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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Taylor series of Lagrangian

Take a look at the Lagrangian defined here. $L=\frac12 a(q)\dot q^2 - V(q)$. You can think of $a$ and $V$ as functions. It seems as though $L$ depends only on $q$. If $q_0$ is a point for which ...
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Moment of inertia and its perpendicular distance question.

Boundary of plate is an ellipse with semi-axes a and b. L is a line that passes through the center of the ellipse and makes angle j with the axis of length 2a. Density function is constant and its ...
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physics tension force [on hold]

Two identical blocks, mass 4.0 kg, are tied together with a string as shown. They are sitting on a bench where the coefficient of kinetic friction is 0.80. The string in the middle can support a ...
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24 views

Force between two finite parallel current carrying wires

Most standard physics textbooks compute the force two infinite wires exert on each other, but they remain silent about the case where the wires are finite. Let's say we have two parallel wires ...
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Find electrostatic energy of three-parallel-plates capacitor

Question: A capacitor consists of three identical and parallel thin metal circular plates, area A, in the planes z = −H, z = a and z = H, (−H < a < H), centres on z axis, and at potentials 0, V ...
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52 views

Simple question on group theory

Suppose we have the following system of differential equations: \begin{cases} \frac{dx_{i}}{dt}=f_{i}\left(\boldsymbol{x},\boldsymbol{y}\right), & i=1,\ldots M\\ \\ ...
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1answer
24 views

How can I find out the center of mass of thin plate in the shape of a rectangle?

How can I find out the center of mass of thin plate in the shape of a rectangle ABCD If the density at any point is the product of the distances of the point from two adjacent sides AB and AD? ...
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2answers
29 views

Physics Word Problem Concerning Finding Power

A twin engine jet aircraft is climbing at a 10 degree angle at 260 ft/s. The thrust developed by a jet engine is 1000 lb. The power developed by the aircraft is ______ (2000 lb)* (260 ft/s) = ...
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1answer
35 views

Solid Angle Integrals

I'm currently working on solving a solid angle calculation for my physics project. I'm trying to solve it computationally using python - but my coding skills are basic at best. I need some hints/tips ...
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1answer
19 views

How Do You Solve for Landing Speed?

A ball is thrown eastward into the air from the origin (in the direction of the positive x − axis ). The initial velocity is 50i + 80k. The spin of the ball results in a southward acceleration of 4 ft ...
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1answer
27 views

Formula for 2 vectors

Hello StackExchange people, I have an issue. I need to create a formula to find degrees and size at a problem like the following: So I have $0-359$ degrees and a length of $2$ lines, and I need to ...
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33 views

Finding total distance and time for a bouncing ball - application of series

Problem: A ball is thrown straight upward so that it reaches a height $h$. It falls down and bounces repeatedly. After each bounce, it returns to a certain fraction $f$ of its previous height. Find ...
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2answers
45 views

Second-order nonlinear ordinary differential equation. Newton's second law.

I have this problem, I will try to give as close translation to English as possible: "A point with mass m moves towards a center due to force $m·k^2/(r^3)$, where $r$ is the distance from the center. ...
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41 views

Sum over all compositions of $n$ into exactly $k$ terms (overlaps of $SU(k)$ coherent states)

Given two $k$-dimensional vectors, $\vec{x}$ and $\vec{y}$, I would like to find, $$ d(\vec{x}, \vec{y}) = \left|\sum_{t_1 + t_2 + \cdots + t_k = n} \prod_{1\leq q \leq k} (x_q\times ...
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1answer
38 views

Why does the vector field $(\sin (\theta), - \cos(\theta), 0)$ indicate sideways motion?

If I study a physical system, such as a car, and let it drive forward a little bit, say a distance $m$, then I can draw out the right triangle and find the car's position at $(m\cos \theta, ...
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12 views

Meaning of Torricelli's Equation ($v^2=u^2+2as$)

The equation of motion $v^2=u^2+2as$ is usually presented as the particular formulation of the SUVAT system which doesn't involve t. It is derived from the others using some (perhaps well-motivated) ...
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30 views

partial differential equation applicational problem

As a Maths student with not much knowledge in physics, I dont understand how the "string" can be "cut" into half at x=L/2. Also, how many initial conditions(data) does this question have apart from ...
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1answer
14 views

Find out the angular speed in terms of time.

Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star: $u=\frac el\cos\theta+\frac 1l$, where $u$ is the reciprocal of the radial distance ...
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59 views

Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm ...
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3answers
94 views

infinitesimal intervals in physics

The density of states of a system in an interval $[E, E+dE]$ is given implicity by $dV = D(E)dE$ (Or I suppose explicitly, by $D(E) = \frac {dV}{dE}$, but we'll be integrating it anyway, so it doesn't ...
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0answers
12 views

Thermodynamics: find exit temp and velocity of air out of a nozzle?

I don't know if I can really ask a thermodynamics question here on this math site but I need help and this was the best site for when I needed help in math class. Concerning thermodynamics, I have ...
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1answer
46 views

Eigenvectors of almost-Toeplitz tridiagonal matrix.

I'm reading a book about semi-conductors, and when figuring out how dopants affect the energy-levels, one wishes to find the eigenvalues (and vectors) of a NxN tridiagonal matrix of the form ...
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1answer
22 views

re-arrange equation $L=2^{10(v-1)} v^2$

Is it possible to re-arrange this equation to make v the subject? $$L=v^2 . 2^{10(v-1)}$$ If so, what is the answer? If it helps (which by excluding zero it should)... $$0<v<1$$ I have tried ...
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1answer
23 views

How to find direction of velocity V2 to reach an object travelling at velocity V1, such that it takes least time?

If an object A is currently at point P1 moving with constant velocity V1, and there is another object, object B which currently at point P2 which can move with velocity v2, then what should be the ...
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1answer
19 views

Work done in a vector field

Say a particle is moving along a path $\gamma$ in a vector field, then the total work done by the force $\vec{F}$ on the particle is $\displaystyle \int_{\gamma}{\vec F}.d\vec{r}$. Say if this value ...
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46 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
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1answer
54 views

Can an integral of a function that is not well behaved be finite?

Consider the following integral which gives the time period of simple pendulum where $\theta_0$ is the initial inclination of pendulum with vertical. ...
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14 views

Projectile landing on an elevated point

A golfer is standing on a fairway and hits a shot to a green that is elevated 6.0 m above the point where she is standing. If the ball leaves her club with a velocity of 43 m/s at an angle of 40.0 ...
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15 views

Tangential and normal components of acceleration of a point moving along a curve

If a point is moving along a curve in polar coordinates, is the tangential component of its acceleration given by $r\left(d^2\theta \over dt^2\right)$ and the normal component by $r\left(d\theta \over ...
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0answers
61 views

Index notation confusion in tensor algebra

I have some confusions regarding index notation in tensor algebra. Let's assume $\vec{v}$ is a vector belonging to vector space $V$. Choosing a basis set $\{\vec{e}_\nu\}$, $\vec{v}=x^\nu\vec{e}_\nu$ ...
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1answer
42 views

Olbers' Paradox in an Euclidean universe with randomly located stars

This question is inspired by Olbers' Paradox. Imagine a universe $U$ shaped like $R^3$ with points $P$ randomly distributed in a Poisson fashion throughout, with density parameter $\lambda$ such that ...
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77 views

Setting up a differential equation to find time constant for RC-circuit

Problem: Calculate the time constant for charging the capacitor in the circuit shown in the figure. What is the maximum charge on the capacitor? Attempt at solution: Let current $I_1$ flow from the ...
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2answers
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Water Density and Fluid Force (question below) [closed]

I've been trying to study the question and the answer below. Can someone tell me how to start this problem myself? I don't understand why they named one fourth of the circle equation the whole ...
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Approximations of the kind $x<<y$

I have an expression for a force due to charged particle given as $$F=\frac{kQq}{2L}\left(\frac{1}{\sqrt{R^2+(H+L)^2}}-\frac{1}{\sqrt{R^2+(H-L)^2}}\right)$$ where $R$, $L$ and $H$ are distance ...
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How do you calculate certain variables of two or more events that occur simultaneously compared to the same events happening subsequently.

