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

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3
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16 views

Arbitrary factors for the (modified) Mathieu equation

I am currently confronted with a physical equation that, after a fair amount of reworking, can be recast in the form of the modified Mathieu equation : \begin{equation} y(x)'' - (a - 2q \cosh(2x)) ...
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0answers
35 views

rope extending under own weight - hooke's law

I am trying to solve following physics exercise but I cannot find my error. A rope with a cross section $A$, density $\rho$ and a length $l$ is hanging (staticly). I want to calculate the total ...
3
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0answers
36 views

The behavior of the 3D wave equation close to the origin

The general solution to the three dimensional wave equation is \begin{equation}u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r} \end{equation} where $F$ and $G$ are arbitrary functions. I want to ...
1
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0answers
28 views

Solution for a harmonic oscillator driven by a sweeping frequency

Given a harmonic oscillator driven by a sweeping frequency: $$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin\left(\omega(t) ...
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votes
1answer
33 views

Elimating ODE constant of a extended surface problem

--- Already managed to solve it. Just to give the whole context: Since the fin has constant cross-section along its length and the area and perimeter are also constant, I can use: ...
1
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1answer
115 views

Why is $|\cos\theta d\omega|$ the projection of the differential solid angle $d\omega$ onto the $(x,y)$-plane?

Let $B\subseteq\mathbb R^3$ be the ball with radius $r>0$ around $0$ and $S_{\partial B}$ be the surface measure of the boundary $\partial B$. Given a piece of the surface $A\subseteq\partial B$, ...
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0answers
28 views

The amplitude of oscillation for a driven harmonic oscillator

What is the amplitude $G(\omega_0,p)$ of oscillations for a driven harmonic oscillator $$x''(t) + 2 \gamma \omega_0 x'(t) + \omega_0^2 x(t) = F_0 \cos (pt)$$ as function of $p$ and ...
7
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1answer
66 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
1
vote
1answer
36 views

How can I solve $\beta^2=\frac{m^2g}{h}\left(-\frac{\beta t}{m}+e^{\frac{\beta t}{m}}-1\right)$ for $\beta$?

This equation arose when I tried to find out how to derive $\beta$ in Stokes' Drag Force $F=\beta v$ as a function of the time $t$ it takes a mass $m$ to hit the ground after falling from a height ...
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1answer
64 views

Mathematical Physics: Differential equation of a raindrop

I hope this is a suited question for this site since it contains a mix of physics and mathematics. In case I should post this on the physics stackexchange site, please let me know. A spherical ...
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0answers
18 views

Given max force due to friction and change in mass, calculating angle of inclination. Paradox?

Given an object resting on an inclined plane, we have the following equations to determine the forces acting on said object: $$\sum f_x = mg\sin(\theta) - f_s = 0 $$ Which says that the sum of the ...
0
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1answer
38 views

How does the Pauli principle work?

Let $H$ be some Hilbert space. Now in general, in quantum mechanics, the vector space representing states of $n$ (non-interacting) particles is $H^{\otimes n}$, but if I consider these particles of be ...
0
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1answer
11 views

Collision point of two vehicles with constant velocity

Given the constanct velocities of two vehicles that are on a frontal collison line and their distance from each other: How can I calculate the collision point?
3
votes
1answer
54 views

Diffraction and Fresnel Integrals

Migrated from Physics SE due to mathematical content I am trying to derive the intensity variation function for a single slit diffraction. Sorry for the poor diagram... So I decided to take the ...
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0answers
13 views

Accelerometer sensitivity

I'm using an accelerometer which is for a defined FS of $\pm2$g, I have a linear acceleration sensitivity of $1$ mg/digit. If my calculation is correct, does this mean that the digit values won't ...
0
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0answers
26 views

Throwing a baseball on top of a cliff

A baseball is thrown from the stands 32 ft above the field at an angle of 30 degrees up from the horizontal. When and how far away will the ball strike the ground if its initial speed is 32ft/sec? ...
2
votes
1answer
35 views

Show that gravity is described by this 1-form

From Harold Edwards' Advanced Calculus: A Differential Forms Approach, section 2.1, exercise 1: The central force field. Newton's law of gravitational attraction states that the force exerted by ...
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1answer
44 views

Non uniform circular motion, can you find the error?

The vector function and its derivatives for a non uniform circular motion is: $$ \vec s(t) = r \cdot \begin{bmatrix} \cos(\omega t) \\ \sin(\omega t) \end{bmatrix}, \qquad \omega = \omega(t) $$ $$ ...
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0answers
8 views

Re-introducing dimensions into a dimensionless equation?

