00:01
In this problem, you have a shelf supported by wire and also by a frictionless hinge at the wall itself.
00:16
We have the angle that the wire makes with the wall and the masses of this extra mass that the shelf is supporting and the mass of the shelf itself, which we assume is uniform.
00:29
So let's draw the free by diagram for the shelf.
00:36
Now, the only question we are asked is to find the tension.
00:41
And so here is the mass, the weight of the shelf, i should say.
00:49
Here is the weight.
00:51
Technically, this would be a normal force up on the mb.
00:56
And from the third law, that you'd have a normal force down, but it's equilibrium, what's the normal force value? it's equal to the weight mb times g.
01:08
So, a lot of times you don't go through all that elaborate where you write the n and the n down and show the mbg, but that's where it comes from.
01:18
Technically, it's not a weight force directly.
01:21
The weight force x on mb, but because of the third law, it appears here also.
01:28
Now there is this hinge.
01:34
There's a force fx, which could be to the right, could be to the left.
01:37
That would have to be determined by the equations.
01:42
Have tension force in the wire.
01:46
Now, i should mention, if this angle is 60 degrees, this is 90, that makes this angle has to be 30 degrees.
01:57
So this angle here is 30 degrees.
02:01
We'll need that a little later.
02:04
And now let me mark some dimensions here.
02:07
So we have it.
02:09
Uniform shelf.
02:11
So that means its weight is in the center.
02:16
And this is 2l over 3.
02:18
Okay, so that's our free by diagram.
02:24
If you were to, you have at most three equations you could write.
02:31
And let me write f net x.
02:35
That'd be fx plus tx equals zero.
02:43
Or that'd be fx minus t cosine 30 degrees, is equal to zero.
02:50
But we don't know fx and we don't know t.
02:53
So that equation is no use to us to find t.
02:58
It'd be used to us if i don't know if there's more to this question.
03:02
If they want you to find fx afterward, then you'd use this equation once you had t.
03:09
But you can't do anything with it now to get t.
03:12
F -net y is going to be f -y minus msg ,g, minus m bg plus t sine 30 degrees is equal to zero again f y is unknown t so these equations for finding t are no use to us so what's the third equation it's a torque equation that torque equals zero so this is what we want to write next torque net is equal to zero that's our third equation but this requires a little more work this requires a little more work and i'm going to do that work up here so you can see the diagram.
03:56
First off, first off, the equation for the torque for any particular force.
04:05
The plus, if that force by itself would turn the object counterclockwise, minus the clockwise.
04:12
F is the magnitude of the force.
04:15
Our perpendicular is called the moment arm.
04:17
It's the perpendicular distance between a line of action of the force, and i'll draw those in a second, and the rotation axis, or equivalently the shortest distance from the rotation.
04:26
Axis to line of action.
04:28
Both give you the same result.
04:31
So let's draw a little table here.
04:35
Force, moment arm, sign.
04:43
In the free by diagram, we have these forces, fx, fy, msg, mbg, t.
04:55
Now, line of action.
04:59
It's an infinite line that goes through the force.
05:04
Now for these two, if you were standing in the rotation axis and you asked, for say, fx, what's the shortest walk to go from your rotation axis to the dash line going through fx? you look down, you're standing on it.
05:19
Zero.
05:21
So any force that has a line of action through the rotation axis contributes nothing.
05:26
So the same thing would be true of fy.
05:28
So they don't contribute anything in the net torque equation, not at all.
05:33
That's why we use the net torque to get t, because these were the other two unknowns.
05:38
These are, well, you could obviously do it.
05:40
If you want to have three equations, three unknowns, you could choose some other point.
05:44
You have t and fx and fy...