00:01
Okay, we have a uniform bar of mass 0 .0, 1, 2 kilograms, that's 30 centimeters long.
00:10
Pivots without friction.
00:14
Gravitational force is in negative y, and it's in a uniform magnetic field.
00:18
There's a current passing through the bar, and we want to know what's the current needed to keep this thing in rotational equilibrium.
00:26
So let's draw it real quick.
00:27
Got our x, our y and x axes here.
00:36
Our field is into the page.
00:41
Here is our bar.
00:44
Here's our angle theta.
00:47
And then we got point a and point b.
00:55
Okay, and they even give us the torque due to the magnetic field or the magnetic force, which we could derive easily, but they gave it to us, so that's nice.
01:06
So first off, rotational equilibrium.
01:09
Rotational equilibrium means the sum of the torques equals zero, right? so it's not rotating.
01:15
It's not accelerating, it's just rotating.
01:17
So in this case, since it's not moving to start with, it means it's not rotating at all.
01:24
So this is the condition that means rotational equilibrium.
01:28
Some of the torque equals zero.
01:30
Well, what torques do we have? so we have, well let's just look at the forces acting on our bar.
01:38
Here's our bar.
01:39
What forces do we have of gravitational force, pulling it down? we have some sort of force here at the pivot point, right? that's holding the bar in place.
01:54
We don't, like maybe we know what direction it is.
01:57
We'd have to probably look at the, all the forces and we could figure out the exact direction.
02:06
For simplicity, i'm just going to guess it's in this direction.
02:09
So some sort of contact force, right? that's keeping the bar connected to whatever it's pivoting about.
02:19
And then we also have a magnetic force, which we don't know the direction, but we can find out from rotational equilibrium that the magnetic force needs to be in this direction.
02:34
Yeah, so excuse me the rotation point is this blue dot which means that the force holding the bar in place does not cause, it does not produce a torque, right? the torque is equal to a force acting through a distance, or at a distance i guess.
02:59
So here's our, and that distance is from the rotation axis.
03:03
If the force is acting through the rotation axis, then the distance is zero, and it produces no torque.
03:10
But these other two forces act at a distance of l over 2.
03:14
Well, the magnetic force doesn't, but i've just drawn it that way.
03:18
The magnetic force is acting all along the bar.
03:22
We could place it at a single point.
03:26
It probably wouldn't be this point i've drawn, but just for simplicity, i've drawn it there.
03:32
But they act through some distance.
03:36
Okay, so to get the magnetic torque, you actually have to integrate.
03:39
And they've done that for us.
03:41
So let's just say the sum of the torques equal zero.
03:46
So what torque do we have? well, i have the magnetic torque.
03:49
One half, i, b, l squared.
03:55
And let's just guess it's going to be the direction that i've drawn, the magnetic force, which is going to cause a torque this way.
04:02
Right? we'll call that positive, which means gravity is going to produce a negative torque.
04:08
The gravitational force is mg, and the distance from the rotation axis is l -halfs, the weight of an object.
04:17
You can always draw it acting through the center of mass.
04:21
So the weight is distributed along the bar, right? but since it's equally distributed, you can, in your free body diagram and in your calculations, you can pretend that all of the mass is constant, at the center of mass and the torque will be acting through that point.
04:41
The force, yeah, the gravitational force will be acting through that point.
04:47
Okay, so minus mgl and then we're at an angle here, right? so we need some sort of angle because this force is not perpendicular to the distance.
05:02
So if we were to draw kind of what i'm talking about...