23

What is wrong with your argument is this paragraph: If we imagine the chain as having many small segments, then the potential energy of each segment is $E_p=mgh$. As the number of small segments approaches infinity, their masses equalize because the difference in mass between any two segments goes to $0$ as the number of segments goes to infinity. ...


22

To put the accepted answer in mathematical terms, if you have a curve $y(x)$, hanging fixed at $x_0$ and $x_L$ at an height $h=y(x_0)=y(x_L)$, of total length $L$ and mass $M$ then then linear mass density is going to be $\lambda = M/L$. The length of the curve is given by the integral of the arc-length $$L=\int_{x_0}^{x_L} \sqrt{ 1+\left({dy \over dx}\right)...


4

This limit of a sum is by definition the integral of the chain curve. There's no such thing as "the integral of the chain curve". Curves don't have integrals. For an integral, you need three things: an integrand, measure function, and a set over which those are defined (in basic integrals, these are the function you're integrating, the ...


2

The argument goes the other way: The energy of the system is the sum of the energies of its parts and the interactions between them - this is why it is a state function. The energy can be changed by work or transferring heat, which is why their sum is a state function, even though separately neither of them is.


1

I would only add to @Vadim's answer that the internal energy $E=Q+W$ being the sum of the absorbed heat $Q$ and work $W$ is a state function because this is what the experiments are saying going back to at least 200 years from Count Rumford to Joule. It does not matter what $Q$ and $W$ individually are, if their sum is the same $Q+W$ value then the specimen'...


1

But the sum of mechanical and thermal power is the variation of total energy of the system, and the total energy is considered a state function. That is true because heat and work are the only two basic means for transferring energy to or from a system. So while neither heat nor work are state functions the combination of the two must necessarily equal the ...


1

I can think of an example to understand the difference between the 2 parts. A hot bar comes from a roller table to a pair of rolls of a stand in a rolling mill. During the rolling process its thickness is reduced and its width is a little increased. But the increase of the width doesn't compensate for the decrease of thickness, so that its cross sectional ...


1

If you break the velocity into two components, the normal component crosses the surface and the other one does not.


1

It doesn't matter whether the surface is internal or external, all what matters is that a traction vector is a unit surface vector that acts on the surface of a body ( particles at the surface) and not on the interior of the body. Otherwise it would a body force vector that acts on the interior of a body (example: gravity force vector).


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