[tag:classical-mechanics] entails the study of the trajectory of bodies under the influence of forces. More specific subtopics are: [tag:newtonian-mechanics], [tag:lagrangian-mechanics], [tag:hamiltonian-mechanics] for point particles and [tag:fluid-dynamics], [tag:statistical-mechanics] and ...
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0answers
44 views
Closed-form equation for orientation and angular velocity over time
If a rigid body, rotating freely in 3d, experiences no friction or other external forces and has an initially diagonal inertia matrix $\mathbf{I}_0$ (with $I_{11}>I_{22}>I_{33}>0$) and ...
0
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0answers
62 views
Small oscillations [closed]
I am asked to consider a fixed homogeneous rod of length $2L$ and mass density $\rho$ It is centered around $O$. A particle with mass M is moving in the same plane. The attractive force between the ...
2
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1answer
62 views
Invariance, covariance and symmetry
Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory? ...
1
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0answers
27 views
Doubling the energy of an oscillating mass on a spring [closed]
From this question:
Question 1.
What do we need to change in order to double the total energy of a mass oscillating at the end of a spring?
(a) increase the angular frequency by $\sqrt{2}$.
...
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2answers
47 views
Force applied in a body moving at high speed
Consider a rod of length $l$ and uniform density is moving at high speed. I want to deflect the rod where should I need to apply the minimum force, so that the rod is deflected..?
2
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1answer
84 views
Statics of Rigid Bodies — Can there be two possible solutions?
I've been working on a question and there seem to be two possible solutions. My own solution does not match the one given in the book. However, after resolving forces and taking moments with both ...
2
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0answers
40 views
When can a center of mechanical momentum frame be found for an electromagnetic system?
In classical mechanics, a center of mechanical momentum frame can always be found for a system of particles interacting with one another locally. For an electromagnetic system where the charges ...
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1answer
278 views
Goldstein's Classical Mechanics exercises solutions [duplicate]
Does anyone know where I can find some (good) solution of Goldstein's book Classical Mechanics?
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1answer
41 views
what's the center of mass for triatomic-molecule system
My text use the following example to explain the center of mass. There are three balls (mass $m$) sitting in the origin, at $x=l$ and $x=2l$, each two mass are connected with a spring of constant $k$. ...
3
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2answers
137 views
what's the physical significance of the off-diagonal element in the matrix of moment of inertia
In classical mechanics about rotation of rigid object, the general problem is to study the rotation on a given axis so we need to figure out the moment of inertia around some axes. In 3-dimensional ...
1
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4answers
64 views
is frictional force right or wrong
an experiment to disprove the statement--"frictional force is irrespective of the surface area in contact."
take a x rs note. fold it in a half and put it in the pocket of a shirt. then invert the ...
2
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0answers
32 views
When is classical mechanics valid for describing motion of atoms?
In Molecular Dynamics simulations, the Newton's equation of motion is used to calculate the time evolution of system. Once, I read in an introductory text that when the thermal de Broglie wavelength ...
12
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1answer
231 views
Classical results proved using quantum mechanics
Are there any results in classical mechanics that are easier to show by deriving a corresponding result in quantum mechanics and then taking the limit as $\hbar\rightarrow0$?
(Are there classical ...
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2answers
66 views
Resolution of vectors
What is the fundamental basis of resolution of vector. Suppose we have a vector $\vec{mg}$, now we resolve it into two components, horizontal and vertical. My question is what is the basis for telling ...
2
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3answers
139 views
Lagrangian mechanics and time derivative on general coordinates
I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
15
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5answers
274 views
Does the mass point move?
There is a question regarding basic physical understanding. Assume you have a mass point (or just a ball if you like) that is constrained on a line. You know that at $t=0$ its position is $0$, i.e., ...
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1answer
55 views
Hollow stone columns provide more support?
In history class in elementary school I remember learning that the Greeks would build their stone columns hollow because they thought this provided more support. Is it true that a hollow column is ...
