Classical mechanics refers to the classical (i.e., non-relativistic, non-quantum) study of physics. Three major formulations of classical mechanics are newtonian mechanics, lagrangian mechanics, and hamiltonian mechanics. The latter two are rather useful in extensions to Classical Mechanics; ...

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First integrals for a particle in a central-force field

Consider an arbitrary dimension $n>3$. What are the independent first integrals for a particle? The Hamiltonian is $$ H = \frac{p^2}{2m} +V (|r|) . $$
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136 views

Mathematics of the Virtual Displacement

So I'm pretty certain this question has been asked to death here, but I still can't find a good explanation of a very particular aspect of the virtual displacements in physics. Background For ...
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142 views

Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let $...
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1answer
102 views

Find the separation distance for a line of oil being squashed between two flat plates [closed]

I was wondering if someone could give me some help on how to start this problem, I'm really struggling to get my head around it. A long line of oil is being squashed between two flat plates of length ...
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37 views

What is special about quantum entanglement? [duplicate]

Get two pieces of paper. In secret write the same number on both papers. Transfer one paper to the Moon. Look to the paper which is left at the Earth. Voila! We know what is on the Moon paper. The ...
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76 views

Efficiency of a gravity feed hose

If I am trying to fill a large cylindrical container with a gravity feed hose, would it be more efficient, time-wise, to drill a hole in the bottom of the container for a greater initial pressure ...
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176 views

Where does the energy go when you stretch a rubber band?

There is resistance when you stretch a rubber band. That makes sense to me because the energy you exert is turned into potential energy of the rubber band, but if you hold the rubber band in the same ...
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What exactly is a virtual displacement in classical mechanics?

I'm reading Goldstein's Classical Mechanics and he says the following: A virtual (infinitesimal) displacement of a system refers to a change in the configuration of the system as the result of any ...
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52 views

Hausdorff spaces and finite elements

Must the shape functions and the interpolation functions (which are the same in an isoparametric element) in a finite element model be elements of a Hausdorff space? If so, is this necessary to ...
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146 views

Poisson brackets and magnetic field [closed]

I'm a maths student trying to teach myself some physics so sorry if I'm missing something simple here. I think the main problem is lack of experience with the Levi-Cevita symbol. We have a particle ...
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115 views

How can we explain the difference in change of kinetic energy, due to different frames of reference?

Imagine a ball ($m= 1\,{\rm kg}$) moving at a velocity $2\,{\rm m}/{\rm s}$ towards a wall. When it hits the wall, it suddenly stops, thereby liberating all its ${\rm KE}$ as heat. Here, the initial ...
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1answer
367 views

Does a thermally expanding torus experience internal stress?

I'm trying to learn continuum mechanics and thermo-mechanics. As we know, heating an object increases the mean atomic distance $a_0$ of the atoms in a rigid body. Let's assume it is a linear elastic ...
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167 views

What was the motivation behind the work formula?

Surely there must be a reason we decided to use this as a metric for mechanical energy.How was it developed and what made it more acceptable than other work formula candidates (Like force over time, ...
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124 views

Scaling arguments for the Contact mechanics between two elastic spheres

I am studying a bit granular dynamics and I have seen that two spheres of radius $R$ in contact with a contact area of radius $a$ would need an applied force $F$ on this two spheres that is nonlinear ...
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1answer
117 views

Does a Buckyball spin like an electron or like a baseball?

Does a Buckyball spin like an electron or like a baseball? We are often told that an electron does not really spin like a baseball. Only one (or two, if you count up and down) spin states, for ...
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3answers
295 views

Scalar and vector defined by transformation properties

In Classical Mechanics, we are defining scalars as objects that are invariant under any coordinate transformation. Vectors are defined as objects that can be transformed by some transformation matrix $...
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385 views

Liouville's theorem and the preservation of topology

What might be a simple proof showing that the time evolution of the phase space volume can't lead to splitting off of the phase space volume? By Liouville's theorem, the total phase space volume is ...
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2answers
111 views

Is configuration space in any similar to vector spaces?

The question may sound silly. If it is I'm sorry for it but I just couldn't find an answer anywhere else. I have just learned about vector spaces and their properties and on the other hand have also ...
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2answers
2k views

Standing wave velocity

My question is simple: How is it that a standing wave has velocity? I mean, it's not travelling... A lot of equations depend on this concept, for example: $f_n = \frac{nv}{2L}$ Here we're ...
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409 views

Speed of sound in air

Quick question. I thought that the speed of sound in air was constant, say in the right conditions of pressure and temperature, and humidity... 300 m/s. Now, if I have a sound source that moves ...
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1answer
163 views

Why is centre of mass taken as integral of x.dm and not m.dx?

Forgive me if I'm being naive, but, I don't understand why the X-coordinate of the Centre of mass is taken as an integral of x.dm and not m.dx. I understand the summation part, but how do we convert ...
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3answers
139 views

Does sound have a “louder” direction?

I have a question about the propagation of sound waves. We have two TV's in our house that are almost right on top of each other. One is located on the first floor and the other one is located on the ...
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3answers
11k views

How to calculate the moment of inertia of a solid cube

How do I calculate the moment of inertia of a uniform solid cube about an axis passing through its center of mass? I also wanted to know if the moment of inertia ...
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81 views

why do things stick to surfaces when left for a really long time?

I tried to move a cardboard box off the top of a cabinet, which left it on for probably years, and it was insanely difficult to remove. I don't know why this happens, but it also seems that my feet ...
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1answer
212 views

How does height of a parachute affect air resistance compared to circumference or diameter?

