Find the Center of Mass of a Lamina

[jabar616]

Homework Statement

Find the x-coordinate of the center of mass of the lamina that occupies the region cut from the first quadrant by the lines x = 6 and y = 1 if the density function is p(x,y) = x + y + 1

Homework Equations

Mass = ∫∫_spdσ
X Coordinate of center of Mass = M_yz/M
M_yz = ∫∫_sxpdσ

The Attempt at a Solution

I am not sure how to set this one up, as I have not seen one like this without a bounding equation (ie x² + y² + z² = a² or similar.)

I attempted simply setting up the integral as being from 0≤x≤6 and 0≤y≤1 so M=∫∫x+y+1dydx = 27 and M_yz = ∫∫x² +xy +x dydx = 126

However, dividing 126 by 27 results in 14/3, not 11/3 as the answer theoretically should be.

Any help would be greatly appreciated!

 

[SteamKing]

The problem appears to be in your calculation of M_yz. If you can't find the problem, please post your work here so it can be reviewed in detail.

 

[Mark44]

Homework Statement

Find the x-coordinate of the center of mass of the lamina that occupies the region cut from the first quadrant by the lines x = 6 and y = 1 if the density function is p(x,y) = x + y + 1

Homework Equations

Mass = ∫∫_spdσ
X Coordinate of center of Mass = M_yz/M
M_yz = ∫∫_sxpdσ

Your two formulas above are a bit confusing. You have a two-dimensional object, so the two moments are with respect to the x-axis or the y axis.

The notation M_yz is the moment about the y-z plane, which isn't applicable here because the object is two-dimensional.

The formula to use for the x-coordinate is M_y/M

The Attempt at a Solution

I am not sure how to set this one up, as I have not seen one like this without a bounding equation (ie x² + y² + z² = a² or similar.)

Here the boundaries are a lot simpler: the region is a rectangle, with x ranging between 0 and 6 and y ranging between 0 and 1.

I attempted simply setting up the integral as being from 0≤x≤6 and 0≤y≤1 so M=∫∫x+y+1dydx = 27 and M_yz = ∫∫x² +xy +x dydx = 126

However, dividing 126 by 27 results in 14/3, not 11/3 as the answer theoretically should be.

Any help would be greatly appreciated!

 

[Mark44]

The problem appears to be in your calculation of M_yz.

Yes, I agree. BTW, I get 11/3 for the x-coord. at CM.

Please show us your calculations for M_y.

 

[jabar616]

I did end up finding my error, of course it was just a simple issue that I had missed. Thank you all!