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Center of Mas of a planar Lamina

  1. Jul 24, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the center of mass of a planar lamina, in the form of a triangle with vertices (0,0),(0,a),(a,a),
    if ρ=k


    2. Relevant equations

    m = ∫∫f dA

    xbar = My/m

    ybar = Mx/m


    3. The attempt at a solution

    mass = ka²/2

    Mx = ∫∫yk dy dx

    My = ∫∫xk dy dx


    **Side Question, how do we use math type or some other type of symbolic language on physics forums? ***

    Any thoughts would be appreciated.
    Below is my photo:
     

    Attached Files:

  2. jcsd
  3. Jul 25, 2014 #2

    Simon Bridge

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    You can use LaTeX to type maths.
    i.e. $$M_x = \iint ky\;dy\;dx$$

    Did you have a question?
     
  4. Jul 25, 2014 #3

    HallsofIvy

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    If the density is a constant, the "center of mass" is the "centroid", the geometric center of the figure. And for a triangle that is particularly simple to find. If you really want to use the double integral, x goes from 0 to a and, for each x, y goes from 0 to x:

    [tex]m= \int_0^a\int_0^x k dydx[/tex]
    [tex]M_x= \int_0^a\int_0^x kx dydx[/tex]
    [tex]M_y= \int_0^a\int_0^x ky dydx[/tex]
     
  5. Jul 25, 2014 #4
    Aw sweet, turns out I had the wrong variable in my Mx & My formulas, kept getting the thing wrong:)


    And for symbols, my screen just shows "quick symbols", none of the fancy LaTex y'all are using ^^
     
  6. Jul 25, 2014 #5
    Okay actually I do have a question, in my book the formula for Mx is double integral of ky dA

    And for My is double integral kx dA


    Which is correct?
     
  7. Jul 26, 2014 #6

    HallsofIvy

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    What do you mean "which" is correct? [itex]M_x[/itex] is the moment of inertia about the x axis and is [itex]\int\int ky dA[/itex]. The y coordinate of the center of mass is [itex]\overline{y}= M_x/m= \int\int ky dA/m[/itex] and the x coordinate of the center of mass is [itex]\overline{x}= M_y/m= \int\int kx dA/m[/itex].

    Personally, I find the "[itex]M_x[/itex]", "[itex]M_y[/itex]" notation "non-intuitive" and prefer the [itex]\overline{x}= \int\int kx dA[/itex], [itex]\overline{y}= \int\int ky dA[/itex] notation.
     
  8. Jul 28, 2014 #7
    oh!! Yes :D Excellent, thank you for clearing that up.
     
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