A wastebasket, designed to fit in the corner of a room, is 400 mm high and has a base in the sha

A wastebasket, designed to fit in the corner of a room, is...

Key Concepts

Uniform Density

When an object is made of a material with uniform density, every region of the object contributes equally to its mass per unit area (or volume). This assumption means that the center of gravity coincides with the geometric centroid, allowing one to compute it using only geometric properties without needing to consider variable mass distributions.

Centroid of a Planar Region

The centroid is the geometric center or the mean position of all the points in a shape, and for planar regions, it is often computed using integration. For standard shapes like a quarter circle, there are established results that give the location of the centroid relative to the boundaries of the shape.

Extrusion of a Planar Shape

When a two?dimensional shape is extended uniformly in the third dimension (as with the wastebasket, which has a uniform height), the center of gravity in the vertical direction lies at the midpoint of the height. Thus, the overall centroid of the object is determined by combining the centroid of the base shape with the mid-height position.

Geometric Symmetry

Exploiting symmetry can greatly simplify the analysis of an object's center of gravity. In cases where the object or its base has symmetric properties (even when partially formed, such as a quarter circle in a corner), these symmetries ensure that the centroid lies along specific, predictable lines, reducing the complexity of the calculations.

Transcript

00:06 For problem 5 .109, we are looking for the center of gravity of this basket, which means we'll need to find the centroid in the x, y, and z direction.

00:18 And to do that, we'll use this equation, where the centroid times the total area of the basket, since it's a sheet metal, so we can use simply the area instead of the volume, is equal to the sum of the centroid times the area of each component.

00:40 We have four components.

00:42 We have the quarter circle base.

00:44 We have the curved piece at the front.

00:47 And then we also have two rectangular pieces, which i'll label one and two.

00:55 And i think the easiest way to go about this is to make a table where we find the centroid of each piece and sum them together.

01:34 Now the centroid in the x direction for the first rectangle is simply going to be half of its length in the x direction, which is 5 inches since it is a rectangle.

01:45 The second rectangle lies completely on the z axis, so it's going to be at 0 inches for its centroid in the x direction.

01:57 Now the curved piece is a little bit trickier.

02:04 If we look at this curved piece in the xz plane, so we're looking from the top, we can find it centroid by integrating, looking at a little sliver and adding them all together, so looking for centroid in the x direction of this curved piece it'll be equal to the integral of a section of x times the change in length over the integral of the change in length now to find x at any point along that curve it'll be equal to the radius times a cosine of its angle and the change in length is equal to the radius times the change in angle, because that's the arc length formula.

03:11 And plugging that in, we get that, i'm going to make a little more room here.

03:26 So plugging both of those into the equation, we get at the integral of r squared cosine theta over the integral of the radius d theta is equal to the centroid.

03:40 And since it's a quarter circle, it's going to go from 0 to pi over 4.

03:49 And taking the integral, we get that the centroid is going to equal 2 times the radius over pi.

03:56 So then going back to our table, plugging in 2 times the radius over pi, we get that as equal to 6 .366 inches.

04:10 And for a quarter circle, the formula is 4 times the radius over 3 pi for a centroid...

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