Let us say we have a charged non-conducting sphere having a spherically symmetrical charge density $\rho (r)$ that decreases as $r$
Ignoring all other properties, If we positon ourselves in the center of the sphere, the medium will appear isotropic because of the spherical symmetry charge distr...
at that point, one expands the binomial and interchanges the order of summation to get $$\sum_{n=0}^\infty \sum_{k=0}^n \binom{n}{k}\frac{y^k (-x^2)^{n-k}}{(2n+1)!} =\sum_{k=0}^\infty \sum_{n=k}^\infty \binom{n}{k}\frac{y^k (-x^2)^{n-k}}{(2n+1)!}=\sum_{k=0}^\infty \sum_{n=0}^\infty \binom{n+k}{k}\frac{y^k (-x^2)^{n}}{(2n+2k+1)!}$$
I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}_1F_1}
\ff(b+k;b;z)\textrm{, for }k\in\mathbb{N}.$$
Numerical tests suggest that this is always a polynomial of degree $k$...
A pity that $x^k (-x^{-1} D_x)^k\neq x(-x^{-1} D_x)x(-x^{-1} D_x)\cdots x (-x^{-1} D_x)$, or one could do something cute with that
Ideally, I'd be able to write down a cute operator exponential for $\sum_{k=0}^\infty x^k(-x^{-1}D_x)^k (z^k/k!)$
another cute identity, and one I don't have a good explanation for (besides hurr mathematica says so): $\int_0^\infty x^{-n} j_n(x)\,dx=\pi/(2^{n+1} n!)$
OH-ve Can't be reduced....
It may only Oxidized by giving electron at anode. while H+'ve Reduced instead of OH-ve.
Secondly Reducing property of H2O Is Greater than That of H+'ve ion,Moreover It is greater in number than other ions like H+'ve Ions etc.....So It will be preferentially discharged A...
I have an inquiry about a simple classical mechanics problem. Let's see if I can paste this picture
Here I am, it's problem 29. I've done everything right, except determining the $k$ spring constant.
My reasoning was: if the spring can be compressed by $L1 = 2.0cm$, then during that process it accumulates $\frac{1}{2} k L1^2$ energy. If this energy is drawn from force $F1$ which acts along a distance $L1$, then we have the equation
$\frac{1}{2} k L1^2 = F1 L1$.
But as it turns out, something must be wrong in the reasoning. But what exactly?
Someone is going to suggest to rely on the $F = kx$ equation, but then the question still applies. Why doesn't the reasoning above lead to the same result?
@Acsor the equation for the energy $U = \tfrac12 kx^2$ assumes that the applied force is equal to the spring force i.e. the applied force is $F = kx$, not a constant force.
The equation comes from the expression $U = \int F dx$. Since $F = kx$ we get $\int kx dx$ which is $\tfrac12 kx^2$.
So you can't equate the energy to the work done by a constant force.
Probably I have done the wrong assumption of equating the spring potential energy with the work from the constan force
What does "a spring that can be compressed 2.0 cm by a force of 270 N" mean then? 270N at the end? If it is, it sounded hugely ambigous to me :/
Well it looks it all boils down to how I interpreted the statement, which is "The spring can be held compressed at L1 by an application of a constant force F1". Basic Newton 2nd law yields the answer
what theories of physics are there that don't rely on universal mechanism (the philosophy that everything can be reduced to the motion and collision of matter)? I can think of the "physics" of Aristotle and Native Americans, but they are very limited in what they can predict and are sometimes just wrong. What other theories qualify that do better at predicting the world? Quantum mechanics might fit but I don't have a strong enough understanding of it to say that definitively.
Really one of the main things we learned in the past 100 years is that phrases like “the motion and collision of matter” are too vague to mean anything.
@ACuriousMind If modern physics don't rely on the motion and collision of matter then here is a follow up question. What theories of physics don't rely on fields (I believe QM and GR use fields)?
@roobee "Field" is really just a name for a function that assigns some value to each point in spacetime. Since physics uses math, you won't really find any theories without that.
Wouldn't that only be a scalar field...do you consider multi-valued functions to be functions as well or are you on the side of the fence that functions should be single valued?
And would this count as a "field"? A theory of physics using sound as the building blocks. Upon hearing sound A, after waiting a period of time equal to how long it would take sound B to start and finish, sound C will occur 50% of the time. with inputs being A and outputs being (1, B, .5, C)?
@roobee That's another question that's a bit too vague to give a useful answer. E.g. a point in spacetime may be represented by some tuple of real coordinates in some coordinate chart, but it "really" is just a point in a manifold, which can be represented by many different tuples depending on the chosen chart.
@roobee Note that I said that a field is a function on space(time), so the inputs are always generalized "locations". You can of course decide to call every function a field, but what would be the point of that?
I had originally thought the below two theories had some fundamental difference. but upon contemplation I think that there only difference is that life experience shows one to work while the other doesn't. Is this accurate?
if there is a new material and you wish to judge it's toughness you could 1) Use theory A which says to determine how much it is imbued with earth via techniques like licking and smelling it. the more earth it is imbued with the tougher it is 2) Use structural analysis according to Newtonian Physics.
so the two theories above use fields, so that is not what differentiates them fundamentally. I don't know what else would fundamentally differentiate them, so is the answer there is no fundamental difference? Only that one is right and the other is wrong learned by experience?
so an object imbued with earth could emanate an earth field within a short distance from it. When it interacts with your nose field or tongue field it would give a characteristic taste or smell. Or is the answer that the above is not conventially viewed as a field, and that the absence of using fields to describe things in theory A what fundamentally differentiates it from modern physics?
like i'm trying to learn about theories that seem to have very different foundations from modern physics, like theory A. And I know that they will not be as useful/accurate, otherwise they would be popular. But I do want to know what's the best they have to offer in terms of predictions, to see just how much worse the best of them are. So I originally asked about theories that don't use universal mechanism or fields, but apparently those words were vague or inaccurate.
