Moments and Centres of Gravity - Mass

Quick Refresher

$\text{Mass}=\text{Density}\cdot \text{Volume OR Area OR Point}$
$\text{Moment}=\text{Mass}\cdot (\text{Mass’s Coordinate})$

Involves the use of Integration to solve the centres of Gravity.

What is a Moment?

The turning force of a point about a specific point

Centres of Gravity / Mass on a Line:

Centre of Gravity / Mass

It’s the point on a line (or plane), for most intents and purposes, where all the weight may as well be located. It would have the same moment as the current weights in their current location.
The formula is as follows:
For masses $m_1,m_2,...,m_n$ located at $x_1,x_2,...,x_n$:

$$\bar{x}=\frac{\sum _{i=1}^n{m}_i{x}_i}{{\sum }_{i=1}^n{m}_i}$$

Basically, sum up all the moments and divide them by all the masses.

Centres of Gravity / Mass on a Plane:

Ok. So you can successfully and easily get the COGs of a line. What about a plane? Well what is a plane if nothing but 2 dimensional lines… No seriously. The formula for getting the the $\bar{x},\bar{y}$ coordinates is literally the same as the individual line.

Now just wait until you have to get the Centre of Mass of a Uniform Plane Lamina