A uniform piece of sheet metal is shaped as shown in Figure P9.38. Compute the x and y coordinat

A uniform piece of sheet metal is shaped as shown in Figure...

Key Concepts

Center of Mass

The center of mass is the point at which the entire mass of an object can be considered to be concentrated. For uniform objects, it coincides with the geometric centroid, and it can be found by taking the weighted averages of the x and y coordinates across the entire area.

Moment Calculations

Calculating the center of mass involves computing the moments of the area with respect to the coordinate axes. This is done by integrating or summing the products of each small area element and its corresponding distance from a reference axis, and then dividing by the total area.

Composite Shapes Method

When dealing with complex or irregular shapes, it is often practical to divide the shape into simpler geometric figures with known centroids. The overall center of mass is then determined by taking the area-weighted average of the centroids of these individual components.

Uniform Density

Uniform density implies that mass is evenly distributed over the object, which means that the computation for the center of mass can focus solely on the geometry of the shape without needing additional mass or density factors in the equations.

Transcript

00:01 Okay, so in this case, you have a mass that looks something like this where you have, so this thing, you are told, is a uniform piece of metal that has, so this is some y distance, and that is some x distance.

00:32 And it says, so they tell you that this sheet is just uniform, and they want you to locate.

00:37 The center mass given the coordinates of so you know that this is 10 that's 20 and that's 30 same thing here 10 20 30 and i want you to find the center of mass of this whole thing so you know that the center of mass to find the center mass you want to calculate the product of the masses so essentially the sum of products of the masses and then you divide by the total of the masses.

01:20 So this will give you the location of the center of mass in this direction.

01:25 So in this case, i can, since this is in two dimensions here, i can use x and y separately for r here.

01:34 So in this case, because you are told that this sheet of metal, but this is a uniform sheep, of metal.

01:41 So you can actually just transform this problem into a problem where you have uniform masses located at the center of these squares, right? so if you assume that you have uniform mass there, uniform mass there, uniform mass there, another one here, another one here, and another one here.

02:05 Because this whole sheet is uniform.

02:08 So if you take each block separately, it just it looks like you have a mass that's right at the center of each of these blocks here.

02:16 So this system here is going to be slightly easier to analyze in this.

02:21 So notice now you can start writing xy coordinates for each and every one of these masses.

02:26 Because we know that to get to here, it's 5x plus 5y.

02:33 Here it's 15x plus 5y.

02:35 Here it's 25x plus 5y.

02:38 So we know that, so this total thing weighs mass m.

02:46 So this summation here is equal to m.

02:51 But we split it into six equal squares here.

02:57 So that means that the top here is going to be, so each mass is going to be m over six.

03:07 So let's say start with this one.

03:11 One mass here for example so to find this expression for this mass is going to be m over six times the distance in x direction for that m of six so i'm starting with the x here so this is again up here so that's five x so that's going to be times and then i go to this mass here which is 15 away from zero so that's 15 x so i'm going to add because the summation again m over 6 times 15 then i go to this mass now 25 i'm now going to add again in the same mass because they're this uniform times 25 so that's these three on the row so now you go up notice again now this the x is changing but the mass doesn't change so this is going to be the mass times the x coordinate here which is five again plus this one, the mass, times its coordinate, which is now up here it's 25.

04:28 No, it's 5 again because we're going in the x direction only...

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