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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Natural Frequency and Damping Ratio

I cannot find a simple explanation of the damping ratio formula. The natural frequency for a spring mass system seems pretty simple: position, velocity and acceleration are given by: $$x(t)=Acos(\...
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1answer
19 views

uniform rod revolving around a vertical axis with given angular velocity and given length of rod

A uniform rod of given length and given angular velocity is revolving around a vertical axis. Clearly it can do so in a horizontal plane with respect to vertical axis. At what other angle can it do ...
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25 views

Predicting accelerating object collisions in a friction-less environment

Background I'm building a simple game where two ships are launching missiles at each other in space. Stuff got complicated when the ships started moving in a friction-less environment and the ...
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1answer
25 views

How to interpret summation convention?

In Landau and Lifschitz Mechanics, p. 99, we have (implicit) the equality $$\Omega_i^2 x_i^2 = \Omega_i \Omega_k \delta_{ik} x_{\ell}^2 $$ written with Einstein summation convention. The left hand ...
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1answer
62 views

Water flowing from a vessel with curved sides

Suppose a hole is drilled perpendicularly into the side of the beaker which is full to the brim with a fluid (say water). This will result in water spurting out, travelling in a parabolic trajectory ...
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3answers
26 views

How to show that $\frac{R_1R_2}{R_1+r_2}<(R_1,R_2)$ strictly using AM-GM inequality?

I was reading about parallel circuits in Physics.Equivalent resistance of $n$ resistors in parallel is given by $\displaystyle\frac1{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}$. I tried to ...
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0answers
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Problem of rectilinear motion [closed]

$a=-{\dfrac{C}{s^2}}$ where $C=gr^2$. Neglect all resistance.(a) Now,let a body start from rest at a distance $h$ from the surface of the earth (radius r). Choose the centre of the earth as origin. ...
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2answers
39 views

S0lve for $v(t)$ when there is a quadratic drag force [closed]

How do I get $v(t)$ in this problem? The motion of a falling body is governed by Newton's second law $\frac{d(mv)}{dt} = mg- F_d.$ Find $v(t)$ for $m=1kg$, $g = 4 \frac{m}{s^2}$ and $F_d =...
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1answer
44 views

Simple rocket model

I have a problem creating a model for a horizontal rocket flight. I want to model a rocket with constant force, drag constant and gravity. I also have to account for a changing mass and drag. I know ...
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3answers
117 views

Can $E=mc^2$ be also expressed as $mc=\sqrt E$? [closed]

As the title suggests, can the equation $E=mc^2$ be also expressed as $mc=\sqrt{E}$?
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3answers
59 views

Given a rocket with constant acceleration after t = 4, when will it hit the ground?

A rocket is launched straight upward. During the first four seconds of powered flight, its height is given by: $h(t) = 16.1t^2 − 1.75t^3$ The function is valid when $0 ≤ t ≤ 4$ $t$ in seconds and $...
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1answer
27 views

Finding acceleration of an object given height

The height of a moving object is given as a function of time. $$h(t) = 3.0 + 2.7 \sin(1.3t + 0.9)$$ $t$ is measured in seconds and $h$ is measured in feet. Given this, I've found the velocity and ...
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1answer
47 views

How can one tell if a PDE describes wave behaviour?

I have been looking at a lot of different non-linear PDEs which describe waves lately and have come to the realisation that I don't know what it is about these PDEs that make them behave like waves. ...
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0answers
20 views

Physical Object to Pseudo-Riemannian Manifold

It is well known that Lorentzian mainfold is studied in general relativity. So this raises my curiosity about How about the classical mechanics? Does it correspond to the manifold $\mathbb{R}\times ...
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1answer
25 views

Concerning the notation $\chi (U)$ in one of the hypothesis for some properties of curl and divergence

I have the following exercise: Let $U \subset \mathbb{R}^3$ be open, $X \in \chi (U)$ and $f \in C^{\infty}(U)$, prove the following: $$curl(\nabla f)=0 \\ div(curl(X))=0 \\ curl(f.X)= f.curl(X)+(\...
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1answer
48 views

Magnetic field by current in an infinite cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder of radius $R$, with its axis coinciding with the $z$ axis, entirely enclosed by the cylinder's lateral surface. Then, for any constant $\...
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2answers
41 views

Trigonometry word problem (involving wires)

