Determine the x- and y-coordinates of the mass center of the...

Determine the x- and y-coordinates of the mass center of the homogeneous plate of uniform thickness t = 0.25 in. Assume b = 19 in., h = 12 in. Answers: x = \text{ky}^2 x = in. y = in.

Transcript

00:02 All right, so you're doing a mass and center of mass problem with a variable density.

00:08 So we'll start with a mass.

00:11 The mass here is going to be our integral from zero to one.

00:15 We're going to do the density function times the area between the two, which is just the difference between f and g.

00:21 So that's radical x minus one half of x, dx, and that equals 0 to 1 of x of x to the 2, or 5 halves rather, sorry.

00:33 I wrote that upside down.

00:37 X to the five halves minus one half x to the third, which will equal x to the seven halves over two, over seven halves, which makes it two sevenths, minus one fourth, x to the fourth, evaluated from zero to one, which equals two sevenths minus one.

01:09 I'm sorry, this is one eighth here, not one fourth, that's one eighth.

01:11 One eighth, which is a final result of 9 over 56 when you do your common denominators.

01:19 So there's a mass, and the mass will be used later to find the center of mass.

01:23 Next, we need to find the two moments, the one around the x and one around the y.

01:27 So the moment around x is equal to the density function, or integrating from 0 to 1, the density function, times the sum of the functions.

01:38 So radical x plus one half of x over two times the difference.

01:44 The functions, radical x minus one half of x, and all of that with respect to x.

01:52 Okay, so we'll clean this up a little bit.

01:54 We'll take the one half out in front.

01:56 So go from zero to one...

Question

Determine the x- and y-coordinates of the mass center of the homogeneous plate of uniform thickness t = 0.25 in. Assume b = 19 in., h = 12 in. Answers: x = \text{ky}^2 x = in. y = in.

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