this is equal to M two multiplied by 80 centimeters. Now, we're gonna move the 22 centimeters multiplied by M two. That's on the left hand side to the right hand side. In order to do that, we need to subtract. So we have 550 centimeter grams on the left hand side, on the right hand side, we can factor out that M two. We have M two multiplied by 80 centimeters minus 22 centimeters. OK? 550 centimeter grams is equal to 58 centimeters multiplied by the mass M two. OK. Just simplifying on the right hand side. Now we're gonna divide both sides by 58 centimeters to isolate for M two, the unit of centimeters is gonna divide out and we're gonna be left with gramps when we get a mass M two of 9.48 g. OK? And that is that mass we were looking for. Now, we're gonna go up and compare this to the answer choices in just a minute. But I want to make one comment here about units. OK? We were given the units in centimeters and grams which are not our standard unit. OK? We did not convert them and actually use them anyway. And in this case, it was OK. Because when we did our calculation, we ended up taking the centimeters on the right divided by the centimeters on the left. So the unit of centimeter actually canceled and we were left with the unit of grams, which is the unit. The question was asking us to solve the problem for. But you do need to be really, really careful. This is not the case in every problem that you will solve. And it is important that you check and make sure whether it, whether you need to convert your units or not. All right. So going up to our answer choices, we are going round to the nearest gram and the correct answer is gonna be option C nine g. Thanks everyone for watching. I hope this video helped see you in the next one.