please help with using double integrals to solve for mass and center of mass

**Skittles8768** · November 15th 2009, 04:10 PM

Find the mass and center of mass of the lamina for density = k
where R = a triangle with vertices (0,0), (b/2,h), (b,0)

(use double integrals to solve for both the mass and center of mass!)

ive tried this problem a million times and i cant get the right answer :-( (and yeah i checked calcchat but i need a more detailed explanation....walk me through it please? :-) )

**Chris L T521** · November 15th 2009, 05:04 PM

Originally Posted by Skittles8768

Find the mass and center of mass of the lamina for density = k
where R = a triangle with vertices (0,0), (b/2,h), (b,0)

(use double integrals to solve for both the mass and center of mass!)

ive tried this problem a million times and i cant get the right answer :-( (and yeah i checked calcchat but i need a more detailed explanation....walk me through it please? :-) )

First off, if $\delta\!\left(x,y\right)$ is the density of the lamina, then $M=\iint\limits_{R}\delta\!\left(x,y\right)\,dA$ .

In our case, $R$ is a triangle with vertices $(0,0)$ , $(b,0)$ and $(b/2,h)$ . From this, we can come up with the equation of the line that connects the points $(b,0)$ and $(b/2,h)$ and the points $(0,0)$ and $(b/2,h)$ . I leave it for you to verify that the equations of the lines are $y=-\frac{2h}{b}x+2h$ and $y=\frac{2h}{b}x$ respectively.

Thus, it follows that if $\delta\!\left(x,y\right)=k$ , then

$M=\iint\limits_{R}k\,dA=k\cdot\text{Area of $R$}=k\cdot\left(\tfrac{1}{2}bh\right)=\tfrac12 kbh$

Now, to find center of mass, we need to evaluate $M_x=\iint\limits_{R}x\delta\!\left(x,y\right)\,dA$ and $M_y=\iint\limits_{R}y\delta\!\left(x,y\right)\,dA$ .

From there, it follows that

$\bar{x}=\frac{M_x}{M}=\frac{\displaystyle\iint\lim its_{R}x\delta\!\left(x,y\right)\,dA}{\displaystyl e\iint\limits_{R}\delta\!\left(x,y\right)\,dA}$ and $\bar{y}=\frac{M_y}{M}=\frac{\displaystyle\iint\lim its_{R}y\delta\!\left(x,y\right)\,dA}{\displaystyl e\iint\limits_{R}\delta\!\left(x,y\right)\,dA}$

Note that $M_x=k\int_0^h\int_{\frac{b}{2h}y}^{-\frac{b}{2h}y+b}x\,dx\,dy$ and $M_y=k\int_0^h\int_{\frac{b}{2h}y}^{-\frac{b}{2h}y+b}y\,dx\,dy$ .

I leave it for you to simplify the two integrals and figure out the coordinates for the center of mass.

Can you take it from here?

Thread: please help with using double integrals to solve for mass and center of mass

LinkBack

Thread Tools

Display

please help with using double integrals to solve for mass and center of mass

Similar Math Help Forum Discussions

Question about a center of mass/integrals problem

Find mass and center of mass of the solid S....

Center of Mass

center of mass

Center of Mass

Search Tags