Center of Mass

Finding the center of mass is essentially finding the weighted average of the component masses. Of course this can get more complicated depending on the number of masses and the amount of dimensions (1, 2 or 3).

Definition:

Here cm means “Center of Mass”. You can expand this calculation out to x y and z, just consider the position (distance) of your mass from the origin in relation to the plane you are looking at.

For example an equilateral triangle, made from three masses connected by three straws. If you balance This contraption on a piece of cardboard, The center of mass along the x axis is Length over two and the center of pass along the y axis is 1/3(sqrt(3)/2) times L. The center of mass does not have to be where mass is.

The next part of the course combines lateral and circular motion and the center of mass is important for this.

Suppose you have a distributed mass of length L. And it’s a meter stick (just for fun). It has even linear density along the meter sticks, suppose you look at a very small quantity of mass, a little dm x distance from the origin (The start of the meter stick). If you add up all of those little bits of mass, and use the ratio dm/dx = M/L, you can find the linear density.

So, what this says, is that the center of mass, at some distance from the origin is the integral of the small change in mass times the distance, divided by the total mass. Which in the end figures out to the center of mass, is at one half the length.

Now think of a triangle. An Isosceles Right Triangle. Now to find the center of mass, divide it up into many little slices. (little dx’s and dy’s). Once again we can use the handy ratio. The little mass divided by the little area is equal to the big mass over the total area. dm/da = M/A.

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