Find Center of Mass of Thin Plate in Region Bounded by y-Axis, x=y-y^3

[whatlifeforme]

Homework Statement

find the center of mass of a thin plate with constant density in the given region.
region bounded by y-axis, x=y-y^3 ; 0<=y<=1

Homework Equations

x(bar) = (integral)(a to b) α(x) * x * (f(x) - g(x))
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(integral)(a to b) α(x) * (f(x) - g(x))y(bar) = (integral)(a to b) (1/2) α(x) * x * (f(x)^2 - g(x)^2)
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(integral)(a to b) α(x) * (f(x) - g(x))

The Attempt at a Solution

how do i solve this since it is f(y) and not f(x). would i replace the "X" in x(bar) integral with a "y"?

 

[HallsofIvy]

Your "x= y= y^3" is "shorthand" (and not very good if you ask me) for "x= y" and x= y^3. Solve the latter for y: y= x^(1/3).

 

[whatlifeforme]

are the limits of integration from 0 to 1 still?

 

[whatlifeforme]

Your "x= y= y^3" is "shorthand" (and not very good if you ask me) for "x= y" and x= y^3. Solve the latter for y: y= x^(1/3).

update: it is x=y-y^3

sorry for the misunderstanding.

 

[Dick]

update: it is x=y-y^3

sorry for the misunderstanding.

Then you had the start of a good idea. Take the function y=x-x^3 bounded by the x-axis for 0<=x<=1. It looks the same as your original region, just has x and y reversed. So work out it's center of mass then exchange x and y again.

 

[whatlifeforme]

not getting the right answer.

 

[whatlifeforme]

made a mistake on one of the integrals. now I'm getting the correct answers.
but I'm getting the two backwards the X value of center of mass is the y value for the center of mass on the answer key.

 

[Dick]

made a mistake on one of the integrals. now I'm getting the correct answers.
but I'm getting the two backwards the X value of center of mass is the y value for the center of mass on the answer key.

Without seeing what you are doing, it's hard to say. But if you are getting say, (1/2,1/3) for y=x-x^3, then the answer for x=y-y^3 should be (1/3,1/2). You need to interchange the center of mass coordinates to go from one to the other.

 

[whatlifeforme]

Without seeing what you are doing, it's hard to say. But if you are getting say, (1/2,1/3) for y=x-x^3, then the answer for x=y-y^3 should be (1/3,1/2). You need to interchange the center of mass coordinates to go from one to the other.

what do you mean?

 

[Dick]

what do you mean?

I mean that the regions bounded by x=y-y^3 and y=x-x^3 don't have the same center of mass but there is a simple relation between them. What did you get for the center of mass of y=x-x^3?

 

[whatlifeforme]

i got (8/15, 16/105) whereas the answer is (16/105, 8/15).

should i just note that whenever solving a center of mass with x= to switch the values at the end?

 

[Dick]

i got (8/15, 16/105) whereas the answer is (16/105, 8/15).

should i just note that whenever solving a center of mass with x= to switch the values at the end?

You could make a note of it, but it would better if you clearly understood why. As I've said, switching x and y to turn x=y-y^3 into y=x-x^3 is going to switch the x and y coordinates of the center of mass. You need to switch them back.

 

[whatlifeforme]

how would i solve this where it would not involve switching the coordinates?

 

[Dick]

how would i solve this where it would not involve switching the coordinates?

You can't solve x=y-y^3 to get y as a function of x in any useful way. So you can't use your formulas directly. Either get similar formulas for the case where you have x expressed as a function of y or set up the center of mass formulas as double integrals, if you've covered that.

 

[whatlifeforme]

we haven't covered double integrals.

 

[Dick]

we haven't covered double integrals.

Then stick with the switching approach.