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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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
53 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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2answers
19 views

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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37 views

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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24 views

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
51 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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2answers
46 views

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
57 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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43 views

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
25 views

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
55 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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83 views

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) ...
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0answers
20 views

Are 1-D line sections of 2-D point source-invoked potential distributions positive definite?

Consider the 2-D potential distribution induced on the plane $y=0$ by a point source positioned at $(0, -y_0, 0)$ in the open halfspace below that plane. The material below the plane is assumed ...
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2answers
140 views

Calculating speed and distance travelled by an object using quadratic function

I'm actually getting stuck with a tricky part of a math problem using a quadratic function. An object propelled with an initial speed, has its altitude given after $t$ seconds by the quadratic ...
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1answer
32 views

time until gas is depleted cylinder

If one has a $10$ liter (water volume) tank of gas at $15$ Mpa, then how long until gas is depleted if gas is exiting the cylinder at a rate of $15$ liters per minute? ATM i have used ideal gas law, ...
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2answers
108 views

Can someone clarify the definition of flux?

I am confused by the concept of flux as used in vector calculus. Suppose I have a sphere. On the inside of this sphere is a spherically symmetric electric charge distribution. Now I want to find the ...
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2answers
54 views

How do I find torque on an axis when no weight or force is given?

I need some help with the following problem: Find the torque around the x axis of the triangle with vertices (0, 0), (1, 4) and (1, 0). Assume the density equals one. — Life of Fred Calculus, ...
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21 views

Question about DE related to physics; includes Hooke's Law and Newton's Second Law as well as system of DE equations and solutions, and a phase plane.

I mainly need help with part A, and a little bit on B and C. Thank you in advance for your answer or any comment or edit that helps!!! A second-order DE can be sometimes solved with clever ...
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0answers
29 views

Why is the difference of a stream function at two points the flow rate PER UNIT WIDTH?

For a velocity vector field $\bf{u}$$ = (u,w)$ in two dimensions ($x$ and $z$), we define the stream function $\psi$ to be; $$\psi(P_1) = \int_{P_0}^{P_1} \mathbf{u}\cdot\mathbf{n}dl$$ where $P_0$ is ...
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1answer
62 views

Force between parallel conductors using amperes law

Two parallel conductors are 0.3m long, and 0.15m apart. They each carry 2.5A of current in the same direction. Calculate the force between them. I did (2 * 10^-7) (2.5^2) / (0.15) and got the ...
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1answer
105 views

Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
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2answers
43 views

Why do we use the word “energy” for this integral?

I came across the following and I would like to know the motivation or idea behind using the word "energy". Suppose $u(x)$ is a $C^1$ function on some domain $\Omega \subset R^3$. Then, we can think ...
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1answer
46 views

Trying derive $a=v^2/R$ with complex numbers.

I'm trying to derive the radial acceleration formula for uniform circular motion: $a =v^2/R$. Here's what I've $\mathbf r(t)= Re^{-it/2}\mathbf e_1e^{it/2}$ $\implies \ddot {\mathbf r}(t) =\frac ...
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1answer
51 views

Wondering if my reasoning is correct for a tricky calculus limit word problem

and thanks for reading my post. The problem reads like this: If a book starts to fall off a building from 120 feet in the air and the position of the book after t seconds is given by s(t) = -16t2) + ...
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0answers
18 views

Does 'variance' have a physical meaning?

Let's say we have some probability density function $f(x)$, we can find the mean by solving $\mu = \int_{-\infty}^{\infty}xf(x)dx$. In a similar way, we can find the center of mass of an infinite wire ...
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2answers
90 views

Volume integral of electric field (hemisphere solid)

Let $S$ be a hemisphere of radius $R$, and let $\sigma$ be the constant charge density at each point $(x',y',z')$ in $S$. The electric field generated by the hemisphere is a vector function: ...
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66 views

Charge distribution in a conducting sphere

This is a math problem that appeared in a physics one: "Evaluate the density of electrical charge in the surface of a hollow sphere such that the electric field is constant (in direction and absolute ...
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0answers
61 views

Find the magnitude, angle of acceleration and the angle of a particle's direction of travel

So I'm doing a Physics problem and have run into a snag with the online homework - every answer I've tried keeps getting rejected and I'm down to my last shot and I don't know what I'm doing wrong. ...
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1answer
44 views

how do you find position vector?

So at time zero a particle is at x= 4 m and y= 3 m and has a velocity of $${ v= \left( 2.0 \ \boldsymbol{\hat{\imath}}-9.0 \ \boldsymbol{\hat{\jmath}} \right) \text{m/s}}$$ The acceleration of the ...
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2answers
46 views

Help in Differential Equations - find velocity at t seconds…

Hello Everyone I am stuck where I am , Would like to know if I'm going the write path, and any hints on how to proceed would be greatly appreciated !! Question: A boat carrying 7 people is being ...
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1answer
55 views

Harmonic Oscillation using Gaussian Quadrature [closed]

Assume that the potential is symmetric with respect to zero and the system has amplitude $a$ suppose that the potential $V(x)=x^4$ and the mass of the particle is $m=1.$ Write a java function that ...
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1answer
47 views

Where should the fire man place the hose for the water to reach its maximum height?

