00:01
I'm going to draw a diagram of what's going on here.
00:04
We've got a rod.
00:10
There are eight kips.
00:19
That's kilopounds, 8 ,000 pounds here.
00:27
We have a force at b, so i'm going to put a, b, c.
00:35
This force is in this direction where we have a point d up here, which is a distance of 2.
01:06
So i'm going to call this d, and i'm going to say that it equals 2 feet.
01:17
All right.
01:19
Theta is here.
01:23
I'm going to call this l, where l is 1 .5 feet.
01:33
And so i'm going to call this 2l.
01:39
All right, i'm going to set up my coordinate system so that x and y are here.
01:51
So, y at a, is positive, 3l, and sign is opposite over hypotenuse, that would be sine theta.
02:15
The y value at b, actually i'm going to need the y value and the x value at b.
02:27
The y value at b is going to be 2l, sine theta, and the x value at b is going to be 2l, sine theta, 2l, cosine theta.
02:46
So i can take the derivatives.
02:48
Delta, y, a, is 3l cosine theta.
02:58
Delta yb is 2l, sine theta, and x, x was actually negative.
03:14
It was negative 2l ,0.
03:17
X .3divative of the cosine is the opposite of the sign.
03:22
So delta xb is going to be 2l sine theta.
03:31
And i forgot to write my delta theta's.
03:34
Delta theta.
03:37
Delta theta.
03:38
All right.
03:40
Now i need to figure out the force, which i'm going to call, i'm going to call this, p here.
03:54
So i'm going to call this q.
03:58
So q in the x direction is going to be q times something.
04:06
So let's try to figure out that something.
04:08
If i make a horizontal line here, and if that's two, so this is 90, and this would be 90 minus theta.
04:56
I'm thinking about the law of cosines.
05:01
And if i had the law of cosines, i could get this length here.
05:08
And if i had that length, i could use the law of signs to get this angle.
05:19
I'm just trying to decide if there is an easier way.
05:28
If i use the pythagorean theorem, i could figure out this length, and then i could figure out the upper length.
06:00
I could figure out that length.
06:03
Oh, man.
06:04
So the pythagorean theorem method is going to be even a little longer.
06:08
All right.
06:08
So let's just use the law of cosine.
06:10
I'm going to call this distance up here d2.
06:18
So, d2 squared equals 2l squared plus d squared plus d squared minus 2l d squared, cosine theta.
06:54
And i'm going to do this.
06:56
I'm going to write that.
06:58
Okay, so i know d2.
07:08
Now, let me just double check here.
07:13
I just want to look at the law of cosines.
07:15
Okay, so i already had a two here.
07:19
So it's actually two twos.
07:21
It's c squared equals a squared plus b squared minus 2ab cosine c.
07:29
So i need another 2.
07:33
A, b, cosine c.
07:38
All right.
07:40
So now, the sign of that green angle, which i'm going to call phi, over d is going to equal the sign of theta over d sub 2.
08:23
And so now, i can get phi and phi minus 90 minus theta which would be the same phi minus oh 90 minus theta there we go that's what i was thinking so that would be the same as phi plus theta minus 90 that is this little angle in here and i'm running out of colors, i'm going to use red.
09:15
Phi, i can't write it that small.
09:22
Call that gamma.
09:25
Gamma is phi plus theta minus 90.
09:29
Phi plus theta minus 90.
09:34
Okay.
09:36
So q in the x direction is just q, cosine gamma.
09:46
And q in the y direction is just q.
09:53
Sine gamma and they are both positive.
09:58
It's going positive in the x direction and positive in the y direction.
10:01
All right.
10:03
So, virtual momentum equation.
10:10
The change in potential energy is zero, which is p, which is negative, times delta y.
10:23
Della y is positive.
10:27
3l cosine theta delta theta.
10:43
Let's do qy, which is q, xan gamma times delta y, 2l sine theta delta theta.
11:12
And then we've got qx, which is q cosine gamma, times delta x, which is 2l sine theta delta theta.
11:41
All right.
11:45
The delta theta's cancel out.
11:51
Do the ls cancel out? yes, they do.
11:54
L, l, l.
11:59
Okay.
12:01
So now i have p cosine theta.
12:11
Now there's a three there also.
12:13
3 p cosine theta equals q times two.
12:29
There's a two in both of those.
12:30
2q, sine gamma plus cosine gamma.
12:47
Sign gamma plus cosine gamma.
12:50
Now, i'm just making sure that i didn't make any mistakes.
12:58
And i definitely did because i'm missing some thetas here.
13:03
So let's erase that.
13:07
I'm definitely missing something here.
13:09
It's 2q, sine gamma, sine theta, plus 2q, to cosine gamma, sine theta.
13:28
But if i'm going to write that, i might as well factor out the sine theta.
13:43
But why do i have a sine theta? did i put the wrong thing in? okay, qx is q cosine.
13:51
So this is the x one over here.
13:54
So delta x is going to be, oh, this is where i missed it.
14:03
Derivative of the sign is the cosine right here.
14:08
Noticing that there was a problem.
14:12
Okay, so the y value, which would be this one, needs to have a cosine.
14:24
And so now it's sine gamma, erase, cosine theta, and then cosine gamma, sine theta.
14:39
Okay, this looks better.
14:44
But i did want to double check to make sure that i have my negatives and positives correctly.
14:49
Correct.
14:50
So the y value at point a is positive.
14:53
That's good.
14:54
The y value at point b is also positive and the x value at point b is negative.
14:59
Good.
14:59
When you take the derivatives of that, you get this.
15:02
All right, p is negative, and i believe that i indicated p to be negative.
15:06
Yes.
15:08
Qx and qy are both positive, and i indicated them as positive.
15:15
Now, utilizing a trigonometric identity, sine alpha cosine theta plus cosine alpha sine theta is just the sign of, that's not alpha, that's gamma plus theta.
15:42
So 3p cosine theta equals 2q sine.
15:51
Gamma plus theta.
15:54
But gamma is phi plus theta minus 90.
16:01
Sign of gamma is phi plus theta minus 90.
16:10
Phi plus theta minus 90.
16:21
Okay.
16:22
So that is the sign of five.
16:22
So that is the sign of phi phi plus 2 theta minus 90.
16:33
Just write everything out again.
16:41
Okay, but now we have up here that the sign of phi is d over d2 sine theta.
16:59
Let's move that down here.
17:07
D over d2 sine theta...