Say you have two hoses, A and B, that fill up a pool of equal size at different rates. Hose A fills up a pool in 10 mins, hose B in 20 mins. Thus A = 1p/10m, B = 1p/20m. Lets say that Hose A filling ...
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1answer
60 views

Solutions to Kepler problem/Motion in central field

It is clear that particals in a repulsive central field given by a point mass at the origin are moving along hyperbolas, e.g. given by the expression ${x^2 \over a^2} - {y^2 \over b^2} = 1$ (after a ...
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4answers
69 views

How to solve $a \cos \alpha + b \sin \alpha = c$ for $\alpha$?

I'm solving a physics problem and I came down to solving an equation of the form $$a \cos \alpha + b \sin \alpha = c$$ Can someone help me to solve this? Thanks in advance!
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50 views

Vectors problem, Please Help!

A buoy is floating in water and is tied to a post. The water is creating a force of $3$ N on a bearing of $125^\circ$ and the wind is creating a force of $2$ N on a bearing of $230^\circ$ What force ...
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53 views

Attempt to solve the brachistochrone problem

I am attempting to solve the brachistochrone problem for fun. I don't have too much experience with differential equations and wanted to see if I am on the right path. My attempt Assumptions ...
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1answer
47 views

Integration of partial derivative $\frac{dL}{dq}$ with respect to $t$ where $q$ is implicitly a function of $t$

Is $\int_{t1}^{t2} \frac{\partial L}{\partial q}\delta{q} dt$ equal to $\left[\frac{\partial L}{\partial \dot{q}}\delta{q}\right]_{t1}^{t2} $ if $q$ implicitly depends on $t$ ? If not I ...
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Finding the derivative Maxwell Boltmann distribution

I need to basically show that the slope is $0$ when $\epsilon = kT$ $f(\epsilon)=\left(\frac{8 \pi}{m}\right)\left(\frac{m}{2 \pi kT}\right)^{\frac{3}{2}}\epsilon e^{\frac{-\epsilon}{kT}}$ So I ...
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According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general

I recently read Khinchin's derivation of Liouville's theorem. I was able to follow the math for the most part, however I was hoping for an intuitive understanding about why the form of the measure on ...
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Continuity of the divergence of a static electric field

Let $\rho:\Bbb R^3\to\Bbb R$ be a continuous charge density function. Define the electric field $\vec E:\Bbb R^3\to\Bbb R^3$ by $$\vec E(\vec r)=k\cdot\int_{\Bbb R^3}\rho(\vec{r}')\cdot\frac{\vec ...
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1answer
45 views

Velocity vector

We suppose that a ship, that is at the position $(1, 0)$ of a nautical map (with the North at the positive direction $y$) and it "sees" a rock at the position $(2, 4)$, is directed to North and is ...
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Angular velocity

The angular velocity $\omega$ of rotation of a rigid body has the direction of the rotaion axis and magintude equal to the rotation rate in rad per second. The orientation of $\omega$ is determined by ...
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1answer
23 views

Torque of a force

The torque $M$ of a force $\overrightarrow{F}$ as for the point $O$ is defined as the product of the magnitude of the force $\overrightarrow{F}$ and the perpendicular distance of the point $O$ and the ...
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I need help with divergence and gradient?

$$A_z = \mu{\frac{e^{-jBr}}{4\pi r}}∫I(z')e^{jBz'\cos\theta}dz'$$ Midway into my question, I want to compute: $$-j\left( \frac{\nabla(\nabla\cdot A) }{w\mu\varepsilon} \right).$$ Symbols like $ w, ...
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1answer
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Can someone explain why acceleration is not negative in this problem?

Studying for a Physics exam and I'm going back through the homework and redoing the problems. I've come to a problem and am a little confused on the equations. The problem is: With what speed ...
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1answer
50 views

Solve this system of equations for acceleration

I'm dealing with a double Atwood's machine problem, and so far I got these $F=ma$ equations: $T - m_2g = m_2 a_2$ $T - m_3g = m_3 a_3$ $2T - m_1g = m_1 a_1$ $a_1 = -\frac{(a_2 + a_3)}{2}$ ...
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Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) ...