Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

learn more… | top users | synonyms

0
votes
1answer
26 views

Locus Equation $f(r) = \frac{-h^2}{r^3}$?

For the Locus equation $$\frac{\mathrm{d^2}u }{\mathrm{d} \theta^2} + u = - \frac{1}{h^2u^2}f\left(\frac{1}{u} \right )$$ How do I find the solution for $f(r)= \frac{-h^2}{r^3}$ and sketch the ...
0
votes
1answer
16 views

I need help with variable expression

Good day! I have this coirdinate equation: $$\frac{gt^2}{2}+{v_y}t-\frac{5}{3}R=0$$ $$v=\sqrt{\frac{10}{9}*gR}$$ How i can express variable $t$ from this equation? I calculated this as quadratic ...
1
vote
5answers
36 views

find distance traveled of object that is slowing down

I apologize in advance because I feel like this problem is fairly simple but I can't seem to figure out what the formula would be. Essentially, if I had an object that were traveling 60cm/second, but ...
0
votes
1answer
29 views

Finding a composite solution to an ODE (boundary layer problem)

Given $\epsilon \frac{d^2u}{dt^2}-a(t)\frac{du}{dt}+b(t)u=0$, where $a(t)>0$, $u(0)=1$, $u(1)=1$, and assuming that the boundary layer is at $t=1$, and the boundary layer variable is ...
0
votes
2answers
21 views

Constant of motion that is not the hamiltonian

Given the lagrangian $L(x,\dot x)=\frac12 (\dot x_1^2+\dot x_2^2)-\frac k 2(x_1-x_2)^2$, we know that its hamiltonian is a constant of motion. (See here) Is there another function $f$, not of the ...
2
votes
1answer
32 views

Show that for small oscillations, the period of the resulting motion is $\frac{20 \pi}{7} \sqrt{\frac{a}{g}}$

Apologies if math stackexchange is the wrong place to post this, I'm stuck on a differential equations/physics problem in a DE textbook. An object of mass $m$ is attached to the midpoint of a ...
1
vote
2answers
42 views

How to subtract two equations?

I am working on a physics problem where I am told that there are two questions: EQ1: $$ W = \left(\frac{\mu_0}{2\pi}\right)\left(\frac{L}{R}\right)I_1^2+I_1LB_{e,z} $$ EQ2: $$ W = ...
0
votes
3answers
55 views

Justification of $dV = r dz dr d \theta$

Context: I'm taking calc based physics, and we are supposed to be able to integrate moment of inertia for a cylinder. I referenced a mit vid, and though I have no education on multiple integrals, I ...
2
votes
1answer
35 views

Shape Of A Blimp.

Was playing around with solids of revolution, the shape given by rotating $y=\sqrt{\sin x}$ about the $x-$axis seems to resemble a blimp. The only thing I can find out about the natural shape of ...
0
votes
0answers
35 views

Rope wrapped around pole Friction

A rope is wrapped $M$ whole turns round a cylindrical post, the two ends of the rope going in opposite directions. The coefficient of friction between rope and post is $0.25$. It is desired that by ...
0
votes
2answers
86 views

conservation law for the trajectories

Is it possible to find the conservation equation as the form of $Q=h(x,y)$, given that $$\dot{x}=x-xy$$ $$\dot{y}=5xy-5y$$ I am not sure how to start with.
0
votes
0answers
26 views

Definition of Work Done

I am trying to make sense of the meaning of the definition of work. The original definition of work was also known as "the weight lifted through a height." I was hoping that our mathematical ...
1
vote
1answer
24 views

Unit step function differential equation

Suppose we model a physical phenomenon with a 2nd order linear differential equation: $a_2$(t)$y''$+$a_1$(t)$y'$+$a_0$(t)$y$=$f(t)$, where 't' stands for time. In choosing an appropriate driving ...
0
votes
2answers
32 views

Factoring out an equation

I found this equation in a book: $$ m_0 v_0^2 + m_1 v_1^2 = m_0 v_{0_{Final}}^2 + m_1 v_{1_{Final}}^2.$$ It says that Notice that you have a different equation with the same two unknown ...
3
votes
1answer
29 views

Recursive integral definition for the dynamics of a DC motor

I have been trying to build a simulator of motor dynamics, but am running into the wall of a seemingly recursive integral definition. Currently I am trying to derive a formula for the rpm of a motor ...
2
votes
1answer
68 views

What is happening in this integration?

I found in Peskin-Schroeder, while reading Quantum Field Theory. the following integration. $$\frac{1}{4\pi^2 r}\int_m^\infty \frac{se^{-sr}}{\sqrt{s^2-m^2}} = e^{-mr}$$ at the limit $r \rightarrow ...
2
votes
0answers
40 views

Degenerate solutions to polynomial in one variable, parameterised by two variables.

