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Moment and center of mass

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Intros
Lessons
  1. Moments & Center of Mass Overview:
  2. Mass of the Region
    • Imagine a thin plate in region with density ρ\rho
    • Calculate using double integral
    • An example
  3. Moments of the Region
    • Tendency of region about xx-axis, and yy-axis
    • MxM_x & MyM_y
    • Calculate using double Integrals
    • An example
  4. Center of Mass
    • The point that balances the plate horizontally
    • Calculate by using formulas xˉ=Mym,yˉ=Mxm\bar{x} = \frac{M_y}{m}, \bar{y} = \frac{M_x}{m}
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Examples
Lessons
  1. Find the mass of region R, R, where R,  R, \, is the top part of the circle  x2+y2=4,\, x^{2} + y^{2} = 4, with density  p(x,y)=x+y\, p(x,y) = x + y.
    1. Finding the Moment
      Find the moment of region R, R, where R, R, is the region between  y=sin x  \, y = sin\, x \, and the x− x-axis, between  x=0  \, x = 0 \, and  x=π \, x = \pi . Suppose the density is  p(x,y)=1 \, p(x,y) = 1 .
      1. Finding the Center of Mass
        Find the center of mass of R, R, where R, R, is the quarter circle  x2+y2≤4  \, x^{2} + y^{2} \leq 4 \, in the second quadrant and has a density of  2x2+2y2 \, 2x^{2} + 2y^{2} .
        1. Find the center of mass of R, R, where R, R, is the triangle  0≤x≤2,≤y≤x, \, 0 \leq x \leq 2, \leq y \leq x, and has a density of kxy, kxy, where  k  \, k \, is a constant.
          Topic Notes
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          Notes:

          In this section, we will learn how to calculate the center of mass of a thin plate.

          Mass of the Region

          Suppose we have a region RR, with density ρ(x,y)\rho (x,y) . Then we calculate the mass by the following:

          m=∫∫Rp(x,y)dA m = \int \int_R p(x,y)dA


          Moments of the Region

          Moments are used to measure the tendency of the region about the xx-axis, and yy-axis. There are two moments: MxM_x & MyM_y. They can be computed as follows:

          Mx=∫∫Ry⋅p(x,y)dAM_x = \int \int_R y \cdot p(x,y)dA
          My=∫∫Rx⋅p(x,y)dAM_y = \int \int_R x \cdot p(x,y)dA

          Center of Mass

          The center of mass is a point where if you put a pencil underneath the plate at that point, then the plate would balance without falling.

          The x-coordinate of the point (denoted as xˉ\bar{x}), and the y-coordinate of the point (denoted as yˉ\bar{y}) is calculated as follows:

          xˉ=Mym\bar{x} = \frac{M_y}{m}
          yˉ=Mxm\bar{y} = \frac{M_x}{m}