The three masses shown in FIGURE EX12.15 are connected by massless, rigid rods.(a) Find the coordinates of the center of mass.

Video transcript

Hey, everyone in this problem, we have four distinct spherical balls that are connected by a glass rod of negligible mass in the shape of a rectangle. OK. The balls are located at the vertices of the rectangle which is shown in the figure. So we have ball A which is located at the origin. It has a mass of 200 g. Ball B is located to the right of ball A in the bottom, right with a mass of 400 g. In the top right of our rectangle, we have ball C with a mass of 100 g. And in the top left of our rectangle, we have ball D with a mass of 400 g. Now, the width of this rectangle is 12 centimeters and the height is six centimeters. And we're asked to determine the coordinates of the center of mass for this configuration. Now, we're given four answer choices options A through D and each of them just contain a different combination of the coordinate for the center of mass for the X and the Y direction. So like these answer traces show we have a two dimensional object here. So we need to consider two different directions. For a center of math. We need to figure out the X component and we need to figure out the Y component and we're gonna do those separately. So starting with the X component, we have X C M K and recall that to calculate the center of mass, we're gonna take the sum of the mass multiplied by its position relative to some point. OK. So we have four objects here, four balls. So we get MA X A course M B X B course MC X C plus MD X D. And all of that is divided by the sum of the masses, MA plus M B plus M C++ MD. OK. So M is the mass here XX is the position of that mass relative to a particular point which we're gonna come back to in just a second. And then the subscripts ABC and D just indicate which ball we're talking about. Now, when we're calculating the center of mass, we need to do about a particular point. OK. We're looking for the coordinate of the center of mass. So we want to calculate this relative to the origin, which means that we're gonna calculate our center of mass relative to the point A, all right. So if we go to substitute in our values, the position, the exposition of ball a relative to itself is just gonna be zero. OK. The distance between A and itself is zero. So this first term is gonna go to zero and we're gonna skip writing that on the next line for the sake of saving space. And we're gonna move on with the other terms. So we have the massive bul be, which is 400 g multiplied by its exposition relative to a OK. Well, the width of this rectangle is 12 centimeters. So that's gonna be 12 centimeters. Yeah. Moving to MC, we add the mass of C 100 g multiplied by its exposition relative to A OK. Well, again, C is on the right hand side of this rectangle. So this is also gonna be 12 centimeters. And it's important to note here, we are not looking for the distance between A and C OK. We're not calculating this middle distance here. What we're looking for is the difference in the X position. OK. So we're only looking in the X direction. All right. And then we move to B D. It has a mass of 400 g. A it's on the left hand side of this rectangle with A. So its exposition is the exact same as A and so X D is gonna be zero centimeters, right? So we have 400 g multiplied by 12 centimeters plus g multiplied by 12 centimeters. OK. The other term goes to zero all divided by the sum of the masses which is g plus 400 g plus 100 g plus 400 g. Now, before we go any further, I wanna make a quick comment on units here. You can see that we've left our units in grams for mass instead of converting to our standard unit of kilogram. That's because we're gonna have grams in the numerator, grams in the denominator. And that unit's gonna divide out anyway. OK. So as long as the unit is consistent, we can use any unit for um the mass here. And then in terms of the position, we can see that the answer choices are in meters. OK? But we're gonna convert at the end. So we only have to convert our final answer instead of converting the position for every single ball. All right. So let's simplify this in the numerator. We're gonna have 6000 and our unit is Graham centimeter in the denominator, we have 1100 g and the unit of gram is gonna divide it like I just mentioned. So we have 6000 divided by 1100 which gives us 5.45 centimeters. OK? And that is the position of the center of mass in the extraction. Now, if we look at our answer choices, we can see that we can already eliminate option A and option B because they do not match what we were given A. We have 5.45 centimeters. We want to convert this to meters. So we can divide by 100. OK? We have 5.45 centimeters, we multiply by one m per 100 centimeters, ok? Because we know that there are 100 centimeters in every meter. And like I said, it's like we're dividing by 100 to get 0.55 if we round to two significant digits. And that is now in meters, ok? So that matches with option um C and D and now we need to move on to the Y position in order to figure out which is the correct answer choice. So moving to the Y direction and the formula is gonna be the exact same except now we're looking at the Y positions instead of the X positions for each ball. So Y A Y B Y C and Y D, we have MA Y A plus M B Y B plus MC, Y C plus MD, Y D and all of this is divided by that sum of masses, MA plus M B plus MC plus MD substituting in our values. We have a little bit more room here. So we're gonna write everything out. We have the mass of A which is 200 g and now its distance relative to itself is gonna be zero centimeters just like it was in the X position. Then we have the mass M B 400 g multiplied by its distance in the Y direction from A. Well, A and B are both along that X axis. They have the same Y value. And so this is gonna be zero centimeters. Then we add the mass of C 100 g, OK? It's at the top of this rectangle. This rectangle has a height of six centimeters. So the distance between its Y position and um sphere A is Y position is gonna be six centimeters. And the same goes for mass D A mass of g multiplied by that distance of six whoops, not meters centimeters. And all of this is divided by that sum of masses. OK? 200 g plus 400 g plus 100 g plus 400 g. Now, the first two terms go to zero and we're gonna deal with the second two terms. If we simplify, we have 3000 g centimeters divided by 1100 g. Again, the unit of Graham divides out and we're left with the position in the Y direction of the center of mass being equal to 2.72 repeated centimeters. We're gonna multiply again, one m divided by 100 centimeters to convert this to meters. And we are left with 0.27 m again, rounding to two significant digits. And now if we compare this to our answer choices, we can see that this corresponds with answer choice C A, the X coordinate of the center of mass is 0.55 m and the Y coordinate is 0.27 m. Thanks everyone. For watching. I hope this video helped see you in the next one.