# Questions tagged [geometry]

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

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### Confusion about a claim in the “Brief History of Time” by Hawking

In the first chapter, while talking about how Aristotle was able to conclude that the Earth is spherical, Hawking says that had the Earth been a flat disk the shadow of the on the Earth on the moon ...
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### If vector $a$ rotate about vector $b$, does vector $b$ also rotate about vector $a$?

For $2$ vectors $\, \vec a,\vec b$, both originate at $[0,0,0]$: If vector $\vec a$ rotates about vector $\vec b$ when observed from a coordinate system fixed to vector $\vec b$, does vector $\vec b$ ...
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### If body's movement is described with x(t) = cos(t), y(t) = sin(t), z(t) = At, why isn't the circumference of this curve (circle) equal to 1? [closed]

Each equation gives information about the body's location on each axis in Cartesian coordinate system (A is some constant and $t$ is time). We know that $\sin^2(x)+\cos^2(x)=1$ (Pythagora's theorem ...
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### Theoretical question regarding two elliptic trajectories

I am new to Newtonian Mechanics, and I was wondering regarding the following: Are $$\vec{r}(t): a \sin(ωt)\hat x +b\cos(ωt)\hat y$$ and $$\vec{r}(t): a \sin(ωt^2)\hat x +b\cos(ωt^2)\hat y$$ ...
33 views

### Consistency with calculating the Solar Azimuth Angle

I am struggling to find consistency regarding the calculation of the Solar Azimuth Angle. The equations across multiple publications have similar terms, but differ significantly. These differences ...
103 views

### Why Air bubble are always sphere in shape? [duplicate]

Everyone of us had noticed air bubbles once in his life. They are Sphere in shape. But I want to know why the are sphere in shape, instead of any other shape.
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### How is the constant “284” derived within the Solar Declination Angle equation?

How is the constant "284" derived within the Solar Declination Angle equation? The earth's axis results in a day-by-day variation of the angle between the earth-sun line and the earth's ...
122 views

### Reason behind vector addition law

What is the reason behind triangle law of vector addition, in other words, how is this really justified?
107 views

### Time after which the 3 particles will meet? [closed]

Problem is from Classical Mechanics: I am a beginner in Physics trying to understand the solution to the given below problem. Problem Statement: 3 particles are placed at the vertices of an ...
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### volumetric density from linear densities in the three directions x,y,z

I have a cube and I know the linear density of particles along each axis (I mean- the number of particles per unit length along each axis). How can I get the volumetric density?
59 views

### Finding angles in three dimensions [closed]

Suppose there is a line making an angle $\theta$ with $z$ axis in three dimensions. Then what will be the angles of that line with $x$ and $y$ axes?
117 views

### Maximum Vertical Velocity of Pendulum

Say there is a 10N pendulum held parallel to the ground. In a frictionless environment, it will continue swinging along a semicircular path. The pendulum tip should have the highest horizontal ...
51 views

### Geometry with differential angles

In the solution to a problem, the author considers the normal force provided by an arc length of string with a differential subtended angle size, $\textrm{d}\theta$. The author reasons that this ...
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### Why did staining marks on these paving slabs form geometric shapes?

[edited to add more examples] Along several streets in London I can see paving slabs that have become stained in the middle of the slab with white-ish marks forming different shaped patterns. The ...
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### Lorentz Factor from Minkowski's Original Paper 'Space and Time'

Consider the following figure: Minkowski, in his paper 'Space and Time', derives the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ from considerations of this figure. He ...
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### Is there a definition for a *geometric entropy*?

In statistical mechanics, entropy of a system is usually defined as a measure of the system's micro-state randomness, or as an averaged "surprise" of its micro-state: \begin{equation}\tag{1} S_{\text{...
1k views

### Is there a proof that the set of real numbers can exactly represent distances? [duplicate]

Mathematicians define real numbers in an abstract way - as an 'ordered field' with 'the least upper bound property'. In physics, we use real numbers to represent distances. For us to be able to do ...
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### What’s the topology of critical region?

Duhem said the aim of physics is natural classification. I think topology and geometry are a wonderful way to link analogous parts among different phenomena. Thus we can classify and predict facts. ...
97 views

### Method to build a polyhedral die with given probabilities [closed]

Let's define a die as a polyhedron that, if rolled over a perfect horizontal plane, ends up being in a physically stable unambiguous state labelled $n$. The die has $N$ states. Each state $n$ has a ...
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### A problem in newtonian physics [closed]

Hello! I have solved problem 89 using analytic way, i.e, the length of the string will always be constant, so repeated substitutions and differentiating will make way(i got answer as "c"). But is ...
66 views

### Can a black hole have a finite perimeter but an infinite radius/diameter?

Recently, I attended a talk at UNC Chapel Hill by Kip Thorne and I recall that he mentioned that black holes can have a finite perimeter but an infinite radius/diameter (since spacetime is curved with ...
41 views

### Change in area on a 3-sphere bounded by a trajectory due to a differential change in trajectory

I have a 3 dimensional spherical topology, and I draw a curve onto the sphere labelled by $\vec{n}(\vec{r},t)$. The area bounded by the curve is termed the "Wess Zumino Action" (Hence my motivation to ...
58 views

### Why is the angular part of the Schwarzschild metric is not affected by function of $r$ and $t$ as opposed to the first two?