A guy wire $78$ feet long runs from the top of a pole $56$ feet high to the ground and pulls on the pole with a force of $290$ pounds. What is the horizontal Pull on the top of the poll? I am not ...
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1answer
17 views

centroid of a right triangle

I'm asked to find the $M_x, M_y$ and the centroid of the shape created by the functions $5x/6$ and $x=6$ that has a density of $5$. I find $$M_y \int_0^5 \frac{5}{6} x^2dx = \left. \frac{5}{18} x^3 \...
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1answer
38 views

Finding when velocity is zero

Given trajectory $s(t) = 2t^3 - 15t^2 + 36t + 2$ find, when velocity $v = 0$. I'm doing this the following way: $$v = \frac{ds}{dt} = 6t^2 - 30t + 36.$$ Then making $v = 0$, i.e. $6t^2 - 30t + 36 = 0$...
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1answer
45 views

A projectile is launched with a velocity of 30 m/s at 40° to the ground. To two decimal places, what is its horizontal velocity in m/s?

A projectile is launched with a velocity of $30 \, m/s$ at $40^\circ$ to the ground. To two decimal places, what is its horizontal velocity in $m/s$? A ball is thrown downward from the top of a $20\,...
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2answers
43 views

What kinds of motion obeys a higher order form of angular motion?

Angular velocity $\vec{\omega}$ can be defined in terms of velocity $\vec{v}$ and position $\vec{s}$ as: $$ \vec{\omega} = \frac{\vec{s} \times \vec{v}}{\left\lvert s\right\rvert^2} $$ Constant ...
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0answers
29 views

Show a beam is in equilibrium given the stress tensor

Having some trouble with this: https://gyazo.com/0835bdaa8e01cb976765aac94555f6ef I know how to show that at x_2 = -h the surface traction is zero, but I'm not sure how to show it's in equilibrium? ...
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1answer
92 views

Integrate second order DE once

Given the vorticity equation $$\frac{D \omega}{Dt}=(\omega \cdot \nabla)\textbf{u}+ν\nabla^2ω$$ and $\textbf{u} = (−αr/2,v(r),αz) $ in cylindrical polars where alpha is positive constant. Find $\...
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0answers
34 views

About periodic trajectories of a Hamiltonian system

Consider a Hamiltonian system with Hamiltoniana $H (\mathbf{q}, \mathbf{p})$, where $H$ doesn't depend on time $t$. It is known that in some domain of phase space the trajectory of system are peiodic. ...
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3answers
268 views

What is the moment of inertia of a Gosper island?

We know that regular hexagons can tile the plane but not in a self-similar fashion. However we can construct a fractal known as a Gosper island, that has the same area as the hexagon but has the ...
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1answer
15 views

Velocity along curve given velocity in one axis

Let's say that I have some particle travelling along a 2D trajectory mapped out by some infinitely differentiable function $f$ (in this case $f(x) = x^{-1}$). I know the velocity of the particle ...
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0answers
36 views

(Differential Galois Theory) Where is the proof that the three-body-problem is unsolvable?

I'm looking for a proof, which shows that "the 3-body-problem" in physics is mathematically unsolvable. Does anyone know some URLs that contain a proof in mathematical detail? You know, in ...
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1answer
9 views

Finding the radius of frustum cone?

I'm trying to understand how can i find the radius from the figure , the solution of the radius is given in the equation but I'm not getting how its derived enter image description here
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1answer
51 views

Finding the distance between two moving objects

In this case there is a missile whose initial position is $A(30,40)$ with a velocity of $[50,30]$ and an asteroid whose initial position is $B(400,250)$ with a velocity of $[-20,-30]$. The position of ...
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2answers
60 views

Circuit Analysis problem (find the problem)

In this question, I know that $\text{C},\text{R},\text{T},\text{A}\in\mathbb{R}^+$ I've this circuit (the bottom of the resitor is connected to earth ($0$)): When I use Laplace transform I can find ...
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2answers
43 views

Second order non linear differential equation: Central force question

The problem is as below: I have derived that the particle satisfies the motion equation $$ \frac{d^2u}{d \theta ^2 } + u = \frac{F(1/u)}{mh^2u^2} $$ by Newton's Law, $u= 1/r$ and $h = r^2 \frac{d \...
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1answer
30 views

Unique ground state of Schrödinger Operators

I'm reading a book and there is an argument that the ground state of a Schrödinger operator is unique. The problem is I think the argument is complete non-sense! These are lecture notes by Witten, I ...
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2answers
18 views

If the height of an upwardly thrown ball can be approximated by $h(t)=h_0+v_0t-\frac{g}{2}t^2$, what is $\frac{h(t_2)-h(t_1)}{t_2-t_1}$ conceptually?

