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}