@enumaris I originally asked what theories don't use field. Curious said basically all theories use them. So I tried finding a different word that would get people mentioning the types of theories I would be interested in, and couldn't think of any.
If theory A indeed doesn't use a field, that's fine. I would then ask again what other theories don't use fields, especially what are the most successfull theories that don't use fields.
That's my difficulty. I'm not sure if I know specifically what differences I'm looking for. In general, I'm looking for theories that make me think "Oh gee. That seems like a fundamentally different view on physics.". I can think of examples that make me think that, and I've stated them. But I don't know what specific property they have that made me think that.
that synthetic geometry sounds potentially interesting
well that's why i was interested in the most successful (in terms of predicting) of the weird theories, even knowing they wouldn't be as succesfull as modern physics.
One thing that has been irrationally frustrating me on SE is how many people seem to misunderstand Archimedes' Principle and assert that it must be true even with no fluid beneath the object, as if buoyancy is some magic force on it's own instead of just the net force due to pressure gradients.
You can show that Archimedes' principle implies the following: If you have an ice cube floating in a glass of water, and it melts into the same kind of water (i.e. no difference in salinity etc) then the water level won't change.
What I've wanted is a microscopic POV, i.e. why does a microscopic change in the volume of ice not result in a change of fluid volume
i guess maybe the way to look at it is that, if you split the ice in two, nothing should change either (as long as the system stays in equillibrium throughout)
@Semiclassical I haven't done the math... but couldn't the net density change when the system approaches equilibrium, even if it's all fresh water/ice?
Basically it's Euclidian geometry, but in addition to notions of congruences and perpendicularity, you also have notions of simultaneity, temporal congruence, congruence of other quantities, etc
@Semiclassical I'm thinking it could, because the bonds change orientation pretty significantly around the 4°C mark, so if the phase change resulted in all the mixture going above 4 degrees, you should have the same mass taking up less space
@JMac so would that mean: If you leave pure ice in pure water in a room temp above 0, wait around a while, and then come back once it's all melted, the fluid level won't have changed
well actually i was wondering whether weird theories could help in areas that modern physics has weaknesses in (namely actually being able to compute a solution like for chaos theory or weather prediction), at the expense of generality. but i didn't put too much hope in that idea
physics.stackexchange.com/questions/499354/… I love when people go around saying you're wrong but can't even remotely support it and then just brush it off because it's "boring"...
@AaronStevens Yeah, the topic of archimedes' principle and no fluid below is a bit of a raw spot for me on Stack Exchange. engineering.stackexchange.com/questions/19159/… The consensus on engineering SE is that you can apply it without thought in every situation, which I find pretty troubling, given that its so trivial to see that it comes directly from the fluid pressure
It's such a common misconception; but I really cant wrap my head around why people don't just do the analysis themselves. It's so easy
@JMac Yeah that is odd. If there is nothing below the object to push it upwards then where is that buoyant force coming from? There is nothing to cancel the pressure acting on the top. Certainly the ground isn't going to help you any more than it would if you tried to pick something up off of the ground in air.
@AaronStevens Yeah, exactly. There seems to be some weird popular idea that Archimedes' principle is just some isolated property related to density and volume; instead of just net force and pressure gradients put into a more digestible package
@JMac Perhaps its because it goes against normal experience. You typically don't have smooth objects and "pool surfaces" so that this effect could be observed. Once even the smallest part of the bottom edge allows for fluid to move underneath that edge then you get an upward force
@JMac But in "ideal land" this is totally reasonable and you cannot bring in an upward force from the fluid
@JMac Haha I just found it odd that they wanted to listen to a line of reasoning that others and the duplicate answer are saying is wrong. It reminded me of "trisectors": people who are still trying to find a way to trisect angles with just a straight edge and a compass even though it has been proven to be impossible
@AaronStevens The problem is that the proof there is much more complex than the problem statement. Galois theory is very much outside the realm of intuition for most people, compass and straightedge are not.
@JMac I was feeling somewhat sympathetic for the most part. The main guy they were interviewing seemed really nice. And some of the people they were showing actually had very inquisitive and scientific minds (up until the point of completely disregarding their own evidence). But I got really frustrated near the end when they all praised a child (pre-teen maybe?) for being at the flat Earth convention.
Like believe what you want, but don't screw other people up with it
@SirCumference That is what I am saying. From a mathematical point of view, there really isn't any correct set of rules. Only correct conclusions drawn from a particular choice of rules
Let us say we have a charged non-conducting sphere having a spherically symmetrical charge density $\rho (r)$ that decreases as $r$
Ignoring all other properties, If we positon ourselves in the center of the sphere, the medium will appear isotropic because of the spherical symmetry charge distr...
I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}_1F_1}
\ff(b+k;b;z)\textrm{, for }k\in\mathbb{N}.$$
Numerical tests suggest that this is always a polynomial of degree $k$...
