To be used for questions on geometry closely pertaining to physics. Includes differential geometry and euclidean geometry.

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2
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1answer
89 views

random particles on a grid: Effect of increasing density on distance between them

Say I have two boxes which both contain, say, 25 red particles (as shown in picture). These particles are randomly placed in a 2d grid, and in one the total area $A_{1}=20000$ and the other has area ...
7
votes
1answer
3k views

How to calculate roll, yaw and pitch angles from 3D co-ordinates (Euler Angles)

I have digitized a video of a flying fly in a 3-dimensional space. At all instants I know the x, y, and z co-oridinates of the following points on the fly's body --- The points are my choice, and ...
2
votes
1answer
162 views

Measuring the nearest order of magnitude

The world's largest ball of a string is about $R=2 m$ in radius. To find the nearest order of magnitude, what is the total length $L$ of the string in the ball? I have tried this in the following ...
2
votes
1answer
203 views

Relative strength of a dome [closed]

Is there a rough way to determine how the height of a dome affects the load that dome could support? For instance, assuming the bases of two domes are 24" in diameter, and one dome is 2" high while ...
2
votes
4answers
1k views

Electric Field due to a disk of charge. (Problem in derivation)

This might be a really silly question, but I don't understand it. In finding the electric field due to a thin disk of charge, we use the known result of the field due to a ring of charge and then ...
5
votes
1answer
222 views

Why is $S^1\times\mathbb{R}^{n-1}$ the topology of $AdS_n$?

Anti-de Sitter $AdS_n$ may be defined by the quadric $$-(x^0)^2-(x^1)^2+\vec{x}^2=-\alpha^2\tag{1}$$ embedded in ${\mathbb{R}^{2,n-1}}$, where I write ${\vec{x}^2}$ as the squared norm ${|\vec{x}|^2}$ ...
1
vote
1answer
425 views

Geometry in diagonal matrix and inertia tensor

For this problem, can anyone explain to me why when $x_1$ axis is aligned with the diagonal of the cube, the resulting inertia tensor will become diagonal? How to interpret this result geometrically? ...
1
vote
1answer
459 views

Moment of Inertia of a sector of a circle [closed]

I am trying to find the moment of intia about its centre of a sector of a circle of radius $a$, mass $m$ and angle $\pi/3$. I have found the answer it is $\frac{1}{2}ma^2$ but originally tried a ...
0
votes
3answers
111 views

How can I compute the angular velocity of a triangle formed by three particles knowing their instantaneous positions and velocities? [closed]

I have a set of trajectories of three particles and their instantaneous velocities. I would like to compute the 3 components of the angular velocity pseudovector of the fictive triangle formed by ...
5
votes
1answer
190 views

What's a geometric explanation for exponential-falloff fundamental forces?

Gravity and electromagnetism are inverse-square laws. This makes geometric sense -- if you build a spherical shell around a lamp then a shell with twice the radius has four times the surface area and ...
4
votes
2answers
198 views

Why do we need non-trivial fibrations?

I am currently reading this paper. I understand how the Bloch sphere $S^2$ is presented as a geometric representation of the observables of a two-state system: $$ \alpha |0\rangle + \beta |1\rangle ...
3
votes
2answers
144 views

How much of Minkowski spacetime structure can be recovered from its causal structure?

A beginner's question: I have always understood that (four-dimensional) Minkowski spacetime can be recovered up to a constant factor—i.e. 'up to a dilation' or 'up to global scale'—from its causal ...
3
votes
0answers
53 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
5
votes
2answers
144 views

What is the notion of a spatial angle in general relativity?

Is there a notion of spatial angles in general relativity? Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror ...
1
vote
1answer
306 views

How to calculate spatial distance in space-time?

Pinning two test particles at two different points in space, how can I calculate their spatial distance, when the geometry is given by the Schwarzschild metric? Let's say particle 1 is pinned at ...
0
votes
1answer
106 views

In Minkowski space, why does the hyperboloid appear to each observer as a circle whose radius is increasing faster than the speed of light?