Consider the Schwarzschild metric $$ds^2 = A(r)dt^2 −B(r)dr^2 −C(r)r^2(d\theta^2 +\sin\theta^2d \phi^2),$$ where $A,B,C$ are some functions of $r$. There is no loss of generality if one takes $C = 1$....
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### Definitions of area and volume in $d$-dim spacetime

I am reading Erik Verlinde's paper "Emergent Gravity and the Dark Universe". Equations 2.12 and 2.13 give the area $A(r)$ and volume $V(r)$ respectively. Do $d$ dimensions comprise of both space and ...
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### Inside a shell full of strings, what are the components of the network, and what the possible defects?

Imagine that we have a (very) large shell full of holes, and flexible strings go (randomly) from one hole to another. Inside the shell, there will be a network of strings. Can one split this network ...
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### What fraction of events goes through a small area $dA$ a distance $r$ from a spherically symmetric point source as $r\to 0$?

I'm not sure how to use limiting principle to estimate the N fraction when r -> 0 and small area dA stays fixed $\lim N_{r\to\infty} = \lim\delta A/(4\pi r^2)_{r\to\infty}$ = 1/2 N ? How?
156 views

### Number of GPS satellites required to give 100% coverage

I am a Belgian school student in sixth grade of secondary school (the Belgian equivalent of High school). Together with two class mates we were assigned to write some kind of essay about the Global ...
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### How is depth perception linked with the separation between our eyes? [duplicate]

I came across a comment online which mentioned that one of the main benefits of having 2 eyes is that we can perceive depth. I tried closing one eye and seeing for myself if somehow everything would ...
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A large plastic ball full of holes is given. (So the holes are in a plastic shell.) Straight threads connect these holes randomly, by passing through the interior of the ball/shell. For a big ball or ...
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### How to find the center of mass of an homogeneous cube? [closed]

How do I find the center of mass of an homogeneous solid cube of side $L$ analytically? I guess that by side $L$ means that the length of the sides is $L$ and the area is $L^2$, but I'm not sure. I ...
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### What makes displacement a vector?

Displacements are vectors because they add like vectors is the answer. It is also an experimental fact. Though rotations are displacements but not vectors. Is there any more fundamental or intuitive ...
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### Why is the idea that two points on a rigid body always correspond to the same distance so important in special relativity?

In Einstein's chapter on physical meaning of geometrical propositions (The special and general theory of relativity) he wrote about supplementing Euclidean geometry with the idea that two points on a ...
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### Similar Triangles in $a=\frac{v^2}{r}$ derivation

Looking at this image, it is said that the two triangles are similar. How are they similar? I can't seem to figure it out. I understand that they're both isosceles triangles, but how can I prove ...
27 views

### Geometrical way to view discretization of energy in quantum mechanics. How commutation relation implies discreteness?

The relation from which discreteness in eigenvalue of the energy of bound state arises is $[x, p]=i\hbar$ followed by the rule that wavefunction should be normalizable. So my question is there a ...
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### Geometric derivation of Lorentz boosts

In two dimensions a very nice parametrization of the rotation group is obtained by the following line of arguments: The group of rotations is connected and compact. Therefore the exponential is ...
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### Calculate thickness of curved membrane from confocal slices

I am imaging a thin, curvy, biological membrane using confocal microscopy. I would like to estimate the thickness of the membrane (assuming it is constant throughout). At every z-slice, I can see the ...
71 views

### Question about the geometric structure of Newtonian mechanics

My point here is about the mathematical structure of Classical pre-relativistic physics and general relativity (GR). It became more clearly, after GR, about the fact that pseudo-riemannian are a nice ...
31 views

### What should be the shape of an axisymmetric solar reflector that provides a uniform flux on the surface of a circular cylinder placed on its axis?

I was wondering about creating a solar concentrator with an antisymmetric concave shape that is able to focus the collected light onto a circular cylinder positioned on the concentrator axis. The ...
1k views

### When does a vector component keep being a vector, exactly?

English is not my native language, so please forgive my errors. Consider this example: This is a classic: an exercise requiring you to calculate the electric field produced by a charged ring on its ...
87 views

### Why do headlights through a window move around a room?

As I lay in bed this evening a few cars have passed by and my blinds are down. As they come down the street, you can see a sliver of light on the wall on the side of the room that is in the direction ...
312 views

### How to calculate the inertia tensor of a spherical cap?

In this question, an attempt is made at calculating the diagonal elements of the inertia tensor of a homogeneous spherical cap, where the $z$-axis is the symmetry axis. The mass moment of inertia ...
218 views

### Different expressions for distance & displacement : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r|, and |$$\int$$d$$\vec r|$

I came across these expressions in my book. And the book says that all these are different from each other. The expressions are : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r|, and |$$\int$$d$$\vec r|$ ...
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### Differential tetrahedron, area of slanted face?

The above diagram shows a cuboid containing a tetrahedron OABC. When the cuboid represents a differential volume element all sides are small in size. With the length of line $OA$ denoted by $|OA|$ ...
661 views

### Is it possible to find the volume of a 4 dimensional object? [closed]

I know there is 4 dimensional objects due to communicating with my friend group. I never found how would you find the volume because it was off the science channel.
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### How do I derive a formula for the water level inside a container when it is filled up at a constant flow rate? [closed]

When a container (e.g. Vase, Water Glass etc.) is filled up at a constant flow rate the water height changes differently over time depending on the shape of the container. This video shows some ...
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### The direction of torque is so confusing [duplicate]

When I was studying cross and dot products, I learned that the cross product of two vectors A and B is perpendicular to both A and B. But my mind is unable to understand that. Since both A and B are ...