I have an implicit function: $$f(\bar{v},\bar{r}) = C $$ where $\bar{v} \space \text{and} \space \bar{r}$ represent velocity and radius (in dimensionless form), respectively. I now, however, need my ...
1
vote
1answer
30 views

Perturbation theory, why are the assumptions of the method satisfied?

I am a undergrad student interested in math taking quantum mechanics. Yesterday I was introduced to what physicists call perturbation theory, non-degenerate case. According to authors Griffiths, ...
1
vote
1answer
61 views

Proof of Kepler's Third Law

Kepler's Third Law states that the square of the time period ($T$) of revolution of a planet about the sun is directly proportional to the cube of the semi-major axis ($a$) of its elliptical orbit. ...
2
votes
1answer
29 views

Related Rates Question with Resistor, Finding rate of change of $R$ (Physics)?

Let $R_1$,$R_2$,$R_3$ be connected in parallel! See circuit bellow: If $R_1$ increasing at $4\:\frac{\Omega }{s}$, If $R_2$ increasing at $2\:\frac{\Omega }{s}$, If $R_3$ decreases at ...
0
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2answers
155 views

Continuity of a certain vector field

Let us define $$\boldsymbol{E}(\boldsymbol{x}):=\lim_{\varepsilon\to 0}\int_{D\setminus ...
0
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1answer
48 views

Time to reach a speed given the acceleration equation

I have modeled the acceleration of a car, and here is the speed of this car with respect to the time : $$\frac{dv}{dt} = \frac{6.21}{v}-0.046-0.000137v^2$$ Now, how can I find the time needed to ...
1
vote
1answer
31 views

approximating $(1-e^{-x})^2$ near $x=0$ with $x^2$ via Taylor expansion

I would like to show that $(1-e^{-x} )^2$ is approximated well near $x=0$ with $x^2$ via Taylor expansion but can't quite seem to complete the job. I know that by expanding the exponential into its ...
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0answers
55 views

Find the friction constant minimizing the duration of the vertical movement of a wheel

Q) The mass of a car that acts on one wheel is $100 kg$. The elasticity (spring) constant in the suspension system of that wheel is $k = 10^{4}N/m$. Design the strut (find the ...
2
votes
1answer
36 views

How to find a sensible approximation of $R\dot{\theta}^2+\ddot{\theta}(R\theta-l)+g\cos\theta=0$

The differential equation $$R\dot{\theta}^2+\ddot{\theta}(R\theta-l)+g\cos\theta=0$$ is derived from considering a pendulum attached to the uppermost part of a disk. As in the picture above (but ...
0
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1answer
181 views

The acceleration due to gravity for the Moon is one sixth the same for the Earth

I have to show the acceleration due to gravity for the moon is one sixth the same for the earth. The question gives the formulae and the variables to plug in for each, thus: Gravitational constant: ...
4
votes
1answer
52 views

Formula for heat in thermodynamics.

happy to join this community. :) I have a simple problem. We have a well known formula $C_v:=(\frac{\partial U}{\partial T})_V=T(\frac{\partial S}{\partial T})_V $. It comes from (we assume ...
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0answers
38 views

What are the steps to write a control function?

I have a system defined like this: $$\delta\delta x(t) = \rho(t) \cos(\alpha(t)) \\ \delta\delta y(t) = -g + \rho(t) \sin(\alpha(t))$$ I need to write a control function to calculate $\rho(t)$ and ...
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0answers
20 views

Acceleration of two points problem (mathematical explanation)

There is a square board with two nails in corner A and B. We apply a torque M to the square board. Nail A can move freely on a track only in the $x$ direction and B only in the $y$ direction. The ...
2
votes
0answers
31 views

Evaluating Hyperbolic Cotangent (coth) Integral

I am working on some simulation, and the paper that I am basing some of the work off of involves several complex integrals. In particular, the one I am trying to solve is $\int_0^\infty ...
0
votes
1answer
20 views

P is an arbitary point , and p=AP/AC. x=AX/AM, where AM is a median. Observe that x=2/3 when p=1/2. Find a general formula for x in terms of p.

