In this video I go over further into applications of integrals in physics and engineering, and this time go over an introduction to moments and centers of mass. The center of mass is defined as the point at which keeps all the weights on a surface balance. The moment is defined as the multiplication of the mass with the distance to a point a given point, such as the fulcrum or origin of the axes. In this video I illustrate how the center of mass is simply the summation of the moments of all the masses divided by the total mass. This can also be interpreted as the point at which a mass equal to the total combined masses is balanced, and has a moment that equals the total combined moments. This is a very important concept to understand because it is used throughout engineering and physics so make sure to watch this introduction video! In later videos I will go over some useful examples illustrating how to apply integrals in solving for the centers of mass so stay tuned!
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Applications of Integrals: Moments and Centers of Mass: Introduction
Another application of integrals in Physics and Engineering is in determining the centers of mass.
Moments and Centers of Mass
The point P on which a thin plate of any given shape balances horizontally is called the center of mass (or center of gravity) of the plate.
We first consider the simpler situation, where two masses m1 and m2 are attached to a rod of negligible mass on opposite sides of a fulcrum and at distances d1 and d2 from the fulcrum.
The rod will balance if:
This is an experimental fact discovered by Archimedes and is called the Law of the Lever.
- Think of a lighter person balancing a heavier one on a seesaw by sitting farther away from the center.
Now suppose that the rod lies along the x-axis with m1 at x1 and m2 at x2 and the center of mass at x̅.
Thus to balance we need:
The numbers m1x1 and m2x2 are called the moments of the masses m1 and m2 (with respect to the origin).
The above equation says that the center of mass x̅ is obtained by adding the moments of the masses and dividing by the total mass m = m1 + m2.
In general, if we have a system of n particles with masses m1, m2, … , mn located at points x1, x2, … , xn on the x-axis, it can be shown that the center of mass of the system is located at:
This equation can be rewritten as:
Which says that if the total mass were considered as being concentrated at the center of mass x̅, then its moment would be the same as the moment of the system.
Now we consider a system of n particles with masses m1, m2, … , mn located at the point:
(x1, y1), (x2 , y2) … (xn , yn) in the xy-plane as shown below:
By analogy with the one-dimensional case, we define the moment of the system about the y-axis to be:
And the moment of the system about the x-axis as:
Then My measures the tendency of the system to rotate about the y-axis and Mx measures the tendency to rotate about the x-axis.
As in the one-dimensional case, the coordinates (x̅ , y̅) of the center of mass are given in terms of the moments by the formulas:
And since mx̅ = My and my̅ = Mx the center of mass (x̅ , y̅) is the point where a single particle of mass m would have the same moments as the system.
