Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Moments and Center of Mass. Mass is a measure of a body's resistance to changes in motion and is independent of the particular gravitational system. I'm going to start with looking at the law of levers. If we have two masses that are distance away, we know that mass one times distance 1 is going to equal mass 2 times distance 2, or that Fulcomb is a balancing point. If we wanted to put this on a system with an origin, we would have X2 being so far away from O and X1 being another distance away from O. If we called our balancing pointer X bar, then the distance between X bar and X sub two would be X sub 2 -, X sub one, and the distance between X bar and X1 would be X bar minus X sub one. Remember, we're going to go farthest to the right minus the left, so farthest to the right minus the left. So now instead of thinking of it as AD one and AD two, we're actually going to put in those distances. So M 1 * X bar minus X1 sub one equaling M2 times X 2 -, X bar. We're going to distribute out the masses. And then we're going to take the X bars all to one side. So if we distribute out the M1 and M2, we get M1X bar minus M1X1 equaling M2X2 minus M2X bar. Taking all the X bars to one side and factoring it out, we get X bar times the quantity M1 plus M2 equaling M2X2 plus M1X1 or X bar equals M2X2 plus M1X1 over M1 plus M2. The M1X1, The M2X2 are called moments about the origin. The M1 and the M2 are masses. That X bar is the center of mass. Now this works if we have two masses, but it also works if we have lots of masses. So if we thought about taking lots and lots of masses, we'd have X bar equaling all of the sum of the moments about the origin or the M sub I * X sub i's over all of the masses. Now if we thought about having lots of them, a summation of them, and then we thought about taking infinitely many, we could think of this as turning it into an integral. So our integral of A to B of X DX, our integral A to B of X, the density of X delta X times our thickness, our DX over our mass AB delta XDX. So the moment about the origin divided by the mass, where X is the distance, delta, X is the density and DX is the thickness, we're going to look at this in terms of planes or plates. So the moment about the X axis MX, we're going to think about taking all of those masses times our Y distance because what we're trying to find is we're trying to find a balance or an X bar along the strip. So if we look at this strip, we realize that we're going to be going up and down and we're going to find a balancing point at some point. So our moment about the X axis is really changing in the Y direction. So we're going to have Y~ DM and the moment about the Y axis or M sub Y is going to be all of these masses summed together times the distance of how far over we're going left and right. So our integral of X~ DM, the center of the mass of a strip is going to be X~ Y~. So the center of mass of a plate X bar, Y bar is going to be the moment about the Y axis divided by mass, the moment about the X axis divided by mass. Let's look at some examples. So we're going to have a plate and we're going to say Y equal 25 -, X ^2 and the X axis. So if we think about Y equal 25 -, X ^2, we have a parabola going down the X axis. So it's going to intersect at -5 and five. If we didn't know that, we would just set the two equal and solve. So 0 equal 25 -, X ^2 X is 5X is -5 S our X~ Y~ We want the average. So if we think about this lower point as being X0 and this upper point is being X, 25 -, X ^2, we would average them X + X / 225 -, X ^2 + 0 / 2 or X, 25 -, X ^2 / 2. So the length of that line, the top minus the bottom or 25 -, X ^2, the width is going to be DX. If we think about just one single strip for a moment. So the area of this one single strip is going to be length times width or 25 -, X ^2 DX. The mass is going to be the area times the density of the strip. In this case, we're just going to use the density being delta, so delta times 25 -, X ^2 DX. So our mass of the plate, we're going to go from -5 to five of the delta times 25 -, X ^2 DX or delta times 20 five X -, 1/3 X cubed from -5 to five 500D over three. If I want the moment about the X axis, it's going to be my Y~ times the density times the area of the plate or the area of the strip. And then we're going to integrate to get the air the plate. So from -5 to 5, the