Moment and center of mass

Moment and center of mass

Lessons

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=Ryp(x,y)dAM_x = \int \int_R y \cdot p(x,y)dA
My=Rxp(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}

  • Introduction
    Moments & Center of Mass Overview:
    a)
    Mass of the Region
    • Imagine a thin plate in region with density ρ\rho
    • Calculate using double integral
    • An example

    b)
    Moments of the Region
    • Tendency of region about xx-axis, and yy-axis
    • MxM_x & MyM_y
    • Calculate using double Integrals
    • An example

    c)
    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}


  • 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.

  • 2.
    Finding the Moment
    Find the moment of region R, R, where R, R, is the region between y=sinx \, 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 .

  • 3.
    Finding the Center of Mass
    Find the center of mass of R, R, where R, R, is the quarter circle x2+y24 \, x^{2} + y^{2} \leq 4 \, in the second quadrant and has a density of 2x2+2y2 \, 2x^{2} + 2y^{2} .

  • 4.
    Find the center of mass of R, R, where R, R, is the triangle 0x2,yx, \, 0 \leq x \leq 2, \leq y \leq x, and has a density of kxy, kxy, where k \, k \, is a constant.