00:01
Here i'll start off by reviewing the concept of center of mass, and then we'll take a look at an interesting example that may seem a little counterintuitive at first.
00:12
So the center of mass, if we look at the formula for calculating it, it is like finding the weighted average position.
00:22
Okay, so if we look at the formula, it is like weighting the position based on the mass density of the object.
00:32
And like any distribution or weighted average, you want to divide by the weight function added up over the entire object.
00:42
So we can say that the center of mass is like a weighted average of position using mass's mass distribution as weight function.
01:11
Most of us are familiar with say weighting your grades.
01:17
So you take each score and you use as your distribution the weight function that your instructor has assigned to each category.
01:28
And of course, the denominator in that case has to add up to 100%.
01:32
So in this case, the denominator has to add up to the mass of the object.
01:45
Okay, so the object, i usually like to imagine the object first.
01:52
We are going to look at a two -dimensional object.
01:56
I've got it shown down here.
01:59
We're basically looking at a flat object that has the red curve bounding one side, the x and y axes on the other two, and then we're going from x equals 0 to e to q minus 1.
02:19
So y looks like 2 over 1 plus x, and x ranges from 0 to e to the cubed minus 1.
02:34
And i don't know, this object looks like it could be a rocket fin, for example.
02:41
That might be an interesting thing.
02:46
All righty, so usually it's a good idea to calculate the denominator first.
02:54
And so we're going to find the mass of this object, and we are going to assume that there is a uniform area density.
03:05
So this may be a fin cut out of cardboard or balsabud, for example.
03:11
Okay, so the mass, the denominator is what we're calling the mass, and it is the integral of sigma knot times d area over the object.
03:28
What we'll be doing, this is a good little practice to get the d area element.
03:35
We're going to imagine the curve, the flat plate broken up into rectangles.
03:42
So d area is the area of one of those.
03:46
So it's the width times the height.
03:51
And the width is dx and the height is just the y coordinate.
03:59
And of course these are going to be infinitesimally.
04:02
Narrow.
04:04
I can't really draw that, which is why my height looks like it varies.
04:09
But if we have an infinitesimal element, the height will be a fixed point quantity.
04:17
Okay, so the mass is sigma knot times y d x from x equal zero to e .d.
04:29
Q minus one.
04:34
Okay, and our y function, we have that.
04:40
So dx over 1 plus x going from 0 to eq minus 1.
04:52
The integral is fairly easy to do.
04:55
It's just the logarithm of 1 plus x.
05:02
And we can evaluate that the upper limit is 3.
05:06
The lower limit is 0.
05:09
So this is just 6 sigma knot.
05:13
And that's what we'll continue to use as our denomination.
05:17
So let's take it one coordinate at a time for the x center of mass and the y center of mass.
05:28
So we simply put x into the integral and we divide by the mass, which we know, and then we have the integral of x times 2 over 1 plus x d x between the two limits.
05:58
We can bring out the factor of 2.
06:02
That helps.
06:06
And that will simplify down.
06:09
But the integral, we're going to call that the integral.
06:13
And i'll just talk a little bit about how we can do this.
06:17
We're going to break up the pieces inside into x plus 1 minus 1 over 1 plus x.
06:29
All we've done is taking the numerator and added zero to it in kind of a weird form.
06:36
And then this can be broken up into 1 plus x over 1 plus x minus 1 over 1 plus x or 1 minus 1 over 1 plus x.
06:58
So that we can integrate.
07:00
The integral of 1 is just x and we've already done the integral of 1 over 1.
07:07
Plus x.
07:08
So that gives us, finally, we'll simplify the stuff out front...