I know that it describes the secant through $(t_1,h(t_1))$ and $(t_2,h(t_2))$, but I wouldn't know how to interpret it in "real life" terms. I also know that if $t_1$ tends towards $t_2$, I get the ...
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1answer
79 views

Employing Newton's Laws with differential equations [closed]

Going through some problem sheets from previous semesters and can't find a full solution for this question so was wondering what the answers might be. A particle of mass $m$ moves on the $x$ axis ...
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28 views

Dissipation term in wave equation

If we're given a string with mass density $\rho$ in units $\frac{M}{L^3}$ with constant cross-section $A$, tension $T$ in units $\frac{F}{L^2}$, and whose length is $L$; and then we assume that the ...
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1answer
45 views

Ellipsoid moment of inertia matrix

Some background info: torque $\tau$ is defined as $$\tau = I*d\omega$$ Where $I$ is the moment of inertia matrix and $d\omega$ is an object's rotational acceleration. As I understand it, the inertia ...
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1answer
28 views

Calculate position with increasing acceleration.

So if calculating the change in an object's position (with a constant acceleration) is done with this equation: $o = vt + (\frac12)a t^2$ $o$ is offset from original position $v$ is starting ...
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1answer
64 views

Parabolic slide

Given a friction-less slide $y=x^2$, place a particle on the slide at $(1,1)$. The particle is acted upon by constant gravity $g= 9.8$ units/s/s. At what time does it reach bottom? The ...
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14 views

Modeling population density with PDE

If we know that the population density $u(x,t)$ in some lake varies as a function of $x>0$ and time $t$, where $x$ increases downwards with depth, and that the population diffuses with constant $D$ ...
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1answer
28 views

Vector Force Application

I have this question for Math StackExchange Site. Suppose you would like to cros a $132 ft$ -wide river in a motor boat. Assume that the motorboat can travel at $7.0mph$ relative to the water and ...
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0answers
21 views

Writing PDE in the form of convervation law

What does one need to know in order to write $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial x^3}=0$ in the form of a conservation law, which contains the ...
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0answers
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Why don't more celestial bodies exhibit higher-order rotations?

It is well known that the Earth spins on its axis. It is also well known that the Earth's axis also precesses, i.e. spins around a secondary axis, much more slowly. Less well known is that we have ...
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1answer
22 views

Computate the commutator $[p^n,x]=-ihnp^{n-1}$

Computate the commutator of $[p^n,x]=-ihnp^{n-1}$. With $p=-ih \frac{\delta}{\delta x}$ the impulse operator. $h$ stands for $\frac{h}{2\pi}$. Answer: I do it with induction over $n$. For $n=1$ it ...
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2answers
33 views

Two trains and a fly using the series

I'm having some difficulties dealing with this problem: A train starts travelling from A towards B. It's velocity is v. Simultaneously train starts travelling at the same velocity from B to A. The ...
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1answer
43 views

Pendulum loss/gain of time per day given : $\ddot{\phi}+\frac{g}{l}\sin{\phi}=0$ and max displacement $5^{\angle}$

Here is what i am given: The oscillations of a pendulum are described by the equation: $$\ddot{\phi}+\frac{g}{l}\sin{\phi}=0$$ where $\phi$ is the angle between the pendulum and the vertical axis, $l$...
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0answers
85 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\...
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1answer
27 views

What direction would a hinge's reaction force point?

In my homework question there is a ladder with the bottom touching the smooth floor and the other end is attached to a hinge. I need to draw a force diagram and use that to find the normal. There is ...
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1answer
41 views

Vortex flow - Surface Integral

Consider the vortex flow of a fluid of density $\rho$ where the fluid rotates with an angular velocity $\omega$ about the $z$-axis. Determine where a unit square $S$ on the $yz$-plane should be placed ...
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1answer
13 views

Dipole-Coupling Tensor: Electrostatic Dipole Moments

I've been struggling with this problem today. Here's an image of the question I'm attempting to answer. I'm relatively new to tensor algebra (I've been studying it for about a week or two), and I've ...
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2answers
14 views

Calculating quantities with regards to a pendulum

The set up is a pendulum of length 2m with a mass on the end at 0.5kg. The mass is released at a small initial angle of 6 degrees moving into harmonic motion. I need to calculate the angular ...