Answer to Question #254223 in Calculus for Susan

Question #254223

Find the mass and center of mass of the thin plate that occupies the triangle with vertices (0,0), (2,1), and (0,3) and has density function p(x,y) = x + y.


1
Expert's answer
2021-10-21T12:02:40-0400
m=Rdm=RρdA=02x/2x+3(x+y)dydxm=\int\int_Rdm=\int\int_R\rho dA=\displaystyle\int_{0}^{2}\displaystyle\int_{x/2}^{-x+3}(x+y)dydx

=02[xy+y22]x+3x/2dx=\displaystyle\int_{0}^{2}\bigg[xy+\dfrac{y^2}{2}\bigg]\begin{matrix} -x+3 \\ x/2 \end{matrix}dx

=02(x2+3x+x2/23x+9/2x2/2x2/8)dx=\displaystyle\int_{0}^{2}(-x^2+3x+x^2/2-3x+9/2-x^2/2-x^2/8)dx

=02(9x2/8+9/2)dx=[3x3/8+9x/2]20=\displaystyle\int_{0}^{2}(-9x^2/8+9/2)dx=[-3x^3/8+9x/2]\begin{matrix} 2 \\ 0 \end{matrix}

=3+9=6=-3+9=6



Mx=RxρdA=02x/2x+3x(x+y)dydxM_x=\int\int_Rx\rho dA=\displaystyle\int_{0}^{2}\displaystyle\int_{x/2}^{-x+3}x(x+y)dydx

=02[x2y+xy22]x+3x/2dx=\displaystyle\int_{0}^{2}\bigg[x^2y+\dfrac{xy^2}{2}\bigg]\begin{matrix} -x+3 \\ x/2 \end{matrix}dx

=02(x3+3x2+x3/23x2+9x/2x3/2x3/8)dx=\displaystyle\int_{0}^{2}(-x^3+3x^2+x^3/2-3x^2+9x/2-x^3/2-x^3/8)dx

=02(9x3/8+9x/2)dx=[9x4/32+9x2/4]20=\displaystyle\int_{0}^{2}(-9x^3/8+9x/2)dx=[-9x^4/32+9x^2/4]\begin{matrix} 2 \\ 0 \end{matrix}

=9/2+9=9/2=-9/2+9=9/2



My=RyρdA=02x/2x+3y(x+y)dydxM_y=\int\int_Ry\rho dA=\displaystyle\int_{0}^{2}\displaystyle\int_{x/2}^{-x+3}y(x+y)dydx

=02[xy2/2+y33]x+3x/2dx=\displaystyle\int_{0}^{2}\bigg[xy^2/2+\dfrac{y^3}{3}\bigg]\begin{matrix} -x+3 \\ x/2 \end{matrix}dx

=02(x3/23x2+9x/2x3/3+3x29x+9)dx=\displaystyle\int_{0}^{2}(x^3/2-3x^2+9x/2-x^3/3+3x^2-9x+9)dx

02(x3/8+x3/24)dx-\displaystyle\int_{0}^{2}(x^3/8+x^3/24)dx

=02(9x/2+9)dx=\displaystyle\int_{0}^{2}(-9x/2+9)dx

=[9x2/4+9x]20=[-9x^2/4+9x]\begin{matrix} 2 \\ 0 \end{matrix}

=9+18=9=-9+18=9


xˉ=Mym=96=32\bar{x}=\dfrac{M_y}{m}=\dfrac{9}{6}=\dfrac{3}{2}

yˉ=Mxm=9/26=34\bar{y}=\dfrac{M_x}{m}=\dfrac{9/2}{6}=\dfrac{3}{4}

m=6,C(32,34)m=6, C\bigg(\dfrac{3}{2}, \dfrac{3}{4}\bigg)

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment