00:01
All righty, this is kind of a long problem, so let's strap in.
00:04
If we have a weight that is hanging at this point b, and this point b is going to be the point, excuse me, that's 0 -0 -2 -3, to 0 -0 -2 root 3.
00:28
And then we have three ropes that are holding it, that are under tension.
00:34
So those ropes are, one is here, and that is the point 1 negative 3 -0, negative square to 3 -0, excuse me.
00:50
So that's this point.
00:52
The other is at 1 positive square to 3 -0, and that's here.
00:58
And then the other is back here at negative 2 -00.
01:12
Negative 2 -0 -0.
01:16
And all of these ropes are applying tension to my weight.
01:22
And what i know about my weight is that it is 500 pounds, if i remember correctly.
01:29
Yes, so this 500 -pound weight is hanging from these three cables.
01:35
So each of these cables has tension in them, right? so each of these cables represents a force vector.
01:46
So we'll call this 1, 2, and 3.
01:50
We can say force from 1, force from 2, well, let me rewind.
01:58
Force from 1 plus the force from 2, plus the force from 3.
02:01
These are all going to have to equal an upward pull of 500 pounds, right? we know that our weight is steady.
02:13
We know that our weight is static, so the sum of the forces must equal to zero.
02:18
It has a downward force of 500 pounds, right? so the sum of the three forces must counteract that downward weight as a upward vector of 500 pounds.
02:34
So this is what we know about our forces.
02:37
We also know that the tension forces are going to be parallel to the directions of these ropes.
02:47
So if we describe the rope, let me call it r, if we describe the vector of the rope, we know that the force is going to be parallel to the direction of this, the this rope.
03:04
So i'll say that r, we look at the difference in x values, and i'm going to go from the weight here to my point.
03:16
You could also solve this by describing the vector in the opposite direction, but i'm choosing to go this way, and i'm going to stay consistent throughout the whole problem.
03:27
So the change in x from my point of attachment to, from the point of the weight to point one here is going to be plus one.
03:40
The change in my y value is going to be negative square of three, and the change in my z value is going to be to the square root of three.
03:50
Right? and i can do the same thing with r2.
03:53
The difference here is that we still, the change in x value is still going to be one.
03:58
The change in y value now is positive square of 3, and the change in z value is 2 squared of 3.
04:05
And lastly, the third rope here, change in my x value is negative 2 now.
04:12
My y value doesn't change, and my z value is going to be still positive 2 squared of 3.
04:20
Let me clean that up a little bit.
04:22
So now that we know each of these vectors, which describe the direction of the rope, we also know that the forces are going to be parallel to these vectors that describe the direction of the rope.
04:40
That is to say that force 1 is going to equal some quantity, which i'll call f1, times the vector r1.
04:53
So this is a vector.
04:55
F1 is a scalar.
04:56
Because the big force, big f1 is parallel to r1, there should be a scalar multiple such that big f1 equals little f1 times r1.
05:09
And this is the same is true for all of these forces.
05:14
So big f2 equals some little f2 times r2, big f3 equals some little f3 times r2.
05:24
And what we want to do is find these little f little fs one two and three and that'll allow us to to find our forces f1 f2 and f3 so let's rewrite the expression that we had up here but now use our vectors for each of our ropes so we can say now that f1 times r1 plus f2 times r2 plus f2 times r2 plus f3 times r3 is going to equal the vector 0500.
06:05
And to expand that out, we would have f1 times the vector 1 negative 3, 2 root 3, negative square of 3, excuse me, 2 score of 3, plus f2 times the vector 1 positive square of 3, 2 score of 3.
06:34
Plus the vector f3 plus f3 times the vector negative 2 right yeah negative 2 0 2 2 2 2 2 2 2 2 3 that's going to equal the vector 0 500 so this is a long expression that on its own just this expression isn't very helpful but we we know that we can break this up into each of the individual components so the x component means that f1 times 1 plus f2 times 1 plus f3 times negative 2 equals 0.
07:21
In our y component, we know that f1 times negative square of 3 plus f2 times the positive square of 3 plus f3 times 0 is going to equal 0.
07:41
And in the z direction, we know that f1 times 2 square of 3 plus f2 times 2 square of 3 plus f3 times 2 squared of 3, it's going to have to equal 500.
08:04
So now we have a system of linear equations, we have three equations and three variables.
08:12
So we know that we're going to be able to solve this.
08:15
Let's first look at our y equation.
08:19
So in our y equation, we have negative square of 3, f1, plus square of 3, f2.
08:29
0 times 0 times f3 goes to 0, of course.
08:35
So this is going to equal 0.
08:36
And if we divide both sides by a square of 3, what we get is negative f1 plus f2 equals 0, which means.
08:47
Means that f1 equals f1 equals f2 cool so that's going to simplify some of these equations a bit right f1 equals f2 that's great so let's plug this into our x equation right our x equation right our x equation was that f1 1 times f1 plus 1 times f2 minus 2 times f3 equals 0 so if we plug f3 f2 in for f1 or f1 in for f2, let's say, what we get is f1 plus f1 minus 2f3 equals 0.
09:35
And we can simplify this to 2f1 minus 2f3 equals 0.
09:42
Again, divide by 2, we see that f1 minus f3 equals 0, which means that f1, equals f3 right if f1 equals f3 and f2 equals f3 that means for f1 equals f2 that means f2 equals f3 so all of our forces are equal to one another or all of excuse me all of the the multiple scalar multiplier is f are equal to one another f1 equals f2 equals f3 perfect this actually simplifies it a lot because now when you look at our z equation let me copy our z equation down here.
10:37
It's f1 times 2 squared of 3 plus f2 times 2 squared of 3 plus f3 plus f3 times 2 squared of 3 equals 500.
10:53
What we get when we notice that f1, f2, and f3 all equal each other, let's call this just f so i don't have to use these subscripts anymore...