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

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Calculating euler number of disk

I'm trying to do exercise 3.1 from Polchinski, which should be a rather easy differential geometry problem. I have to calculate the euler number defined by $$\chi = \frac{1}{4\pi}\int_{M}d^{2}\sigma ...
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0answers
8 views

How to calculate the inertia tensors of an irregular polygonal right prism?

I am attempting to program a physics simulation and need to do calculate the inertia tensors of a mesh constructed of a series of irregular polygonal prisms. Now, I am aware that you can calculate ...
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1answer
36 views

Intersection of two moving objects [on hold]

There are two objects. The first object moves with speed $U_1$ from known point A to known point B. The second object has speed $U_2$ and starts from known point C. What is the direction the second ...
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0answers
48 views

Is a ball noncompact? [migrated]

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
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0answers
31 views

Does “$\mathbf r[~t~]$” correctly denote the trajectory of a point particle wrt. a reference system w/o having to decide on a particular origin? [on hold]

Consider a specific geometric reference system, e.g. a specific inertial frame(1): the suitable set of point particles $\mathcal S := \{ H, J, K, ..., O, P, Q, ... \}$; and consider one other point ...
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0answers
9 views

How is the inertia tensor (just diagonal elements) of a quarter disk related to the full disk one?

Let be a disk of radius $R$ and mass $M$ with an uniform distribution of mass $\sigma$, centered in the origin O, laying in the $OXY$ plane. If I know that $$I_x=I_y=\frac{1}{4}MR^2, \qquad ...
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2answers
418 views

Finding the illuminance from a triangular light source

Since most light sources in games are point-like, it's pretty difficult to approximate area light sources with point sources. As triangles are a universal form to represent 3D models (thus area light ...
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4answers
218 views

Why perpendicular vectors do not share components?

I just can picture it in my mind or on paper. Can someone explain it with examples? This is the key idea behind the uniform circular motion: if the force has a component in direction of the object's ...
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2answers
86 views

How many are the points which are $n$th nearest to a certain point in a hexagonal lattice

Suppose there are infinite points arranged as hexagonal lattice. The question is the one as the title. First we choose a point called $A$. Then when we count the $n$th nearest points to $A$, what ...
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2answers
280 views

Sound source not in a straight line with the sound receiver - does that make a difference?

Hope the graphics will help me explaining my question better. Let's say the box would be a room in the fourth floor, and the sound source would come from cars in the street. It is clear that in ...
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1answer
59 views

Why is the elemental volume of a sphere equal to $4 \pi r^2dr$?

I was doing this question on calculating the electric field at a certain point in a sphere (length $r$ away from the centre), where the charge density is given by an equation. When I checked the ...
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2answers
58 views

Physical Meaning of Cone used in Conic Section for Orbital Mechanics

Does the polar angle (complement of $\theta$ below) of a cone which intersects a plane to yield a conic section have a physical meaning in orbital mechanics? Note that the angle of incident ...
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1answer
125 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 ...
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0answers
33 views

Alternative ways of calculating the principal moment of inertia

Let's say I am given 2x2 masses ($m$ and $m'$ have the same mass, just different coordinates): $m_1: (-a,b,0)$ $m_1':(a,-b,0)$ $m_2: (a,b,0)$ $m_2': (-a,-b,0)$ Due to symmetry the center of mass ...
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1answer
95 views

Trajectory of a swimmer trying to reach the opposite bank [closed]

I came across the following question 2 days ago: $Q$. A swimmer is swimming in a river with a speed $u$ with respect to the speed of water which is $v$. It is assumed that the river has straight ...
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1answer
103 views

How fast would one have to travel in an airplane in order to experience a continuous sunset?

For the sake of simplifying this problem and removing any guess work on how high the plane is flying, we'll say that our airplane is at 30,000 ft. as this is the average altitude for commercial ...
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1answer
34 views

Moment of inertia tensor calculation [closed]

If I have a rigid body consists of three uniform rods, each of mass $m$ and length $2a$, held mutually perpendicular at their midpoints choose a coordinate system with the axes along the rod. So I ...
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4answers
896 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 ...
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1answer
43 views

Standardising shadow length on sundials

The sundial is fundamentally flawed in that the length of each hourly shadow changes with the seasons. If the base of the sundial was engineered to move cyclically on an anual basis however, the ...
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2answers
52 views

What should the length of my wire be in order to get a surface area of 10cm^2?

I would like to make a flat coil out of a steel wire that is $1.22 ~\text{mm}$ in thickness (or $1.22~\text{mm}$ in diameter). I wish to make the coil up the wire into a flat disk that has a surface ...
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1answer
107 views

Constrained Motion of Connected Particles, Small Calculation Question

I am looking at this sample problem from Meriam Kraige - Engineering Mechanics Dynamics: When I do the differentiation, I end up with $\large -2\dot y+\frac{x}{\sqrt{x^2+h^2}}$ Where is the $\dot ...
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3answers
33 views

Foci of elliptical path

I have read the Kepler's laws of motion where I came across the term foci of elliptical path. But what is really a focus? Can anyone please explain what is a focus (maybe with a diagram) and why a ...
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2answers
697 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 ...
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1answer
43 views

Why do you seem to go faster as you hug a turn?

