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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52 views

Work required to align pieces in a plane.

Given two piecewise continuous functions f(x) and g(x) and that $\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$ at all points, find the work used to shift each of the planar slolids ...
3
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0answers
107 views

generating functions for catastrophe theory

I am studying Thom's theorem in catastrophe theory and am having a hard time understanding what the "generating functions" actually do. How exactly are they used to classify generic caustics? The ...
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0answers
23 views

Design with Matlab the equation of the position of a drone

I want to design with Matlab the equation of the position of a drone which is $u=m(\ddot z_{des}+K_pe+K_v\dot e+g)$ where $e$ and $\dot e$ can be calculated from the current and desired states $(z,...
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1answer
38 views

Two blocks and a frictionless pulley problem

Block B ($m_{B}$=0.36 kg) is connected to a lightweight rope that passes over a lightweight, low-friction pulley.The other end of the rope is connected to Block A ($m_{A}$=0.72 kg), which is on a low-...
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1answer
26 views

What is the angular velocity in an inertial frame given the angular velocity in a body fixed frame?

At a given time t, the rotation matrix R has the value: $$R= \begin{pmatrix} 0.675 & −0.1724 &0.7174\\0.2474 & 0.9689& 0 &\\−0.6951& 0.1775&0.6967. \end{pmatrix}$$ The ...
5
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1answer
56 views

Schwarzschild metric, speed of ball as measured by observer who catches the ball, just before ball is caught?

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\...
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4answers
217 views

Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$ \operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases} $$ In physics, the sinc function has ...
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1answer
38 views

Given the parallel and perpendicular component of a vector in terms of another vector, how do you determine the tensor that connects both?

Sorry for the awkwardly phrased title, I wasn't sure how to properly word it. I want to do the following: I have a vector $\vec J$ and a vector $\vec E$ with the following relation (with the ...
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0answers
34 views

Center of a mass for linear $f$ over $E$ with $(x,y,z)\in E \iff (x\cos\theta-y\sin\theta,x\sin \theta+y\cos\theta,z)\in E$

Let $E\subset \Bbb{R}^3$ be a measurable set (i.e. $\int_{\Bbb{R}^n}1_{E}$ exists) and let $v(E)\ne 0$. Let $f$ be a linear function $f:\Bbb{R}^3\to \Bbb{R}$, and let $(x_0,y_0,z_0)$ be the center of ...
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1answer
83 views

$n$ electric charges on a circle

The following problem is of physical nature, but its core consists of pure mathematics, so I ask it here: Suppose we have $n$ electric charges $q$ on a circle. They can move freely around it, but ...
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1answer
35 views

Which is the proper way to calculate average speed.

Let's say we have a graph, a distance-time graph, and there are couple points on the graph. The points are connected to each other with different slopes, so with different speeds between the points. ...
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2answers
66 views

Breaking one stick and balancing it on another

Take two sticks (not necessarily of the same length). Break one of them at a uniformly random point, support the other one at a uniformly random point, and place the pieces of the former on the ends ...
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0answers
29 views

Calculating a rotation matrix: error and properties

Given an axis $u=\begin{bmatrix} \sqrt 3/3, \sqrt 3 /3, \sqrt 3 /3 \end{bmatrix}$ and an angle $\phi =\frac{2\pi}{3}$ I want to calculate the related rotation matrix: Well given Rodriguez’s formula: ...
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1answer
17 views

Verifying that the Rodrigues formula gives the same result as $Rot(x,\phi)$?

How to verify that the Rodrigues formula with $x$ as an axis of rotation and $\phi$ the angle of rotation with $u$ a unit vector along $x$ and $Rot(x,\phi)$ gives the same result? I only know that ...
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1answer
52 views

Mathematical expression of a rotation

I don't understand how my teacher defined an expression for the rotation adding up the two red vectors made up from the strong blue ones after rotation I especially don't understand how does the ...
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0answers
17 views

What is the minimum number of sets of Euler angles to cover $SO(3)$?

This is a question I was asked to answer from a drone-robotics check assignment. What is the minimum number of sets of Euler angles to cover $SO(3)$? $$SO(3)=\{R\in\mathbb{R}^{3\times 3}|R^TR=RR^T=I\...
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1answer
35 views

How to derive the calculus version of the formula for work done by gravity?