I read the assumption in the above question in the paper Hyperbolic geometry on a Hyperboloid by William F. Reynolds (see here, page 444), but it was not clarified further (the discussion was rather ...
3
votes
2answers
97 views

Scattering geometry question

While reading up on light scattering I came across this slide: My vector maths is a bit rusty and I am having trouble understanding the last term (scattering geometry). What is the significance of ...
3
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0answers
267 views

Geodesic distance in de Sitter space

Consider $N$ dimensional de Sitter space embedded in $N+1$ dimensional Minkowski space: $$\eta_{\mu\nu}X^\mu X^\nu=1, \hspace{1cm}\eta_{\mu\nu}=\text{diag}(-1,1,\dots,1)$$ where I set $H=1$ for ...
1
vote
2answers
129 views

Are signal fronts in a beam not at rest to each other?

I'd like to investigate how the notion of "mutual rest" might be applied consistently, but distinctively, in the following thought experiment: Consider a light source ("$A$") which directs a beam ...
0
votes
1answer
194 views

Physics of Skiing: Ideal Carving Equation

I am conducting a research paper on the physics of skiing, specifically how ski parameters affect the ski's ideal carve. I have come across this paper (abstract link to arXiv paper), which is ...
0
votes
1answer
3k views

When to use Cosine or Sine? [closed]

Three blocks of masses $M_A=15.0 \; \text{kg}$, $M_B=12.0 \; \text{kg}$, and $M_C=8.0 \; \text{kg}$ sit next to each other on a frictionless surface. A force $F=54.0 \; \text{N}$ acts on Block $A$ ...
0
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0answers
21 views

Did a racer on a “restless” course race at some particular average velocity? (1: Starting blocks, finish line moving uniformly wrt. each o.)

Consider two participants, $A$ and $B$ moving uniformly wrt. each other; first having approached each other, then having met and passed each other, and eventually having separated from each other such ...
1
vote
2answers
235 views

How far would I have to go to see a fully rounded Earth?

How far would I have to go to see a fully rounded Earth? Recently, I saw a video on Youtube in which a sky diver called Felix Baumgartner ascends to $120,000$ feet (= $39$ miles) in a stratospheric ...
0
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1answer
128 views

find distance using a light line

Let's suppose we have a particular light frequency emitter and relative sensor array, and that there is no external source of this light. This emitter has a know angle respect the receiver, and emit ...
2
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0answers
41 views

Can a value of “length, in meters” be attributed to a pair of ends which are rigid (but not at rest) to each other? [duplicate]

The definition of the SI base unit "metre" [1] doesn't seem to rule out explicitly that a certain value of "length, in meters" could be attributed to a pair of ends which are rigid to each other, but ...
3
votes
2answers
1k views

Differentials in Spherical Shell - Maxwell Distribution

In explaining the Maxwell distribution of molecular speeds, my pchem textbook uses the following figure: We are basically trying to find the probability of having a particle with a speed $u$ between ...
2
votes
0answers
72 views

If two ends were a certain “length” apart were they therefore at rest (or at least rigid) to each other? [closed]

Considering the definition of the SI unit of "length" [1] and [2 (" method a.")] I'm missing any requirements about the two "ends" of the required "path travelled by light" being "at rest to each ...
8
votes
4answers
968 views

What's the dimensionality of a solid angle?

I haven't seen this explained clearly anywhere. Solid angles are described usually as a fraction of the surface area of a unit sphere, similar to how angles are the fraction of the circumference of a ...
4
votes
3answers
476 views

Calculating the Center of Mass

We have a homogeneous body that looks like this: I have tried dividing the body into different parts using the following definition: R g * A = R 1 * A 1 + ... R n * A n I was thinking I could ...
3
votes
1answer
219 views

Can a very small portion of an ellipse be a parabola?

We consider that when a body is projected from any height from the earth surface with a speed lesser than the orbital speed ( tangentially to the earth surface at that point.) it follows an elliptical ...
1
vote
2answers
575 views

Is the triangle stronger?

I have a friend who cedes that a framework that is a triangle would be stronger than a square framework. But he maintains that a square solid would not be stronger than a triangle solid. Ie a set of ...
2
votes
0answers
118 views

Can we derive the relation between proper time and the spacetime interval?