I was driving to work this morning when this question occurred to me. I was going up a clover-leaf entrance ramp to the highway. The person in front of me was lazily floating the outside of the curve, ...
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0answers
22 views

Circumference measurement using a capacitive sensor

I have a question which I hope someone in here can help me answer. Background: I’m building a device where I intend to use a stretchable capacitive sensor to define the radius change of an object. ...
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2answers
229 views

How were the ratios of distances between planets and the Sun first calculated?

I was reading some literature and I found that long before the actual distances between other planets and Earth or distance between Sun and Earth were known, physicists had calculated the ratios ...
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2answers
41 views

How to calculate centre of mass

How do I find the centre of mass with given coordinates? For example if we have four objects with mass $m$ at coordinates of a square $(0,0,0),(0,0,a),(0,a,0),(0,a,a)$ or another example with eight ...
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7answers
4k views

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 ...
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2answers
5k views

How does one calculate the volume of a nucleus and the volume of an atom (in this case hydrogen)?

The hydrogen atom contains 1 proton and 1 electron. The radius of the proton is approximately 1.0 fm (femtometers), and the radius of the hydrogen atom is approximately 53 pm (picometers).
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0answers
20 views

Constructing uniform mesh in reciprocal space?

This is a bit of a mental exercise for me to get comfortable with the math of reciprocal spaces since I am going to start doing some research that requires knowledge of reciprocal spaces. Let's say I ...
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1answer
73 views

How to derive path difference ($\Delta l=d\sin \theta$) for double-slit interference?

The Wikipedia page for the double-slit wave interference experiment states that the path difference between waves diffracting from the two slits is equal to: $$ \Delta l=d\sin \theta $$ where $d$ is ...
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3answers
378 views

Are circles stronger than triangles?

I've often herd in engineering that "there is no shape stronger than a triangle". I also recall that arches are very strong shapes as well, which can be crudely described as a ...
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0answers
172 views

Can the universe be round but still infinite?

Can the universe still be infinite in space if its curvature is > 1? Is a manifold of positive curvature necessarily compact? Does the Tarski paradox have any bearing on the finite or infinite ...
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1answer
279 views

Why are three parameters required to express rotation in 3 dimension?

We know that in spherical coordinates angle $\theta$ and $\phi$ (two angles)are enough to express three dimensional rotation of matrix. But to express rotation mathematically as a transformation ...
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2answers
420 views

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 ...
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0answers
26 views

Radian Zeno-version paradox: Can a particle moving in the circular path reach the angular position of non-terminating decimal form radian value?

Consider a particle P moving in a circle of radius r as shown in the figure. Premise 1: Position of the particle can be described by the angle $\theta$. Premise 2: Particle reaches the position ...
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4answers
2k views

Distinguishing between solid spheres and hollow spheres (equal mass)

If there are two spheres (hollow and solid) with equal mass and radius and we want to find the hollow sphere without using any equipment. What's the best way(s) to recognize the hollow sphere and ...
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1answer
261 views

Center of mass of equilateral triangle

I'm trying find the center of mass ($R$) of a uniform density equilateral triangle without using symmetries to find the x coordinate of the vector $R$, but I can't get the expected $R_x = ...
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1answer
65 views

Calculating the area of region of the sky

I am given the right ascensions ($\alpha_1$ and $\alpha_2$) and declinations ($\delta_1$ and $\delta_2$) of a specific region of the sky. How can we find the area of this region? I know that there ...
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1answer
481 views

Internal forces in a truss and its geometry [closed]

I'm to work out the internal forces in a truss, but I can't get my head around the geometry of the truss itself. I'm starting to think there may have been information on the diagram which I missed. ...
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3answers
1k views

How many times can light revolve around a black hole?

Take a light ray approaching a black hole from infinity which goes out again to infinity. What is the maximum finite rotation it can describe? (I know it can loop around indefinitely in the ...
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2answers
4k views

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 ...
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0answers
54 views

Justifying the use of real numbers for measuring length

I am not sure if this is the most appropriate place to post this but here goes nothing: Assume we were trying to come up with system of numbers $S$ to model our intuition of length. We want $S$ to ...
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10answers
2k views

Is it possible for a physical object to have a 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 ...
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1answer
45 views

Quantum physics and constructable numbers [duplicate]

I do not know much about quantum physics. However, I do know it believes the world is discrete ( has quanta). This seems to contradicts the fact that we can create an object of length root 2 since you ...
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0answers
25 views

Is the total length of the inner sheet of a two ply roll of toilet paper shorter than the outer sheet?

If you look at any individual square of two plies, each ply looks the same. But musn't the inner ply be shorter? Wouldn't the inner ply develop kinks if it was the same length as the outer ply? Or ...
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1answer
34 views

Electric Flux - Vector Components

How is $vcos(\theta)$ the perpendicular vector? I think I'm missing something fundamental in my trig and vector knowledge...
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1answer
94 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, ...
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0answers
36 views

Why is space 3 dimensional and could it be different? [duplicate]

Is there any particular physical law or property which caused space to be 3 dimensional, and is there theoretical possibility of having more or less dimensions in physical space?
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2answers
580 views

Spacetime geometry around two black holes

For space-time geometry we all know images like he one below. But if I were to take a neutron star and put it right next to but not touching a black hole what effect would that have on the geometry of ...