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Dimensions of ##angle\cdot length^2## are correct. There are other factors of ##x_{\pm}^{-1},y_{\pm}^{-1},z_{\pm}^{-1},L^{-1}## and these lead to dimensions of ##[k]charge^2length^{-1}## when all is said and done. I just omitted those prefactors to focus on the integral I have not yet been able...

It is a schoolwork-type problem, but not schoolwork. I was investigating electrostatic properties of an analytically soluble charge distribution when I encountered these integrals.

Sorry that was a typo. Should be ##u(x^-,x^+)##.

Summary:: Could someone please evaluate this double integral over rectangular bounds? Answer only is just fine. [Mentor Note -- thread moved from the technical math forums, so no Homework template is shown] Hi, I'm trying to find the answer to the following integral over the rectangle...

Thank you!

Hey all, So this time I have a different kind of question - namely, "what is this called?" I recall hearing/reading this in at least two places, one of which was YouTube. The idea is the following: A RNG picks an integer uniformly from 1 to N. It picks 4. What is the expected value of N? I'm...

No worries - It's a very tricky integral even (or perhaps especially?) without the Weierstrass substitution. :) I have noticed that integrals arising from a physical scenario (rather than simply composed for mathematics) tend to have "nicer" answers. Without giving away too much, the integrals...

FYI I solved my problem on my own. While my solution is perfectly valid, it uses a Weierstrass substitution which is actually not needed for the integral (although it is quite surprising that it is not necessary!). As luck would have it, the integral can be subdued without the sneaky...

Yes: The formula you quote is just the inverse of the identity ##\tan(A+B)=(\tan A+\tan B)/(1-\tan A\tan B)##. While this seems an extremely obvious way to proceed, the expressions I get just become messier and do not begin to resemble the other side of the tentative equation I wish to prove...

Hello everyone, I have a maths question (for a change). In summary, I would like to reconcile the following two integrals: Integral A: https://www.wolframalpha.com/input/?i=integrate+(a^2tan^2theta)/(a-b+cos+theta)+dtheta \int\frac{x^2\,dx}{\sqrt{x^2+a^2}(\sqrt{x^2+a^2}-b)} =x...

mfb, I've edited the quote below to fill in the other pairs from my spreadsheet: Also, if we no longer restrict to ##m,n<100##, then in fact there is an easy way to construct infinitely many of these pairs. Since ##3^2+4^2=5^2## we have ##\frac{1}{20^2}+\frac{1}{15^2}=\frac{1}{12^2}##...

Wow, good on you guys for finding some working counterexamples! Any chance you can provide the complete list of matches you found (including the trivial multiples)? I would like to see if the domain of validity of my partial proof was right - that is, any match must fall within one of the cases...

Hello all, This is a problem of a different flavour from my usual shenanigans. I'm looking at a function $$f(m,n)=\frac{m^2n^2}{(m+n)(m-n)}$$ and am trying to determine if there are any two pairs of values ##(m_1,n_1)## and ##(m_2,n_2)## which evaluate to the same result. Assume that...

A reflectionless potential?! See, it's posts like yours that make me love PhysicsForums. I've studied a lot of interesting things that branch off of basic physics ideas - the Capstan equation, Maxwell's fish-eye lens, and this problem of late - and it's little gems like what you've just shared...

Hi Jason and Hutch, Thank you both for your in-depth responses. My main problem with this approach was simply that I was expecting the derivation to break when we did not assume a uniform medium. This was true in my freshman mechanics derivation for transverse waves on a string. However, I...

Hi hutchphd, Your effective model is correct. I agree that the energy stored in waves is conserved. However, there is a non-trivial division of how much energy is in the transmitted wave versus how much is in the reflected wave. This makes the effective transmission and reflection coefficients...

Hi A.T., I'm sorry but I don't think this addresses my question. I am familiar with complex refractive indices as they are used for total internal reflection. What I am dealing with is a material with a real, but non-constant refractive index. My assumptions are as follows: 1. We can treat the...