Where p and x represent mass. It suggests putting masses of one at points B and C, and finding an appropriate mass to put at A so that the center of mass will be at X. When p=1/2, x=2/3, which ...
0
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1answer
32 views

Partial differentiation for parallel Resistance Problem

I have this solved problem , and is not clear for me how the teacher got the result. I tried doing someting on the right side of the picture, but i am not sure is correct ... I wonder if someone can ...
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0answers
30 views

What does symmetry imply about the solution in mathematics? (Example: Gauss' law)

Suppose you have an infinite cylinder and are considering a field $\mathbf{D}$ caused by physical elements within the cylinder such that it satisfies $\int \mathbf{D}\cdot d\mathbf{a} = Q_{free}$. ...
4
votes
1answer
47 views

Taking the divergence of a field with a singularity when $\vec{r}=0$ produces a Dirac's delta.

I'm currently taking a classical electrodynamics course. I have a mathematical background and I know that the classical theorems of integral calculus (Stokes, Gauss, ...) are just particular versions ...
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0answers
24 views

Finding the angle when throwing a ball to hit a target

For my game, I need to find the angle to shoot a projectile so that it hits the target (think Angry Birds). Assume that the projectile is launched at origin (0, 0), the target is at (x1, y1), initial ...
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vote
2answers
43 views

Historic reference for Eytelwein or Capstan Equation and assumptions

I'm looking for the actual first reference for what is commonly called the Eytelwein or Capstan equation: $$\frac{T_1}{T_2} = e^{ \, \mu\theta}$$ Is there a book/page where Eytelwein published this? ...
1
vote
1answer
15 views

Deriving Relativistic Force

Newton’s law states that $F=\frac{dp}{dt}$, where $p$ is the momentum of a body. In Newtonian physics, if the body has constant mass $m$, its momentum is $mv$, and Newton’s law becomes the familiar ...
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1answer
65 views

Doppler Effect Related Rates Application

A police car is sounding a siren with a frequency of $1280$ Hz while traveling right towards you. At a certain time, you measure the frequency of the siren to be $1400$ Hz, and increasing at a rate ...
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1answer
16 views

Matrix operation

The below calculation is finding the expectation value of Pauli Matrix. I understood the physics how they got the below term but, I don't get that how the final result is zero in the equation . ...
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votes
2answers
98 views

Find the initial direction and time of flight of a basketball, given initial speed and distance

A player passes a basketball to another player who catches it at the same level from which it was thrown. The initial speed of the ball is 7.1m/s, and it travels a distance of 4.6m. What were (a) ...
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0answers
43 views

Application of the Bernoulli Equation

I'm attempting a question on fluid dynamics and I'm using a rearranged form of the Bernoulli Equation. But I can't prove the equation for the velocity of water leaving the tap. It should equal $$v = ...
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vote
1answer
79 views

Generators of a semi simple lie algebra must be traceless

Consider a semi simple lie algebra. Show that if $T_a$ are the generators of a semi simple Lie algebra then $\text{Tr}T_a=0$. Attempt: $[T_a, T_b] = ic^c_{ab}T_c \Rightarrow \text{Tr}[T_a, T_b] = ...
2
votes
0answers
44 views

What are Einstein's evolution equations for galaxies? [closed]

I'm researching galaxy distributions and have been tasked with solving Einstein's evolution equations for different levels of dark energy and matter. I've been told to do this numerically via Matlab, ...
2
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0answers
22 views

Reference request for complex scalar field, propagators worked out with path integral approach? [closed]

In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can anyone supply me a reference to where the ...
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0answers
68 views

Proving some simple formula with gamma matrices

a) Prove that $\text{Tr}(\gamma^5 \not a \not b) = 0,$ where $\not a = a_{\nu}\gamma^{\nu}$. b) Prove that $\text{Tr}(\gamma^5 \not a \not b \not c \not d) = 4i\epsilon_{\alpha \beta \gamma \delta} ...
1
vote
1answer
57 views

Differential equation free fall in gravitational field

For a physics problem I was told to set up a differential equation for the free fall in the gravitational field of the earth. The equation (via Newton) I've got is following: $$\ddot{r} = - G M \frac ...
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0answers
23 views

Potential energy of a Dipole interaction unexplained sign error.

Hello i've attactched a question as well as my attempt at an answer, I get the correct mathematical expression however I have a sign error for both the terms in the bracket. I can't work out what ...
2
votes
2answers
30 views

Solving Equation of the form $\sqrt{(a+\frac{b}{2})^2+L^2}-\sqrt{(a-\frac{b}{2})^2+L^2}=c$

I have been struggling to find the solution to one of my physics problems mathematically as this is the equation I arrive at where all of the values are known except $a$. I have tried solving for a ...