00:01
This question, basically, we have a beam, right? on this beam, are acting some forces, right? actually, it's a loading, right, which is distributed on the beam, as soon in the diagram.
00:12
So you ask you to find the equivalent resulting force and is location from point a, right? so first, let's find the equivalent resulting force.
00:22
And that force, of course, is given by the total force, right? so that force i'm going to call it a error, and this is simply an integration of the distributed force density times dx from zero to a.
00:36
I will say the, i will state of a coordinate system where is x going this way and arranges at a point a, right? so that's what we'll find.
00:45
And of course, this has two parts.
00:48
One is from zero to a over two.
00:51
And it has distribution, which is given by this, right? and there's another one that's given by this.
00:57
I will call the first part, i will call it w1, okay, is a function of x, dx, and then plus zero, from half air to air, and w2x and dx, right? so the distribution, w1, let me look at it, w1x, which obviously goes from w0 to zero in a period of half air.
01:24
So obviously that is actually given by w a over, air over two times half a a minus x, right? so that when x equals zero, you get w0, but when x equals half a you get zero, right? so while similarly, the distribution in this section, you have w2x, which is obviously similar in a similar way to w0 over a over 2.
01:56
Now in this case, what's happening is that you need to get ao, i think it's a0 minus x, right? so when x equals half a year, you get w0, and when x equals a equals a equals zero.
02:10
So now we substitute these two expressions into the expression for this.
02:15
If all we find this, now we come back to this part, and that will become 0 a0 over 2, w1x is just dx times l half times x and of course times the distribution w0 over l over 2 and similarly for this from half l to l dx w2 which is a l minus x times w0 over 2 right so we need to do the integration next step while the integration for the first part is simply given w0 over 2 the integration you do it, you'll find that there are two parts a over two times air over two, right, and a minus.
03:00
So that in the second part, you get half a a over two squared, right? so that's the first integration.
03:07
Second way, you will get a very similar result, which is w0 over two.
03:11
And the first part you get a times half a and minus.
03:15
In a similar way, you get extra a squared and minus half a squared and divided by two.
03:23
So and you put things in, you would get, for the first part, you get w0 over l over 2 times this.
03:33
If you do it, you find this is a0 over 2 squared over 2...