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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Decomposition of acceleration into normal and tangential components

If the velocity $v=\|\mathbf{v}\|$ of a point having position $\mathbf{x}(t)$ at time $t$ is never null, then acceleration $\mathbf{a}:=\frac{d^2\mathbf{x}}{dt^2}$ can be written ...
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1answer
37 views

Rewrite the system in the form $\dot x = Ax + bu.$

$Ml\ddot\theta = (M + m)\dot x + mL\ddot\theta = u$ $M\ddot x = u - mg\theta$ Using the variables $x_1 = \theta, x_2 = \dot\theta, x_3 = x, x_4 = \dot x$ Rewrite the system in the form $\dot x = Ax ...
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3answers
588 views

Trying to understand the true meaning of integral and Derivative in calculus [duplicate]

I'm solving a physics question, and i just encountered some question i had no idea how to start, i just got the right answer and inside it it has something in math i never thought possible, I know ...
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56 views

Why we have to take the vector $\overrightarrow{e}$ ?

The differential equation of the balance of the momentum is $$\rho \frac{\partial{\overrightarrow{u}}}{\partial{t}}=-\rho (\overrightarrow{u} \cdot \nabla )\overrightarrow{u}-\nabla p+\rho ...
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27 views

Law of hydrostatic pressure

For a calm fluid of uniform density $\rho_0$, that occupies the space $W \subset \mathbb{R}^3$, and is subject to massive forces (per unit of mass) $\overrightarrow{b}(\overrightarrow{x})$, write the ...
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1answer
18 views

Null velocity and piecewise smooth path

On texts of multivariable calculus and real analysis I have always seen the work made by $\mathbf{F}$ along the path $\gamma$ defined as the ...
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1answer
39 views

Stress vector - Stress tensor

Is the definition of the stress vector the following? The stress vector is the force per unit surface. The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the ...
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25 views

Find the streamlines of the velocity field

I have to find the streamlines of the following velocity fields: $$u=x(1+2t), v=y$$ $$u=xy, v=0$$ I have done the following: $$\frac{dx}{u}=\frac{dy}{v} \Rightarrow ...
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1answer
59 views

Dirac delta function with a sum as the argument

I'm reading "First steps in random walks" by Klafter and Sokolov, and I don't understand this step involving the Dirac delta function. They want to obtain the probability density of having a walker at ...
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86 views

Upper bound on the Lipschitz constant of entanglement entropy

I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full ...
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3answers
59 views

How to visualize rotation on a hyperbola?

I am studying Lorentz transform and I do not quite get what it means to use the hyperbolic matrix to rotation a point on a hyperbola, mainly it is because the hyperbola consists of two divergent ...
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0answers
45 views

Streamlines - Pathlines

Construct and draw the streamlines of the velocity field $u=az-bt, v=\frac{b}{4}z-cy, w=2(a-1)$. Calculate $c$ (as a function of the constants $a$, $b$) such that the flow field ...
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26 views

Electric motor fed with constant power.

This problem arose from a challenged to propel a model solar electric car for a distance of 20m in the quickest time. A solar panel is illuminated with constant brightness light to deliver electrical ...
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0answers
30 views

Acceleration/Position signal correction

I have a set of data for a car position, velocity and acceleration. % my data time car_x car_velocity car_acc The problem is that these arrays have error and I ...
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0answers
22 views

Show the relation $W$ is constant

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left ...
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1answer
54 views

What is $\int \omega^2 \ dt$? Integral on a centrifuge.

I was looking at a centrifuge, and I saw the following integral: $\int\omega^2\ dt$. I was wondering if this integral has any significance? Since this is a centrifuge, I would assume that $\omega$ ...
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1answer
44 views

How do we get the last relation?

I am looking at the conservation of momentum. The force at $W$ from the tensions at the boundary $\partial{W}$ is $$\overrightarrow{S}_{\partial{W}}=-\int_{\partial{W}}p \cdot ...
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2answers
34 views

Applying a constant force on pendulum and integrating for velocity

We apply a constant horizontal force $ F $ to a weight that's connected to a solid rod. I'd like to find out the velocity of the weight when it's at its horizontal position. See image below. My ...
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2answers
111 views

Is this correct or completely wrong?

I've bumped onto the problem described below and I couldn't tell if it is wrong. It looks like it could be correct but it makes no physical sense. Thank you! Marcelo! I begin with ...
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0answers
42 views

Concept of a continuum

I have to explain the concept of a continuum that is used for the description of the dynamic behaviour of the fluids, and to explain how this concept is related on the one side with the laboratory ...
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1answer
31 views

Derivitates of parameter-dependent functions of operators

This is a problem from quantum mechanics, but purely mathematical (I've set $\hbar = 1$): Let $\psi(\cdot, t) \in L_2(\mathbb{C}^3)$ for all $t\geq 0$ and $\hat{H}$ a bounded, hermitian linear ...
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2answers
62 views

Pre-calculus - Deriving position from acceleration

Suppose an object is dropped from the tenth floor of a building whose roof is 50 feet above the point of release. Derive the formula for the position of the object t seconds after its release is ...
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0answers
29 views

Angular momentum components in a given direction.