If I have a polynomial equation in one variable $w$: $$\sum_{n=0}^N a_n(x,y) w^n = 0$$ where $a_n(x,y)$ are the coefficients parameterised by the real variables $x$ and $y$, I can obtain solutions ...
0
votes
0answers
20 views

How to find friction?

The question is as follows: A straight uniform beam of length 2h rests in limiting equilibrium, in contact with a rough vertical wall of height h with one end on a rough horizontal plane and with the ...
1
vote
1answer
83 views

Feynman problem on action

It is very weird for me that a newbie can ask a new (may be silly, sorry...) question but must have 50 reputation to comment. When I see a good question like this but have no answer what I have to ...
1
vote
2answers
43 views

Moment of inertia hollow sphere with inner and outer radius

I'm trying to determine the moment of inertia of a hollow sphere, with inner radius 'a' and outer radius 'R'. A lot of websites give me different solutions, so I don't know which one I have to use. ...
1
vote
0answers
44 views

Orthogonality of vectors and its dependence on the inner product.

Consider a set of vectors, $\{{\bf e}_i\}$ in $\mathbb{R}^n$. I am thinking specifically of the standard orthonormal basis. I am having a very difficult time understanding what it means for vectors to ...
0
votes
0answers
29 views

A PDE problem on the Heat-Like differential equations

I came across the following questions in part of my work: Consider the Heat-Like equation of the form $\frac{\partial u}{\partial t}=\hat{H}u + f(x,t)u + g(x,t)$ where $\hat{H}$ is a Sturm-Liouville ...
0
votes
2answers
36 views

Evaluate this integral $ s(t) = \int{\frac{dx}{\sqrt{2G(M+m)(\frac{1}{x} - \frac{1}{d})}} }$

I'm trying to get a displacement-time function from this velocity equation where $x$ represents time. $ v(x) = \sqrt{2G(M+m)(\frac{1}{x} - \frac{1}{d})} $ Since $v = \frac{dx}{dt}$, we can determine ...
1
vote
2answers
38 views

Center of Mass of a Semi-Circle using Cartesian Coordinates

So I'm currently trying to figure this out but am not sure where to start. I know that you can figure the center of mass using polar coordinates, but I know that it's possible to do it using Cartesian ...
0
votes
1answer
8 views

Finding the measured average angular velocities

Supposed I have a couple rows of data with recorded measured ratios $\omega_f/\omega_i$ and they ask me for the "Average Measured $\omega_f/\omega_i$ " This may seem like a really trivial solution but ...
0
votes
2answers
46 views

Calculation of a Fourier Coefficient.

I need some help calculating this Fourier coefficient. Periodic signal, six-steps. Odd periodic signal. I've made the calculations myself, not using any software, and the results are these: ...
0
votes
1answer
51 views

How to evaluate this integral? Fourier transforms

In quantum mechanics, I'm given a wave packet that is described by the Gaussian \begin{align*} \psi(x,t=0) = \sqrt{a} \exp(-x^2 / 2a). \end{align*} I need to compute the Fourier transform of this. By ...
2
votes
0answers
19 views

calorimetry - thermodynamics [closed]

The end of a rod is in thermal contact with a vessel containing a mixture of ice and water .If steady state heat flow through the rod is 140 W , determine how many grams of ice that melts in 1 minute ...
6
votes
0answers
191 views

Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
1
vote
1answer
23 views

Adding waves of equal frequency using phasors

Given the following equation I have to find $a$ and $\phi$: $$a_1 \cos(\omega t + \phi_1) + a_2 \cos(\omega t +\phi_2) = a \cos(\omega t + \phi)$$ Is there an easy way to construct $a$ and $\phi$? I ...
2
votes
0answers
32 views

Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
0
votes
0answers
19 views

What's the position vector of these two forces?

Here is the force diagram I'm not sure of the position vectors of W and T. I would of thought that W=(1/2)Li+(1/2)LWj and T=Li+Tsin(?)+Tcos(?). I put question marks because I'm not sure which ...
1
vote
1answer
51 views

Is it correct to think of the Laplacian as the divergence of a gradient field?

Factoring out the notation, I see that $$\nabla^2(\phi) = \nabla \cdot \nabla(\phi) = \nabla \cdot (\nabla(\phi)) $$ which looks something like the divergence of the gradient of phi. Is it ...
0
votes
0answers
56 views

How is my proof that this vector field is identically zero?