In GR, it's usually taken for granted - or as a definition - that the time measured by an observer's clock is related to the geometry in a very simple way, $d\tau^2 = |ds^2|$. This is easy enough to ...
5
votes
3answers
435 views

Textbook on the Geometry of Special Relativity

I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and ...
3
votes
0answers
100 views

Sound - for purposes of vibration

What is the best way to distribute noise from more than one source (I'm envisioning a system with many), within a dome, with the ground as its primary target, at optimal frequencies and volumes to ...
8
votes
1answer
290 views

Understanding Calculus Notation in Physics

I have just started a first-year calculus-based physics course about electromagnetism and waves. I am having trouble understanding what calculus notation means in the context of physics. Here is a ...
0
votes
2answers
81 views

Spherical coordinate system

I do not understand how is $r^2 = x^2+y^2+z^2$ in spherical coordinate system. Can anybody give a simple derivation? I need to understand this in order to understand the lorentz transformation.
1
vote
2answers
710 views

Tensor of inertia

The tensor of inertia of a solid sphere is $I_{ii}=\frac{2}{5}MR^2$ about an axis passing through its CM. Why would the tensor of inertia of each hemisphere about that axis be ...
3
votes
2answers
235 views

Zwiebach quick calculation 2.5

I am working through Zwiebach's a first course in string theory. It's been a while since I did any math (or physics!), and I am stuck on the following problem (quick calculation 2.5 in the book): ...
0
votes
1answer
257 views

Making a cut trough a center of mass, can the masses of the pieces be equal?

Let's say point $P$ is the center of mass of an irregularly shaped object. If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same ...
1
vote
1answer
96 views

Geometric Interpretation of Rotated basis of Hamiltonian and collective Dicke states

Suppose I start with a basis of states for a two spin-1/2 particle system, namely, $\{\left|\uparrow\uparrow\right\rangle, \left|\downarrow\downarrow\right\rangle, ...
2
votes
1answer
146 views

Where is a closed form also exact?

I'm not very familiar with exterior derivatives. I've some trouble following argument (which is a part of a proof that if the Riemann tensor vanishes, $R^{\,\rho}_{\;\,\sigma \mu \nu}=0$, iff there ...
0
votes
1answer
281 views

Area moment of inertia of regular $n$-gons over polygon center $O$

Is it possible to consider the regular polygons ($n$-gons) as deformed circles and use a pseudo-polar coordinate system to calculate their moment of inertia over its center $O$. Inasmuch as I know (I ...
0
votes
1answer
99 views

virtual boxes mirror reflection

The problem statement Find the lightpath with a given starting and ending points via one or more mirrors. In those kind of problems you try image someone pointing a laser with a very narrow beam ...
4
votes
1answer
564 views

Problem in Youngs double slit experiment

This is from Young Double slit experiment. But How to prove the the two $\theta$ are equal, I meant, how $\angle EAD= \angle PEC$? I see from the both triangle have $90^0$ but what about others?
1
vote
1answer
128 views

Structure factor of crystals (X-ray crystallography)

How can one prove that the degree of each node in a distance graph must be at least four in order to obtain a unique solution to an exact distance geometry problem with sparse distance data? The ...
2
votes
1answer
3k views

Analytic solution for angle of minimum deviation?

Consider a simple prism with a prism angle $A$, angle of incidence $\theta_1$, angle of emergence $\theta_4$ and the first and second angle of refraction as $\theta_2,\theta_3$. the refractive index ...
0
votes
2answers
129 views

Density of a material in different dimensions

Hope this isn't something trivially wrong, I am a beginner in classical mechanics. But anyways, say there is a cube, of side length $a \ \text{cm}$. The volume of the cube is $a^3 \ \text{cm}^3$. If ...
8
votes
1answer
625 views

How did Eratosthenes know the suns rays are parallel?

Eratosthenes famously observed that the suns rays were perpendicular to the ground in one location, yet non-perpendicular to the ground at a location some miles to the north. On the assumption that ...
6
votes
1answer
527 views

Two formulas for a particle's acceleration

While on a class my teacher was taking about particle's motion in space. At some point she said the following: Consider that the particle's path is described by a curve in space defined by the ...
1
vote
2answers
102 views

Paralel light conditions after passing from a sphere

Is it possible to get such result that light will be parallel after passing from the sphere? what is the total condition for such result if possible? Thanks for answers