Y~ we found a minute ago, 25 -, X ^2 / 2, the density is going to be given as delta 25 -, X ^2 DX. I'm going to factor out the delta over 2 because those are constants, and if I multiply 25 -, X ^2 ^2 quantity squared, I get 625 minus fifty X ^2 + X to the 4th or 5000D over three. Now we're going to do the moment about the Y axis, so it's going to be our X~ density times the area. So -5 to 5X delta 25 -, X ^2 DX. When we compute that, we get 0. So our X bar Y bar should be our M y / m, MX over M 0 / 500 delta thirds and 5000 delta thirds over 500 delta thirds or zero 10. If we look at this plate, it makes sense that our X bar needs to be 0 because we have symmetry right and left Y bar if it's zero to 25 along our Y axis. But we have more of the plate occurring from -5 to 5 on the X than we do when X is 0. We know that it should have our center of mass closer or further down South. Zero 10 does seem reasonable. If we look at another example, we get X equal y ^2 -, y and Y equal X. So the first thing we're going to do is figure out where those intersect. So we're going to set those two equal to each other and solve so we know y = 0 and Y equal 2. I need to do my X~ Y~ So if I figure out a strip in here, the point for the 1X, Y would really just be YY. The point for the other X, Y is going to be y ^2 -, y, Y. So if I add the 2X's together, y ^2 -, y + y / 2 because I want the average, and then I add the two YS together and divide by two, we're going to get y ^2 / 2, Y for X~ and Y~ The length of the segment the furthest to the right, Y minus the furthest to the left, the Y ^2 -, y. We're looking at our X values, but they need to be in terms of Y. So two y -, y ^2. The width of each segment is going to be DY. The area is just going to be length times width or two y -, y ^2 d Y The mass of the segment is our density times the area of the segment. So our mass of our plate M is going to equal zero to 2 the bounds we found a moment ago our density or times the mass of the segment. So our density times two y -, y ^2 d Y so our integral of zero to two mass of the segment. So when we evaluate that the density is just a constant, so we get density times y ^2 -, 1/3 Y cubed. Sticking in zero to two, we get 4D over 3 the moment of the plate about the X axis. We're doing the exact same thing, but we're also multiplying by Y~ before we do the integration. So the Y~ was just Y density to y -, y ^2 dy. When we evaluate that, we're going to get 4D over three the moment of the plate about the Y axis. So our M sub Y is going to just be the same thing as our mass times the X~ Our X~ was y ^2 / 2. When we compute that, we get 4D over five. So the center of the mass of the plate is going to be 4D over 5 / 4 delta over three, 4D over three, 4D over 3 or 3/5, one. If we look at a rough sketch, 3/5, one, that seems reasonable for a location. Our next example is going to be assuming that our density is no longer a constant. So our density is going to be given as 12X and we're going to look at Y equal X ^2 y equal X. This thin plate that's being bounded by this. So if we look at our X~ Y~ and we put in strips going along our DX, our upper point here would be X, Y, or we could think of that as X, X. Our lower point here would be X, Y. We could call them X1Y1X2Y2 if we wanted. So here it would be X, X ^2. So take our 2X values, add them together, divide by two. Take our two Y values, add them together, divide by two. The length just the upper minus the lower. So X -, X ^2, the width is going to be DX the area length times width, or X -, X ^2 DX. The mass is going to be the density times the area. But the density this time was actually a formula that was given. So 12X times the quantity X -, X ^2 DX. When we compute that, we get twelve X ^2 -, 12 X cubed DX. So the moment about the X axis MX is going to equal Y~ DM. I'm going to factor out a 1/2 and a 12 so that I can just have the variables left. So when we compute that, we're going to foil it out and we're going to realize that we're going to get 1/2. When we do the moment about the Y axis, we're going to have X~ DM. So when we compute that one, we're going to end up with 3/5. The moment of the plate is just the mass, the integration of the mass, not times a~ at all. So when we evaluate that, we get one. So our X bar, Y bar or our center of mass of this plate is going to be 3/5, 1/2. Thank you and have a wonderful day.