The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods. (a) Find the coordinates of the center of mass.

Question

Diagram showing four masses A, B, and C with coordinates for center of mass calculation.

Accepted Answer

Hey, everyone in this problem, we have a scenario where we have three distinct spherical masses joined together by rigid bars of negligible mass. And we're asked to determine the coordinates of the center of mass for this configuration. So we're showing a diagram of what we have, we have this right angle triangle. So we have a in the bottom left with the mass of 70 g, we have a B in the bottom right with a mass of 190 g. And then sphere C in the top right with a mass of g, we have four answer choices A through D and they each just contain a different combination of the X and Y coordinates for the center of mass. We're gonna come back and go through those as we work through this problem. Now, when we're calculating the center of mass, what we want to do is choose a reference point. OK. So that's the first thing we need to do is to choose a reference point in this problem. OK. Sphere A is at the origin, we want to know the coordinates. So if we were to calculate the center of mass relative to the origin. That's gonna give us exactly the coordinates we're looking for. And we don't have to do any other work after, to convert them from distances into coordinates. What we're gonna do is we're gonna take our reference point about sphere a or about the origin and recall that the position of the center of mass. And we're gonna start with the X position and we need to do both the exposition and the Y position is gonna be the sum of the masses multiplied by their position or distances divided by the sum of the masses. In this case, we have three masses. So we have MA X A what M B X B plus MC X C all divided by MA plus M B plus MC. You know what we're calculating here. If we look at this, it's almost like a weighted average. And so we're kind of taking a weighted average of the X positions of our Sears. Now, when we're talking about X A X B and X C, those are gonna be the distances from our reference point which is point A to the exposition of the sphere we're at and we're talking about the center of mass for the X position. So when we're talking about these distances, we're only talking about the distance between the X component. All right, let's get started. We have the mass of A which is 70 g multiplied by its distance or its position relative to the reference. Well, A is exactly at the reference. And so that's just gonna be zero centimeters plus the mass of B 190 g multiplied by its dis its distance and it's located at 20 centimeters. So it's 20 centimeters directly to the right of A, so that distance is gonna be 20 centimeters, pull off the mass of C 400 g multiplied by its X distance. Now, C is located at the same X value as B and it has an X coordinate of 20 centimeters. A Y coordinate of 12 centimeters that Y coordinate doesn't matter. Remember, we're not taking the complete distance between point A and point C, we're looking at the distance between their X components. So the distance for C is also going to be 20 centimeters, we're gonna divide all of this by the sum of the mass. OK? So we have 70 g plus 190 g plus 400 g. And you'll notice that I used grams and centimeters in this problem for the mass. It does not matter which unit you use as long as you're consistent. OK? Because we're gonna have grams in the numerator, we're gonna have grams in the denominator and those are gonna divide it. OK? So as long as we're consistent with what units we're using, those are gonna go away, we're using centimeters here. Now the answers are in meters, but we're gonna convert at the end So we only have one value to convert instead of converting each of these values. So you can do it either way, whether you wanna convert to meters first or do it after, let's give ourselves some more room to work. This is gonna be equal to 11,800 g centimeters divided by 660 g. If we work this out on our calculator, we get 17.87 repeated seven years. Now, if we want to write this in meters, we're gonna multiply by 100 go from centimeters to meters and we're gonna get 0. m. So that is the X coordinate. If we compare this to answer choices, option A 0.9 m. Option B 0.32 m, option C 0.18 m and option D 0. m. So rounding to two decimal places, we can see that this corresponds with answer choice C but we want to double check the Y coordinate as well. So let's go through, calculate the Y coordinate. Make sure we've made no mistake and see and that both of those options add up. So moving to the Y component and it's gonna be the exact same. The center mass Y C M is gonna be MA Y A plus M B Y B plus MC. Why is he divided by MA Os M B plus MC substituting in our values 70 g multiplied by zero centimeters. OK. A is still our reference point where that sphere is. So the X and the Y distances are going to be zero plus M B 190 g multiplied but it's wide distance. So let's go up, you can see that B lies along the X axis just like A does. So the Y component of each of those coordinates is the same. So the distance in the Y component between A and B is just zero, we go, we use zero centimeters there. Then we have plus C which is 400 g. Going back up to our diagram, it is a wide component of 12 centimeters. OK. So we're going from zero centimeters to 12 centimeters. So that distance is gonna be 12 centimeters and we divide by the sum of the masses again, 70 g plus 190 g plus 400 g. Simplifying the numerator, we get 4800 graham centimeters and this is all gonna be divided by 660 g. Again, the unit of gram divides out and we're left with 7.27 repeated centimeters. We wanna convert to meters, we multiply by 100 and we get that the position of the center of mass. The Y component can be 0.727 repeated meters. All right. Looking at our answer choices. OK. We narrow it down to being option C based off our X coordinate. We can see that for option C we have that X coordinate of 0.18 m. We have a Y coordinate of 0.73 m. And so this matches both of those coordinates we found and that is gonna be the correct answer. Thanks everyone for watching. I hope this video helped see you in the next one.

The four masses shown in FIGURE EX12.13 are connected by massless, rigid rods. (a) Find the coordinates of the center of mass.

Verified Solution

Key Concepts

Center of Mass

Mass Distribution

Coordinate System