0 - 2020-04-01

> The external force F cannot produce a moment either, this is certainly true when we assume a continuous distribution of lines of force; such a distribution can be always considered as parallel in any infinitesimal region (the field of gravity is a simple example for what is meant)

Is that what you want to ask about?

Knight

@JohnRennie Yss

Especially that “lines of force” part

John Rennie

Consider the Earth's gravitational field.

Here we also have lines of force, and when you do basic physics problems you draw these lines of force as parallel lines pointing down. Yes?

Knight

5:20 AM

Yes

John Rennie

But the lines of force aren't parallel. They all point towards the centre of the Earth, so if you zoom out to view at a large scale you see radial lines of force all pointing towards the centre of the Earth.

But when you zoom in to say human scales than the divergence of the lines is so small they appear parallel to us.

And this is what Sommerfeld is saying about lines of force in a liquid.

Knight

Yes

John Rennie

When you look at a large scale the lines of force diverge and/or converge.

But in Sommerfeld's treatment we are assuming infinitesimal cubes in the liquid, and on such a small scale the divergence of the lines is too small to be detected so the lines just look parallel.

Knight

@JohnRennie Yes. So, how that parallelism has to do something with torque or moment?

John Rennie

Let me dig out Sommerfeld's book again.

Knight

5:26 AM

Yes

Sir

John Rennie

Knight

Yes

John Rennie

If you have parallel force lines than they cannot produce a moment because for any point there are the same number of lines either side of the point.

Knight

@JohnRennie But the volume element is not infinite. If we take the right most point then?

John Rennie

So if you calculate $\sum Fr$ for the point the positive and negative values of $r$ cancel and the net moment is zero.

@Knight you're going to be calculating the torque about the centre of the cube.

Knight

5:33 AM

@JohnRennie Sir, I’m sorry but I think I’m missing something, can that cube rotate only about the axis passing though it’s center?

John Rennie

If you take an isolated object it can only rotate about its centre of mass.

Knight

Okay

John Rennie

The cube isn't isolated, but we assume its surroundings are symmetric.

Knight

Symmetry = isolation? I think that requires too too much explanation, let’s just accept it

Okay that parallel lines of force doesn’t permit any rotation because we got equal number of lines on both the sides

John Rennie

5:51 AM

I don't think Sommerfeld is describing this very well. What he's basically saying is that the curl of the vector field is zero.

And what he's leading up to is saying that any vector field that has zero curl can be written as the gradient of a scalar potential.

I would guess you already know this as it's a very standard result e.g. in gravity and EM.

Knight

Yes sir!

John Rennie

It's also not true that a fluid flow always has zero curl unless you impose some extra conditions to make this so.

Knight

There are two possibilities: either he was really obtuse in that explanation or translation messed up something

John Rennie

For example water flowing in a river has a non-zero curl and it can and does make objects in the flow rotate.

Knight

Yes

I see it every day. I think tap water can be assumed to have zero curl

John Rennie

5:58 AM

What are you trying to achieve by reading Sommerfeld? I'm not sure it would be my first choice of physics book ...

Knight

@JohnRennie Hahahahahaahha, He is complete in himself (just like you) that’s why I like him. But I think my reading of him causes you so much trouble :-)

Explaining someone else’s work is very hard (and even to someone like me)

God bless you, God bless your patience !

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