I'm trying to find out how much a double in height (making it more ovular or oblong in shape) of the parachute affects air resistance compared to a double in circumference or diameter. Can someone ...
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3answers
257 views

Is there an intuitive explanation of the work formula?

Upon learning calculus, I decided it was time to derive all of classical mechanics to give myself a good understanding of physics. What I found was that, while trying to do so, I would need some ...
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Classical Statistical thermodynamics phase space and residue $h$

In classical statistical mechanics we have to divide the partition function by a factor of $1/h^n$. In almost every calculation of a real quantity this cancels out and is thought to be a remnant of ...
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117 views

Two Particles in a Harmonic Oscillator with repulsive short-range potential

Do bear with me, I am attempting to learn to write some simulations on the computer and learn some simple MD, so I defined sort of a toy problem. I have two particles confined in a Harmonic Potential ...
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1answer
74 views

Basic Notation Help Needed : Classical Mechanics, Unit Vectors

Can someone help me with some basic notation? Here's a situation where I'm surely missing some trivial piece of the puzzle: Example 1: given $W = \frac{1}{2}cpAv^2$ (air resistance), adding a unit ...
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2answers
501 views

The pressure in a container of water is based on depth. So what happens if I remove the bottom of the container?

So I understand that if we have a system that involves a container of water the pressure will equal atmospheric pressure at the top and as we go further down the container the pressure will increase ...
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2answers
1k views

Why does the Stern–Gerlach quantum spin experiment conflict with classical mechanics?

My understanding of the Stern–Gerlach experiment is that neutral (0 total charge) particles are sent through a non-homogeneous magnetic field, with the expectation that the field will push that ...
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2answers
96 views

How does friction act on a body, if only 2 regions on it are rough? [closed]

While tackling an Olympiad question, it came to my mind that friction need not act in the same direction at all points on a body. I thought of using integration to evaluate the net frictional force, ...
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0answers
71 views

How is the Routhian of classical mechanics defined?

The Hamiltonian is a function on the cotangent bundle to a configuration manifold $H:T^*M\rightarrow \mathbb R$. The Lagrangian is a function on the tangent bundle to the configuration manifold $\...
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1answer
74 views

Why do systems with a fixed gear-ratio still use gears?

From my understanding, there are two uses of a gearing system: to change the speed of output rotation (trading it with torque), and to change the axis of rotation. Now, in a car, for example, it is ...
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Why is sometimes more difficult to lift a baby?

I have a small cousin and she enjoys when I pick her up, which I can do pretty easily. Sometimes though she decides she wants to make my life difficult, and when she decides so, she tells me she is "...
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1answer
165 views

Chaos and integrability in classical mechanics

An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any ...
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1answer
58 views

Extension spring and permanent damage

Is there a way to calculate how far an extension spring can be extended before it suffers permanent damage? There are some online calculators, but how are they done? This calculator is the most ...
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3answers
202 views

What is the time period of an oscillator with varying spring constant?

It is well known that the time period of a harmonic oscillator when mass $m$ and spring constant $k$ are constant is $T=2\pi\sqrt{m/k}$. However, I would be interested to know what the time period ...
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92 views

How fast must a penny roll to remain upright?

I solved this question, however my professor's answer is different from mine. I modelled the penny as circular disc of radius $a$ so its moment of inertia is: $$I_s = \frac{ma^2}{2}$$ so $$I = \frac{...
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2answers
684 views

How do centripetal forces and gravity work for objects in a rotating cylinder?

The following is a question from a past exam paper that I'm working on, as I have an exam coming soon. I would appreciate any help. A fairground ride takes the form of a hollow, cylinder of radius $...
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1answer
290 views

Harmonic oscillator :Two masses are attached to one unfixed spring from both sides (vertically) [closed]

while ($t<0$) the system is still ($\Sigma$ F=0). Mass $m_2$ is held while $t<0$. Mass $m_1$ is located $h_0$ meters above the ground and the spring is currently stretched $L$ meters. The spring'...
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1answer
41 views

What is the Optimal Separation Length for the Tines of a Tuning Fork?

I'm building tuning forks (for fun... why not?), and among one of the design decisions is how far apart should I place the tines (the two long prongs) from each other. I'm not entirely certain whether ...
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1answer
83 views

Every central-force field is integrable, right?

In 3d, there are four independent first integrals, namely, the three components of the angular momentum, and the total energy. So by the Liouville theorem, it is integrable, right?
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207 views

Why is the Hamilton-Jacobi equation important? [closed]

Someone may say it is related to the Schrodinger equation. Okay, let us forget about quantum mechanics. So, if we confine ourself to classical mechanics, why is the Hamilton-Jacobi equation important ...
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1answer
68 views

Limits of integration [closed]

In the following video can someone explain why did he take the limits of integration to be from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ ? https://www.youtube.com/watch?v=bJWFgJTxIFk&index=9&list=...
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11k views

Calculation of the maximum load to the bar

Looking for a way of calculating the maximum weight (W) to the rod with the given length (L) where the rod did not break and that only bend for (b) mm. Need only approximative solution (read: ...
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95 views

Mechanics question I don't understand solution

A simple top consists of a heavy circular disc of mass m and radius a mounted at the center of a thin rod of mass $\frac{m}{2}$ and length a. If the top is set spinning at given rate S, and with the ...
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23 views

spin-orbit coupling for a rigid body

Consider the motion of a coffee cup in the gravitation field of earth. The force acting on the cup apparently depends on the orientation of the cup. Therefore, the internal rotation (with respect to ...
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2k views

Definition of generalised coordinates?

I think the definition of generalised coordinates is something along the following lines: A set of parameters that discribe the configuration of a system with respect to some refrence ...