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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1answer
22 views

Expression for nano scale crack propogation

The following image is of a nano-scale self replicating crack on a thin film/substrate pair. I want to construct a "position" function for the crack front's propagation. So far the only way I can ...
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0answers
16 views

shifting integration variable and taking derivative seemingly giving problem

I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ...
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0answers
31 views

Clarification about consequence of physics' first principle of thermodynamics

I'm reading the book Physics, by Tipler, and I'm confused at the following statement: [...] From the first principle of thermodynamics, $\Delta U = Q+ W$. Suppose an ideal gas is given heat while ...
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0answers
22 views

Deriving E=mc^2 from hollow box with mass M and photon

I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the ...
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0answers
39 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 ...
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0answers
13 views

pressure field calculation from velocity field

Velocity field $v = (kx, -ky, 0)$. Initial condition is that pressure is $P$ at origin. I have to find pressure field, which is easy in this case. My problem is the gravity. In part a of the ...
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1answer
20 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
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0answers
9 views

Centre of pressure [on hold]

I am reading about Centre of pressure and have got one confusion : I read that Centre of pressure is independent of inclination with horizontal for a plane figure. In that case, by rotating the plane ...
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1answer
22 views

Understanding elastic collision between two rocks with unknown masses

I have this problem here that goes like this: Two curling rocks of equal mass, one red and the other yellow, are involved in a perfectly elastic, glancing collision. The yellow rock is initially at ...
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1answer
25 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 ...
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0answers
27 views

Make a mathematical model from the given temperature problem

Find the temperature $u(x,t)$ in a laterally insulated bar of length $L=3.14$ $m$ having the material property $k=c\rho$ whose ends are kept at temperature $0$ degree Celsius, assuming that the ...
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1answer
45 views
+50

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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1answer
55 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 ...
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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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3answers
269 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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0answers
43 views

Simple trigonometric division question

studying for a physics exam and a bit rusty on trigonometry having not done it for four years. I am following the answer to a friction of a car on a banked curve question and I am fine before and ...
4
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4answers
212 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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0answers
38 views

Relation between the tension of a rod and its length [closed]

The middle points of opposite side of a jointed quadrilateral are connected by light rods of length $l,l'$. If the tension in these rods are $t,t'$, then prove that $\frac{t}{l} +\frac{t'}{l'}=0$. I ...
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0answers
18 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,...
1
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1answer
35 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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2answers
60 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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1answer
1k views

Calculating the average force (mathematical physics)

I am trying to calculate the average force with regards to this question: Calculate the average force that USA gold medalist Allyson Felix (mass = $55.3$ kg, height = $168$ cm) exerted backwards on ...
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1answer
97 views

Largest number definable in $n$ symbols [closed]

Let $f(n)$ be the largest number definable using $n$ characters including spaces, in PA or some formal system. Then we can define $g(n)$ to be the largest number definable using n characters ...
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1answer
34 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
26 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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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 ...
2
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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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0answers
13 views

Collision laws (formulas) for colliding discs with rotation [migrated]

Say you have two discs with fixed radii $r_{1}, r_{2}$, positions $q_{1}(t), q_{2}(t)$, momenta $p_{1}(t), p_{2}(t)$, which both depend on time $t$ and fixed masses $m_{1}, m_{2}$. They are assumed to ...
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0answers
133 views

Demystifying invariant measures in probability theory

Just trying to understand, at least conceptually, invariant measures, and specially their role in probability theory. To be brief: I understand that if we have a set $X,$ with $A \subset X$ just a ...
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1answer
33 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. ...
9
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1answer
915 views

Damped Harmonic Oscillator and Response Function

This is another one of those questions that I feel like I am almost there, but not quite, and it's the math that gets me. But here goes: For a driven damped harmonic oscillator, show that the full ...
1
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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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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
14 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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0answers
24 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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2answers
1k views

Displacement vectors

You walk $53$ m to the north, then turn $60$ degrees to your right and walk another $45$ m. Determine the direction of your displacement vector. Express your answer as an angle relative to east. So ...
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1answer
43 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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2answers
411 views

Centre of Mass and Moment of Inertia of a sphere - spherical cap

I have been given a sphere of radius a, from this sphere a cap of hight h is cut off. 1) What is the centre of mass of the rest of the sphere? 2) What is the moment of inertia regarding the axis of ...
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0answers
34 views

differentials in physics [migrated]

Often I find the following expressions in physics books: Say we have a current density $\vec{j}=\rho\vec{v}$ through a surface $\vec{F}$ of particles $N$ in the volume $V$ with the density $\rho=dN/dV$...
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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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1answer
912 views

Acceleration and Tangential Velocity Vector?

Find the resultant acceleration of a particle moving on a circle of radius $0.70\ m$, if its angular speed is 37 rpm and its tangential acceleration is $2.9 \frac m{s^2}$. Express the angle with ...
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0answers
19 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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37 views

Space-time mathematics. Do I get this right?

Trying to understand general relativity like this, do I get this right? distance in $space \neq$ distance in $time$, if $space$ and $time$ = $2$ separate dimensions distance in $space =$ distance in ...
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1answer
27 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
24 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
52 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
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
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{...