# 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 $R$, with density $\rho (x,y)$ . Then we calculate the mass by the following:

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

Moments of the Region

Moments are used to measure the tendency of the region about the $x$-axis, and $y$-axis. There are two moments: $M_x$ & $M_y$. They can be computed as follows:

$M_x = \int \int_R y \cdot p(x,y)dA$
$M_y = \int \int_R x \cdot p(x,y)dA$

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

$\bar{x} = \frac{M_y}{m}$
$\bar{y} = \frac{M_x}{m}$

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

b)
Moments of the Region
• Tendency of region about $x$-axis, and $y$-axis
• $M_x$ & $M_y$
• Calculate using double Integrals
• An example

c)
Center of Mass
• The point that balances the plate horizontally
• Calculate by using formulas $\bar{x} = \frac{M_y}{m}, \bar{y} = \frac{M_x}{m}$

• 1.
Find the mass of region $R,$ where $R, \,$ is the top part of the circle $\, x^{2} + y^{2} = 4,$ with density $\, p(x,y) = x + y$.

• 2.
Finding the Moment
Find the moment of region $R,$ where $R,$ is the region between $\, y = sin\, x \,$ and the $x-$axis, between $\, x = 0 \,$ and $\, x = \pi$. Suppose the density is $\, p(x,y) = 1$.

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

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