Questions tagged [geometry]
To be used for questions on geometry closely pertaining to physics. Includes differential geometry and euclidean geometry.
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What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors?
I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean?
More specifically,
Why is it that dot product of vectors $\vec{A}...
55
votes
8
answers
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Why do we use cross products in physics?
We can define cross products mathematically like if we take two vectors, we can find another vector with certain properties but why do we use it in physics, if we consider a hypothetical physical ...
13
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13
answers
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Is it possible for a physical object to have an irrational length?
Suppose I have a caliper that is infinitely precise. Also suppose that this caliper returns not a number, but rather whether the precise length is rational or irrational.
If I were to use this ...
22
votes
7
answers
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Experimental evidence of a fourth spatial dimension?
As human beings, we observe the world in which we live in three dimensions. However, it is certainly theoretically possible that more dimensions exist.
Is there any direct or indirect evidence ...
7
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2
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Analytic solution for angle of minimum deviation? [closed]
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 ...
13
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9
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Gravitation is not force?
Einstein said that gravity can be looked at as curvature in space- time and not as a force that is acting between bodies. (Actually what Einstein said was that gravity was curvature in space-time and ...
10
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3
answers
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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 ...
26
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4
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Does the rotation of the earth dramatically affect airplane flight time?
Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S).
During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles ...
169
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2
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Why do sunbeams diverge even though the sun is much more than a few kilometers away?
Consider this picture of sun beams streaming onto the valley through the clouds.
Given that the valley is only (at a guess) 3km wide, with simple trigonometry and the angles of the beams, this gives ...
13
votes
5
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Geodesics: Straightest or Shortest? When and Why?
In classical General Relativity (meaning not modified) one can think of geodesics in two ways.
One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) ...
17
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4
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How is Chasles' Theorem, that any rigid displacement can be produced by translating along a line and then rotating about the same line, true?
Chasles' Theorem in its strong form says:
The most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about ...
10
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4
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How far into space does one have to travel to see the entire sphere of earth?
Virgin Galactic will take passengers aboard SpaceShipTwo as high as 65 miles above the surface of the earth. But from this altitude, passengers will only be able to see a certain segment of the ...
1
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2
answers
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Euler Rotations in Ordinary Space
I'm reading LittleJohn's notes on Rotations in Ordinary Space on Quantum Mechanics. Link: http://bohr.physics.berkeley.edu/classes/221/1011/notes/classrot.pdf. I'm trying the last question given in ...
7
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1
answer
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How is the equation of motion on an ellipse derived?
I would like to show that a particle orbiting another will follow the trajectory
\begin{equation}
r = \frac{a(1-e^2)}{1 + e \cos(\theta)}.
\end{equation}
I would like to do this with minimal ...
8
votes
1
answer
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Why is physical space equivalent to $\mathbb{R}^3$?
Why is physical space equivalent to $\mathbb{R}^3$, as opposed to e.g. $\mathbb{Q}^3$?
I am trying to understand what would be the logical reasons behind our assumption that our physical space is ...
2
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4
answers
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Why railroad tracks seem to converge?
I stand up and I look at two parallel railroad tracks. I find that converge away from me. Why? Can someone explain me why parallel lines seem to converge, please?
97
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4
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Seeing something from only one angle means you have only seen (what?)% of its surface area at most?
Is there a logical/mathematical way to derive what the very maximum percentage of surface area you can see from one angle of any physical object?
For instance, if I look at the broad side of a piece ...
14
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5
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Why does the light side of the moon appear not to line up correctly with the evening sun?
I live at roughly $(52.4^\circ,-2.1^\circ)$. On sunny evenings I've often looked at the Moon and the Sun and noticed that the light part of the Moon does not appear to line up with the Sun. For ...
7
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2
answers
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$N$ point charges on a sphere
When charges are released on sphere, what is the shape made by charges?
Two charges are on opposite points of one diameter of the sphere.
Three charges make a shape of an equilateral triangle.
...
13
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2
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Is Dyson Sphere a stable construction?
Suppose that a star is encompassed by a Dyson Sphere. Do we need a position control system for the Dyson Sphere to keep its origin always aligned with the center of the star? Will it stay aligned ...
8
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2
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Rotation in Higher Dimensions
In a world of three spatial dimensions plus time, every atom rotates around a line, the axis of rotation.
In a world of $N$ spatial dimensions where $N$ is greater than 3, must every atom rotate, ...
7
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3
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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 ...
6
votes
2
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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 $r=R$...
5
votes
5
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The effects of Lorentz transformation on shape
Imagine a solid 3D cube. Now imagine that this cube is traveling close to the speed of light. To what degree will the spatial geometric properties of this object (or in general of any 3D object) ...
4
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2
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Can Minkowski spacetime be redefined as a non-flat riemannian manifold?
Minkowski space time is defined in terms of a flat pseudo-Riemannian manifold. I have wondered if it can be redefined as Riamannian manifold and in the case what type of curvature would there appear.
...
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2
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Angular velocity and banking angle
In a swing ride in an amusement park, the angle and speed of a seat in circular motion can be modelled by the banking angle equation:
$$\tan \theta=\frac{r\omega^2}{g}$$
Since tan of $90^{\circ}$ or $\...