Hello all, Apologies in advance for the text-wall; this is a rather involved question. I am trying to compute the effective transmission coefficient for a medium of non-uniform refractive index. For simplicity I am assuming the slab has thickness ##d##, that ##n(0)=1##, and that ##n(d)=n##...

Okay everyone, I think I was able to come up with a solution myself. (I haven't taken a look at any papers other than the one I posted about.) Here's my approach: Assume that force accelerating the Born rigid rod (that is what I meant when I initially said "uniform") is constant in time and...

Okay, I am within tasting distance of the answer I'm looking for. I found this question posted to PSE: https://physics.stackexchange.com/questions/175684/lorentz-contraction-in-continuously-accelerating-rod This is very similar to the question I am asking here. The last element I need comes...

Thanks everybody for your responses. I find Ibix's most helpful: You are making the same assumptions I did. I have heard the terms "Rindler coordinates" and "Rindler observers" before, in the context of the Unruh effect; looking at the Wikipedia page for Rindler coordinates seems to elucidate...

I am trying to push the boundaries of special relativity with a self-imposed challenge problem. A common derivation of relativistic kinetic energy involves an object to which a constant force is applied. I want to consider a similar scenario, but instead of a point object we now have a uniform...

Sorry to bump this thread, but I was able to apply the ideas discussed for the toppling rod (in the event of a perfectly inelastic collision) back to the hemisphere and would like to confirm my answer for reasons that will become obvious. Does this appear correct? If so, it's completely...

In the https://www.physicsforums.com/threads/proving-a-known-limit-of-a-recursive-sequence.976487/#post-6224413 I outline this exact method. Regrouping products and canceling, I got precisely one-half times the inverse of the Wallis product for ##\pi/2##. That is, ##\omega=\omega_0/\pi##. And...

Actually, it is the last part of the original (simplified) problem. You see, if one demands that the collision is not perfectly inelastic, but just inelastic enough that the lowest point is stationary, then this arises as the recurrence relation which converges on the final linear and angular...

All right, I think I have almost everything I need to solve my original problem. My last question is purely a mathematical one: Given the sequence defined recursively by a_{n+1}=\frac{1}{4}\left(1+\frac{a_n}{b_n}\right)^2b_n...

So you're saying that to minimize ##K## one needs to have ##y=r##. Okay. Now, using both our methods (assuming I got yours more or less correct) we both find ##v=\frac{1}{4}R\omega_0##. In your case, this comports with ##\omega=\frac{1}{4}\omega_0## since ##v=R\omega##. In my case, I obtain...

Okay, MSA got me nowhere—literally, to the location of nothing. I’m getting ##v=\omega=0##, which not only seems unphysical, but also suggests I made an arithmetic error since ##v(n+1)>v(n)##. I’d state my recurrence relations for the record, but it’s 2:00 AM and I need some sleep. See you in...

The uniform rod of length ##2r## has moment of inertia ##I_0=\frac{1}{3}Mr^2## and kinetic energy ##\frac{1}{6}Mr^2\omega_0^2## at the moment of collision. We separate the rod into two rods of length ##r## and moment of inertia ##\frac{1}{12}(\frac{1}{2}M)r^2=\frac{1}{24}Mr^2## about their...

I think that you are misunderstanding some of my "quarter-sphere" argument, performing reductio ad absurdum, or both. The argument I am forming is as follows: 1. Separate the hemisphere into two quarter-spheres immediately before the collision. 2. Have the two quarter-spheres reconnect...

I do disagree that the hemisphere stops rotating altogether. What I am imagining is that the lowest point of the hemisphere has no horizontal linear velocity immediately after the collision. Here's why I believe the hemisphere and the ledge do not stick together after the collision. Immediately...

The initial orientation of the hemisphere is flat down. The thread is of infinitesimal length and connects the geometric center to the ledge. The purpose of the thread is just to ensure the flat surface of the hemisphere is pressed against the ledge when it collides - otherwise only the bottom...