EDIT: If my work is fine, I believe that the problem statement (an old exam question from 1992) has given one too many assumptions - namely, divF=0. I think towards the end of my proof, when I ...
0
votes
1answer
15 views

How to use the assumption that a vector field is curl-free in a “convex” region,

I don't seem to need this assumption in one of my proofs, but the problem statement gives it, so I think I had better try to use it. Does a convex region imply that it is simply connected (but that ...
0
votes
0answers
4 views

Determine the $\frac{1}{e^2}$ width of $E(w)=A \cdot e^{\frac{-t^2 \cdot (w-w_0)^2}{16}}$

Let A, t, and $w_0$ be constants. Determine the $\frac{1}{e^2}$ width of $E(w)=A \cdot e^{\frac{-t^2 \cdot (w-w_0)^2}{16}}$. Then determine the Full Width at Half Maximum (FWHM).
1
vote
1answer
57 views

What does this gradient-like symbol mean?

If $\nabla \phi$ denotes the gradient of some scalar field $\phi$, then what does $\nabla^2 (\phi^2)$ mean? I don't think it means taking the gradient of a gradient (of a squared-scalar field), ...
0
votes
0answers
18 views

Understanding the relationship between phase and frequency in a particular equation

Context I have the following equation: \begin{align} a&=1-e^{-\omega t/x}+\frac{m}{\sqrt{1+x^2}}\Bigg[\sin(\alpha+\omega t + \varphi)-\sin(\alpha+\varphi)e^{-\omega t/x}\Bigg]. \end{align} This ...
0
votes
0answers
29 views

Kepler orbital elements from state vectors

Well given the 6 common Kepler orbital elements: Eccentricity $e$ Semimajor axis $a$ inclination $i$ Longitude of ascending node $\Omega$ Argument of periapsis $\omega$ True anomaly $\nu$ As can ...
2
votes
1answer
31 views

Solution to DE: $r^2R(r)'' + [r^2k^2-l(l+1)]R = 0$ (Bessel like)

I was solving a problem stated in this question. The solution led me to the differential equation: $$r^2R(r)'' + [r^2k^2-l(l+1)]R = 0,$$ which arises after solving $c^2\nabla^2v = v_t$ separating ...
0
votes
0answers
25 views

Multidimensional gaussian integral with complex elements.

I am trying to find the answer of this Gaussian integral: \begin{equation} \int{d^{N}\alpha \,\,\,\,e^{-\alpha^{\dagger}M\alpha+\alpha^{\dagger}d}} \end{equation} where $\alpha$ is a vector with ...
1
vote
1answer
61 views

Heat equation with odd boundary conditions

A somewhat similar question to what I'm going to ask is this one. The problem is basically that one has the heat equation $c^2\nabla^2u = u_t$ in which initial and boundary conditions are given. But ...
0
votes
0answers
29 views

Flux in a coaxial transmission line made of perfect conductors

Question: A coaxial transmission line has inner and outer radii of $a$ and $b$, respectively and has the magnetic field intensity, $H_\phi$ = $\frac{A}{\rho}\sin{\omega t}\cos{\beta z}$. The ...
4
votes
1answer
65 views

Lagrangian invariant under left and right multiplication by unitary matrices, slick way to see?

Is there a slick way to see that the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G\partial_\mu G^{-1}),$$where $G$ is an $N \times N$ unitary matrix, is invariant under left and right ...
0
votes
1answer
17 views

A bead is threaded on a friction-less vertical wire loop of radius $R$.

The question is the very last sentence at the end of this post. In this post, I'll demonstrate how I reach to a contradiction(the conditions mentioned in conjecture 1 should be satisfied by all ...
1
vote
2answers
22 views

Optimization of location to find where 2 particles are closest.

A particle moves west at a speed of $25km/hr$ from the origin when $t=0$. Another particle moves north at a speed $20km/hr$ and stops at the origin when $t=1$, where $t$ is in hours. What will the ...
-6
votes
1answer
222 views

Physical interpretation of the Lebesgue integral

Is there a physical interpretation of the Lebesgue measure and an associated experiment that shows that it is the 'right' measure to use to model various phenomena in the physical world? I'm asking ...
0
votes
0answers
27 views

Obtaining current density from magnetic field

Question: Let $\vec{H} = \frac{2}{\pi\rho}[1+\frac{10^7\rho^3}{6}]\hat{a}_\phi + 8\hat{a}_z$ A/m for $0\leq\rho\leq0.01$ m, and $\vec{H} = \frac{16}{3\pi\rho}\hat{a}_\phi + 8\hat{a}_z$ A/m for ...
1
vote
1answer
51 views

Pump water from half-full cylindrical tank from a spigot at height higher than top of tank

I looked at this question, however, it pumps water of a full tank out from the top of the tank. My question is a bit different, but mostly similar. There is a cylindrical tank with a radius of 3ft, ...
0
votes
0answers
10 views

Sommerfield Expansion Taylor Expansion

Ugh... I can't figure this out and I DO NOT understand why. So this has to do with the Sommerfield expansion of the Fermi function (wiki) (another reference) The issue I'm having is we are supposed ...