1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Moments,Center of Mass, & Centroid

  1. Jan 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Find Mx,My, & (x bar, y bar) for the laminas of uniform density ρ bounded by the graphs of the equations. (Use rho for ρ as necessary.)

    2. Relevant equations

    m= ∫f(x)-g(x) dx
    my= ∫x(f(x)-g(x)) dx =>x bar my/m
    mx= 1/2 ∫ (f(x))2-g(x))2dx => y bar=mx/m

    3. The attempt at a solution
    So this is my work

    x=-y <-- g(y)
    x=5y-y^2 <----f(y)


    *note I don't know how to put 0 to 6 on the integral

    m=p ∫ [(5y-y^2)-(y)]dy
    =p [3y^2 -(y^3/3)]= 36 p

    My= p∫[(5y-y^2)+((-y)/2)][(5y-y^2)-(-y)]
    =p/2∫ (4y-y^2)(6y-y^2)dy
    =p/2∫ (y^4-10y^3+24y^2) dy
    = p/2 [(y^5/5)-(5y^4/2)+8y^3]
    =216/5 p is wrong I don't know why :?:
  2. jcsd
  3. Jan 27, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The general form of the relevant equations is
    m &= \iint \rho\,dx\,dy \\
    M_x &= \iint \rho y\,dx\,dy \\
    M_y &= \iint \rho x\,dx\,dy
    When the region of interest is between x=a and x=b and is bounded on the top by f(x) and on the bottom by g(x), you get the equations you cited. For instance, for the moment about the x-axis, you get
    $$M_x = \rho \int_a^b \int_{g(x)}^{f(x)} y\,dy\,dx = \rho \int_a^b \left.\frac{y^2}{2}\right|_{g(x)}^{f(x)} \,dx = \frac{1}{2}\rho \int_a^b [f(x)^2-g(x)^2]\,dx$$
    If you sketch the region for this particular problem, however, you'll see the roles of x and y appear to be reversed, so the formulas you were trying to use don't work. You'll need to derive the correct ones or adapt the ones you have for this particular case.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Moments,Center of Mass, & Centroid