I emphasise again that this is a problem I created myself to better my understanding of moments of inertia and angular momentum. My latest approach: The method of successive approximations - Separate the hemisphere into a part with positive horizontal velocity and a part with negative...

Hello, So I've been toying with the following problem and I'd really appreciate some feedback and advice. The problem statement (as unambiguous as I can make it): A solid hemisphere is placed on a horizontal ledge so that its center is directly above the edge, fixed there by a delicate thread...

I believe this is possible; as an extreme case, consider hemispheres that are a shell of negligible mass and a heavy point mass near the center of the circular face. The CM is then very close to the point mass and the hemisphere will not have traveled far before colliding with the other. Could...

Hold on a second - I think we might both be right. The energy-conservation ODE doesn't apply at r=0 because that's the very small window during which the spheres separate. Then, moments later, the rotation of each hemisphere causes them to collide inelastically, losing some kinetic energy but...

Yes, that is correct. I could be wrong (hence why I'm asking here), but the description you give seems to be valid only for the instant that the split occurs. The hemispheres should continue rotating as well as sliding immediately after they separate. Consider an orange thrown into the air...

Hi Kuruman! Thank you for your response. Yes, I forgot to mention, but angular momentum conservation is implicitly used to derive both of the differential equations above. For the first, I do not assume conservation of energy - merely that there are centrifugal, Coriolis, and Euler forces...

Hi all, The scenario I'm considering is a solid sphere (of uniform density) rotating with constant angular velocity when it abruptly splits into two hemispheres along a cut which contains the rotation axis. The hemispheres will begin to separate; if, for example, we consider the rotation to be...

Homework Statement A mass m attached to a spring of spring constant k emits sound at frequency f, detected by a collinear observer at distance r. If the mass has maximum velocity v_0, what is the total number of waves the observer detects in one period of oscillation? Homework Equations...

I wrote and solved this problem but am having serious doubts about the answer I obtained. Homework Statement Two point charges \pm q move along the z-axis with velocity \pm v. If they are at the origin when t=0, what is the electric field magnitude a distance r from the z-axis? Homework...

My reasoning is best described mathematically as follows (and I admit, rather poorly at that): Suppose we rotate the coordinate axes by \frac{\pi}{6} (i.e., counterclockwise) and label the new grid (x^*,y^*). We could technically have in this new coordinate system a function y^*=f^*(x) that...

Hey everyone, I was just curious about the nature of the cube root function f(x)=x^{1/3}. I know that its derivative is obviously \frac{1}{3}x^{-2/3} which has a discontinuity at x=0. However, in the non-mathematical sense, the graph of y=f(x) looks smooth - I don't see any angles or cusps like...

I'm preparing to start a year-long sequence of 400-level real analysis using Rudin's Principles of Mathematical Analysis 3E in my second undergrad year, and my advisor recommends I take the graduate-level sequence the following year through Real Analysis by Stein and Shakarchi. Since both of...

In a roundabout manner of speaking.

I'm not quite sure I follow...so far I believe your theory is correct through above proof (or probably reproof). I also believe that this is precisely what you requested so that others can begin to accept it. As far as I know my informal proof used only undisputed axioms for components, so...

I can't believe that on Physics Forums, of all places, two and a half pages worth of posters managed to miss this on the first page. Picking at the use of "inside" would have made sense, but he correctly stated that the hexagon was inscribed. Barking up the wrong tree, if you ask me. The...

Ahh! Thanks, that does make sense now. In my haste such a simple observation eluded me! You can see clearly why I am not cut out for contour integration. I will attack the problem with this new integrand now. Thanks again, QM

Thanks, micromass. I'm completely new to this LaTeX equation thing...I've been on Math Equation Object for so long now...

Hello, all: I have a few questions concerning a definite integral where I am meeting with limited success. The most important problem concerns: \int_{0}^{\infty}\frac{\cos{x}}{x^2+1}dx Contour integration always has been my last resort, but differentiating under the integral sign, I soon...