(BTW, I think it should be $r-\Delta r$ in keeping with the axis.) Let me first re derived the formula as in my book, then ask my question. Suppose a object is at some distance r away from center ...
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0answers
28 views

By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...
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1answer
47 views

Central force is planar in n-D

So in 3-D the differential equation $$\ddot{\bf{r}} = -\frac{f(r)}{m}\bf{r}$$ is shown to be planar by noting $$\bf{r} \times \dot{\bf{r}}$$ is constant. But isn't the differential equation planar (...
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1answer
72 views

A friend and I are trying to figure out who is doing more work benchpressing? Me? Him? Or is it the same amount? [closed]

Me and a friend have been working out for some time and have been bench pressing each week, increasing our weight. Once we started getting to heavier weight we noticed that he was able to get more ...
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0answers
27 views

Question about G-Forces and Circular Motion on a Human Centrifuge

I don't really understand what a G-force is and how it can be used to solve problems using the formula: $$T=mv^2/r$$ T is tension, m is mass (in kg), v is velocity (m/s) and r is radius of the circle....
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1answer
25 views

Vector magnitude notation

Is the follow equality true? $$\left| \overrightarrow {u}\right| =u$$ I wonder, because on AP Physics formula sheets, sometimes the magnitude of a vector is clearly denoted, while other times the ...
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0answers
25 views

Instantaneous rate of change of a three-dimensional parametric

In physics, it is common to define the horizontal position $x$ and the vertical position $y$ of an object as functions of $t$ and then us the formula $$\dfrac {dy}{dx}=\dfrac {\dfrac {dy}{dt}}{\dfrac {...
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1answer
46 views

The radius of the track.

A racing car completes $5$ rounds of circle in $2$ mins . It has uniform centripetal acceleration $\pi^2 t^{-2}$ then the radius of circle is?. I asked it on physics $SE$ but I dont know how to ask ...
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1answer
30 views

Accelerate to Max velocity, then decelerate to known velocity

I have an object traveling at a known velocity (Vi). It then accelerates (known A) to a known maximum velocity (Vmax), then decelerates (-A) to another known velocity (Vf). The total distance ...
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1answer
53 views

a question about circuits and diferential equation [closed]

Consider the DC circuit of Figure $ 4$. Inductance L satisfies $L = (R^2) C / 2$. Calculate: a) The differential equation for the charge $ Q (t) $ contained in the capacitor; b) The solution of ...
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1answer
25 views

Velocity as a function of position given acceleration as a function of position?

$v(x)$ given $a(x)$, where $v_0 = 0$ and $x_0 = 0$ I'm clueless. This is what I thought: $dv = v, dx = x$ $a(x) = \frac{dv}{dt} \frac{dx}{dx} = \frac{dx}{dt} \frac{dv}{dx} = v\frac{dv}{dx} = \frac{...
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1answer
34 views

Distance dependent acceleration evaluation

I have been working on a derivation that deals with 2 bodies of equal mass being attracted by an effect similar to gravity. I have gotten to a point of attempting to numerically evaluate parameters ...
0
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1answer
29 views

How to compute the pivot point of a rectangular cuboid to achieve a certain rotation?

Summary: For a video game project, I have an object (craft) that hovers the ground using a soft constraint. Imagine that on the picture below there is an invisible point above the craft whose ...
2
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1answer
58 views

Two mathematically similar shapes…

I've researched this question but explanations that I have found were either not thorough enough or simply seemed downright incorrect. I thought it looked quite simple when I first saw it but it ...
3
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1answer
53 views

Gauge condition equivalent to condition that coordinate functions satisfy wave equation to first order

Let $\eta_{ab}$ be the metric of special relativity and let $x^\mu$ be global inertial coordinates of $\eta_{ab}$. Let $\gamma_{ab}$ be a small perturbation of $\eta_{ab}$. How do I see that the gauge ...
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0answers
47 views

ODE inside ODE question

Given the equations of motion: $$ x'' = \frac{F - .375(\theta'' \cos\theta - (\theta')^2 \sin\theta)}2$$ and $$\theta'' = \frac{2g\sin\theta - \cos\theta (F+.375(\theta')^2 \sin\theta)}{1.5 - .375\...
2
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1answer
19 views

Finding equation of a bent sufficiently flexible cardboard of length $l$ fitting into a gap of width $m<l$

I was thinking about how the walls of a barrel is made then I realized it is someone like fitting a piece of wood of length $l$ in between some "gap" of length $m<l$. This would cause the piece of ...
2
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0answers
25 views

Related Rates: rate of change of the speed of sound with respect to temperature.

The speed of sound, v, in air is a function of the temperature, T, of the air: $v = 331.4 + 0.6(T − 273)$ with v in meters per second and T in kelvins Suppose the rate of change of air temperature ...
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1answer
20 views

Which formula is appropriate to use when calculating the displacement in this case?

The "reaction time" of the average automobile driver is about 0.7s. If an automobile can slow down with an acceleration of 12.0 ft/s^2, compute the total distance covered in coming to a stop after a ...
1
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1answer
17 views

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
21 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 ...
0
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1answer
29 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
26 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 ...
4
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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 ...
3
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3answers
27 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
23 views

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
47 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
119 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 $mc=\sqrt{E}$?
2
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3answers
60 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 $...
0
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1answer
29 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
50 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)+(\...