Picture a seesaw and two children are placed at opposite ends, an equal distance from the fulcrum.
Centroid is the Point of Balance
Now if one child is bigger than the other than the seesaw will tip, but if the distance and mass of both children are equal than the seesaw will balance.
Why?
Because the mass is concentrated about one point, which creates equilibrium. This point is called the center of mass, or the center of gravity or the centroid.
And when we multiply the mass of a particle by its directed distance from this point is called the moment of the particle, and it measures the tendency of the pass to produce a rotation about that point.
Moment of the System about the y-axis and x-axis
So our goal is to use definite integrals to find a point on a thin disk of any shape that balances horizontally.
We will look at how a lever or seesaw balances objects, and how the fulcrum must be moved in order to balance objects of different weights and mass.
Next, we will then look at the formulas needed to calculate mass, moments, and center of mass given uniform density.
And we will see the connection they have to how we calculate Area Between Two Curves and Volume of Solids of Revolution (Disk/Washer and Shell Methods).
X-Coordinate of the Center of Mass of the Plate
Y-Coordinate of the Center of Mass of the Plate
Finally, we will look at two examples of how to calculate the Center of Mass using various integration techniques, including integration by parts and the half-angle identity
Moments and Center of Mass Video
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