1
vote
2
answers
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Are signal fronts in a beam not at rest to each other?
I'd like to investigate how the notion of "mutual rest" might be applied consistently, but distinctively, in the following thought experiment:
Consider a light source ("$A$") which directs a beam ...
1
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0
answers
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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 ...
64
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12
answers
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Mathematically prove that a round wheel roll faster than a square wheel
Let's say I have these equal size objects (for now thinking in 2D) on a flat surface.
At the center of those objects I add equal positive angular torque (just enough to make the square tire to move ...
23
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9
answers
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Why can the cross product of two vectors be calculated as the determinant of a matrix?
The cross product $\vec{a} \times \vec{b}$ can be written as the determinant of the matrix:
$$\left| \begin{matrix}
\vec{i} & \vec{j} & \vec{k} \\
a_i & a_j & a_k \\
b_i & b_j &...
12
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2
answers
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Why sphere minimizes surface area for a given volume?
I was studying surface tension recently. Rain drops or bubbles of any kind which form are always of a spherical shape.
This is because the liquid tries to minimize the surface area as the molecules ...
9
votes
4
answers
1k
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Is there a geometric interpretation of the spacetime interval?
In Euclidean space, the invariant $s^2 = x^2+ y^2+ z^2$ is equal to the length square of the position vector $r$. This is easily understood and can be represented geometrically in a graph.
On the ...
6
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5
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Do perfect spheres exist in nature?
Often in physics, Objects are approximated as spherical. However do any perfectly spherical objects actually exist in nature?
5
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2
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Which causal structures are absent from any "nice" patch of Minkowski space?
Which "causal separation structures" (or "interval structures") can not be found among the events in "any nice patch ($P$) of Minkowski space"?,
where "causal separation structure" ($s$) should be ...
5
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4
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Why is the Pythagorean Theorem used for error calculation? [duplicate]
They say that if $A = X \times Y$, with $X$ statistically independent of $Y$, then
$$\frac{\Delta{A}}{A}=\sqrt{ \left(\frac{\Delta{X}}{X}\right)^2 + \left(\frac{\Delta{Y}}{Y}\right)^2 }$$
I can't ...
5
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2
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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 ...
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1
answer
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How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other
Given a flat region of spacetime as set $\mathcal S$ of events together with values of spacetime intervals (up to a common non-zero constant) for each pair of events,
$s^2 : \mathcal S \times \mathcal ...
1
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1
answer
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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
5
answers
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Why do floating water drops form spheres?
Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere.
How would one prove that? Is it equivalent to showing ...
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1
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Understanding Euler's rotation theorem
According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will ...
0
votes
2
answers
522
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If we shift vectors any which way in a Cartesian coordinate system, isn't the meaning of the vector changed?
According to 3Blue1Brown, as long as the magnitude and direction of a given vector are the same, even if we move it around in the Cartesian coordinate system, it is still the same vector.
Here, we ...
28
votes
2
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Experiment to show that we need two eyes to determine depth
I'm trying to understand Manishearth's experiment in the answer here,
To try this out, close one eye. Now hold your arms straight out, nearly stretched, but not completely stretched. Now extend your ...
21
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3
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Why can't a piece of paper (of non-zero thickness) be folded more than $N$ times?
Updated:
In order to fold anything in half, it must be $\pi$ times longer than its thickness, and that depending on how something is folded, the amount its length decreases with each fold differs. –...
20
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9
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Why do objects appear smaller when viewed from a distance? [duplicate]
Yes, I know all about perspective (I'm an artist). I even have some basic knowledge of descriptive geometry. I know how it works. My question is more about why it works.
I have a sneaking suspicion ...
17
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2
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2k
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The Reeh-Schlieder theorem and quantum geometry
There have been some very nice discussions recently centered around the question of whether gravity and the geometry and topology of the classical world we see about us, could be phenomena which ...
11
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2
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Can an angle be defined as a vector?
In Classical Mechanics angular velocity, angular acceleration, torque and angular momentum can be defined as vectors with clear advantages such as the possibility to use vector product to simplify ...
8
votes
3
answers
26k
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Why do far away objects appear to move slowly in comparison to nearby objects?
When we are in a moving train, nearby stationary objects appear to go backwards. In Physics, relative velocity can be employed to explain the phenomenon:
velocity of object w.r.t train = velocity ...
7
votes
2
answers
686
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Is Newton-Cartan theory really equivalent to Newton's theory of gravity?
It is often said that Newton-Cartan theory is a reformulation or perhaps a generalization of Newton's theory of gravity, and it is said that (given certain conditions/assumptions) the two theories are ...
7
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9
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Understanding Parallelogram Law of Vector Addition?
Recently I've been adding vectors using the Parallelogram Law and the maths is trivial. However, I can't understand the underlying principals. What allows us to move a vector such that the tail meets ...
7
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3
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Can I calculate the size of a real object by just looking at the picture taken by a Camera?
Can I calculate the size of a real object by just looking at the picture taken by a Camera? (I think people do that) i dont understand how? (from physics point of view)