The exam question of interest is the following: Suppose the angular momentum of $\mathbf{v} = |1\ m \rangle$ ($j=1$, $|j \ m \rangle$ denoting the usual eigenstate of $J_3$ and $\mathbf{J}$) is ...
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1answer
50 views

Mathematics of contamination [closed]

I want to know the distribution of residual material (contamination) in subsequent refills. For example, suppose a cup normally used for transferring salt is used, without cleaning, for transferring ...
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1answer
68 views

Asymmetrical Center of Gravity Problems

Calculating center of gravity asymmetrical objects find the position of the center of gravity of relative to the edges AB and AC ...
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2answers
47 views

Projectile to clear a hemispherical mound

I have only just stumbled across this site a few days ago while searching for other things. This is a great resource. I have a question for nearly 30 years now when me and my friend were trying to ...
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1answer
32 views

Light attenuation through water at an angle

I know that light intensity decreases exponentially governed by \begin{equation*} \frac{dy}{dx} = -ky \end{equation*} where $y$ is the intensity and $x$ is the distance. Now what happens when light ...
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27 views

Finding the magnetic field

A mechanical pendulum is made of a metal rod, one end of which is attached to a string. The mass of the rod is $m$ and the string length is $l$. The oscillation period of the thus obtained pendulum ...
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1answer
58 views

Integration — Work done in pulling an elevator using a rope

An elevator weighing $3000$ lbs. is supported by a $12$ ft. cable that weights $14$ lbs./ft. Find the work a winch has to do by pulling the rope to lift the elevator $9$ feet. I eventually ...
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1answer
68 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
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2answers
104 views

Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
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1answer
27 views

Calculus Velocity and Acceleration

Here's the question: I know that: $a(t) = -9.8$ So I integrated the acceleration function to find the velocity: $v(t) = -9.8t + c$ And because $v(0) = -5$, I can determine that $c = -5$, thus: ...
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1answer
93 views

Example of using the Hadamard's matrix to determine the superposition

I've came across those notes for Quantum computation from John Watrous. I am having troubles understanding the last example. We have those two vectors, or if I understood correctly, from now on ...
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2answers
32 views

Freefall with terminal velocity - expressions for velocity and position

A body weighing 29 kg is dropped from a height of 30 m with an initial velocity of 3m/sec. Assume that the air resistance is proportional to the velocity of the body. if the limiting velocity is known ...
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1answer
53 views

Prooving Kepler's Second Law through vectors.

I am taking a multivariable calculus lecture online provided by MIT OpenCourseWare. ...
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1answer
38 views

Integral equation involving Planck radiation formula

I am stuck in solving the following integral equation: $$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$ where: ...
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0answers
45 views

Tension on the string of a drawn bow.

An archer's bow is drawn at its midpoint until the tension in the string is 0.881 times the force exerted by the archer. What is the angle between the two halves of the string? *Edit: some context, ...
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21 views

What force should be applied to each axis given a forward angle and a left/right direction?

I have three parameters: up/down angle, left/right angle, and power. The up/down angle can range from 0-90. It determines the forward force applied to my object. 0 means it moves straight forward ...
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1answer
42 views

Invertibility of inertia matrices

The inertia matrix is defined, for a discrete body composed of material points $P_i$ of mass $m_i$ whose position vector $\overrightarrow{CP_i}$ with respect to the centre of mass $C$ has Cartesian ...
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2answers
95 views

Velocity and Acceleration

I'm currently studying applications of calculus to the physical world in mathematics. I don't understand how a particle with zero velocity can have a positive acceleration. If the particle is at rest, ...
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1answer
48 views

Physics/Math conversion

I'm having difficulties solving this conversion. I know what to do at first but the way to set this up is a bit confusing:
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2answers
73 views

Mechanics problem. [closed]

A bottle is dropped from a blimp whose height is 1200m. Assume that a= 9.81m/s/s, find the time it takes the bottle to reach the ground.
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0answers
37 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
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0answers
29 views

Algebraic determination of asymmetric unit (aka irreducible wedge) in Brillouin zone of lattice

In Solid State physics the reciprocal space is of utmost importance to predict the band structure and thus most of the electrical transport parameters like effective mass, etc. The First Brillouin ...
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1answer
42 views

Impact of weight of the dice

If you throw a 6-sided die there is a probability of 1/6 to throw any specific value. However, this assumes that dice are exactly symmetrical, and we all know that they are not, in reality. Let's ...
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1answer
25 views

Calculate a jetplane's distance from my location

So i was sitting outside my workplace and saw this jet flying. I was really curious if there is a way to calculate the jet's distance between the jet and my location. (I have very little knowledge ...
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1answer
77 views

Trouble with a Statement in Arnold's “Mathematical Methods of Classical Mechanics”

On Pg 6 of Arnold's Mathematical Methods of Classical Mechanics (2nd Edition), there is a line which reads One can speak of two events occuring simultaneously in different places, but the ...
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2answers
48 views

Physics- Horizontal Displacement

Currently working my way through a maths problem, and I've hit a brick wall. A circus acrobat is launched from a cannon that is aimed at 75 degrees above the ground. He leaves the barrel at a ...
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1answer
38 views

Uniform Circular robot motion

Here is the initial diagram: The robot is moving counter-clockwise around the circle. I am having trouble seeing how they derived the centers of the circles. They have defined the center of a ...
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1answer
120 views

2 Second Pendulum Length on The Moon [closed]

This was a physics bonus assignment. I have been trying to solve it for the last 2 hours. Let's say that the density of The Moon and The Earth are the same. Moon radius:Earth radius is $0.273:1$! How ...