Water leaves a fireman’s hose (held near the ground) with an initial velocity ${v_0 = 22.5 m/s}$ at an angle θ = 35° above horizontal. Assume the water acts as a projectile that moves without air ...
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1answer
31 views

Compute the specific heat capacity of ideal gas under constant $V$ and $p$

Compute the specific heat capacities at constant volume and constant pressure for air at standard temperature and pressure, assuming it is diatomic ideal gas and a molecular mass of 28u. I have ...
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1answer
69 views

Water leaking from box and the relationship of volume and height.

Suppose we have a container that has a base of area $b$ and we fill it up with water. Volume of water = $b \cdot h$, where $h$ is height. Hence, $\mathrm{d}v/\mathrm{d}t = b \cdot ...
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1answer
19 views

How do you calculate a percent of precision?

What does it mean when "calculating a percent of precision can be found from the ratio of your mass sensitivity to the equilibrant mass for each case." My ratio of sensitivity is .010 kg. My ...
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1answer
35 views

Problem Analysis - Answer but no procedure - Differential Eq.

I stumbled with this problem in a notebook that has been bothering me...The answer is written but there's no explanation nor a steb-by-step procedure or anything. If you know how to analyse the ...
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1answer
248 views

Derivation of the Boltzmann factor in statistical mechanics

I have seen similar derivation of the Boltzmann factor many times before, (http://micro.stanford.edu/~caiwei/me334/Chap8_Canonical_Ensemble_v04.pdf , just for example), which I think is incomplete. ...
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1answer
37 views

what is the position and acceleration at the instant when its velocity is zero?

The position of an object is given by $${x(t)= 2.17+4.80t^2-0.100t^6}$$ Would I have to take the derivative of that in order to find velocity? But from there how would I find the position and ...
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2answers
26 views

How much is does it take for the stone to reach the bottom?

So a stone is thrown vertically upward with a speed of 10.0 m/s from the edge of a cliff 65.0 m high. So, just making sure I have all my variables right Vinitial: 10 m/s V: 0 m/s g: -9.8 m/s^2 ...
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1answer
25 views

How do I correctly introduce a time parameter into this equation?

So, for the past few years it's been my goal to create an equation that would give me the position of an object in a gravitational field at time $t$, given it's initial position and velocity. At first ...
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1answer
43 views

Differential Equation Concerning Torricelli'$s^2$ Principle

We have $v = \sqrt{2gh}$ where $g$ is acceleration due to gravity and $h$ is height. By equating the rate of outflow to the rate of change of liquid in the tank, show that $h(t)$ satisfies the ...
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2answers
125 views

Moment of inertia of semi circular disc with hole

Suppose we have semi circular disc of radius $R$ with semi circular hole of radius $R/2$ , how can I find moment of inertia from axis thru $H$ I fail to see any symmetry so I cannot integrate , ...
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51 views

Angle at which a body bounces off a sphere

Suppose a solid body approaches a sphere of radius $R = 1$ and height $z$, how do I calculate the angle $\theta$ at which the body bounces off the sphere? I am writing a java code where the ...
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1answer
28 views

Solving for $T$ in this Algebra problem

I'm taking Physics E&M and I'm trying to solve the following equation for $T$. I feel like the algebra I'm doing is correct but I cannot get the solution in the book which is $53.1 $C ...
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42 views

How do you take divergence of this field?

I forgot how to do divergences 3 years ago, this one is very confusing. We used to take them with respect to $x,y,z$ but this one doesn't have them. Help me with this question Prove that $∇.E = 0$ ...
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1answer
25 views

Relation between Diffusion equation and 1st order linear ODE

I'm studying PDE with a book written by Walter A Strauss. In sec 3.3, we tried to guess what is the solution of diffusion equation with a source. In analogy, they brought 1st order linear ODE ...
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2answers
138 views

How do I deal with reflections inside an ellipse?

Suppose I have an ellipse with foci $F_1$ and $F_2$. How do I show that any ray of light which intersects the segment connecting the foci will have subsequent reflections that always are tangent to ...
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1answer
44 views

Help deriving third order gaussian optics formula.

We are given: $$\frac{n_1}{l_0}+\frac{n_2}{l_i}=\frac{1}{R}(\frac{n_2s_i}{l_i}-\frac{n_1s_0}{l_0})$$ $$l_0=\sqrt{R^2+(s_0+R)^2-2R(s_0+R)cos(\phi)}$$ $$l_i=\sqrt{R^2+(s_i-R)^2+2R(s_i-R)cos(\phi)}$$ ...
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52 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
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28 views

Calculus Problems - Analysis [closed]

Thank you for your time! I have come across 3 problems I can't put together but I would like to understand. (The book has the solution, but not the actual step of how to get the solution...hence me ...