3
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2answers
77 views
Runge-Lenz vector and Keplerian Orbits
Is the loss of closed Keplerian orbits in relativistic mechanics directly tied to the absence of the Runge-Lenz vector?
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1answer
39 views
Finding the acceleration at an angle
"What's the maximum acceleration you can achieve in a a water-slide at a 34 degree angle (If you can't use your arms and legs)"?
This is the free-body-diagram that I drew, assuming $g = 10m/s^2$:
...
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2answers
3k views
What is the exact definition of center of gravity?
I've come across many definitions. Is it
1) The point from which the weight of the body acts, i.e., the point at which if the entire mass of the body is assumed to be concentrated, the gravitational ...
5
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5answers
1k views
What sustains the rotation of earth's core (faster than surface)?
I recently read that the earth's core rotates faster than the surface.
Well, firstly, it's easier to digest the concept of planetary bodies, stars, galaxies in rotation and/or orbital motion.
But, ...
16
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4answers
1k views
What symmetry causes the Runge-Lenz vector to be conserved?
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...
3
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1answer
176 views
Can any physical rigid body be represented by an ellipsoid with the same angular dynamics?
According to wikipedia, the inertia tensor of an ellipsoid with semi-axes $a,b,c$ and mass $m$ is
$\left[\begin{array}{ccc}
\frac{m}{5}(b^2+c^2)&0&0\\
0&\frac{m}{5}(a^2+c^2)&0\\
...
17
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3answers
611 views
What's the real fundamental definition of energy?
Some physical quantities like position, velocity, momentum and force, have precise definition even on basic textbooks, however energy is a little confusing for me. My point here is: using our ...
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2answers
122 views
Hooke's Law, Phase Space and Classical Geometry
Hooke's Law tells us that $m\ddot{x} = -kx$. We can apply the chain rule to rewrite $\ddot{x}$ as follows:
$$\frac{\operatorname{d}\!^2x}{\operatorname{d}\!t^2} = ...
2
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0answers
59 views
Approximations in simple pendulum
In the approximation $$-(g/ \ell) \sin \theta \approx -(g/ \ell) \theta $$ we make an error $R$ which is $O(\theta ^3)$. If i did well my calculations it is estimated by $$R\leq|(g / ...
3
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2answers
106 views
Is a heavier skier faster? [duplicate]
Is it true that a heavier skier goes faster? If it is, why is that?
My intuition would be that the speed gained by a skier should be independent from its mass, since both its acceleration and the ...
0
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1answer
76 views
FWHM in resonance amplitude square derivation
Consider a linear harmonic oscillator subject to a periodic force:
$$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$
The solution tends to:
$$A \cos (\omega t - \delta)$$
where:
...
4
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2answers
177 views
Why is the Lagrangian quadratic in $\dot{q}$? [duplicate]
My teacher said we only consider Lagrangians which are quadratic in $\dot{q}$, and we don't take other Lagrangians. I couldn't understand why. Can anyone please explain this?
3
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2answers
75 views
Conservation of Linear Momentum at the point of collision
This is a pretty basic conceptual question about the conservation of linear momentum.
Consider an isolated system of 2 fixed-mass particles of masses $m_1$ and $m_2$ moving toward each other with ...
3
votes
1answer
280 views
Writing equation for amplitude of driven harmonic oscillator in Lorentzian form
This harmonic oscillator is driven and damped, with the form:
$$\ddot{x} + \lambda \dot{x} + \omega_0^2 x = A \cos(\omega_d t)$$
Now, I have used the ansatz (guess): $x(t) = B \cos(\omega_d t + ...
3
votes
1answer
88 views
Calculating the path of a ball with spin moving across a table
A ping pong ball is rolling over a smooth (but not frictionless) table. During its travel, a clockwise spin is placed on the ball. The ball's path is changed to move to the right (in perspective from ...
2
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1answer
123 views
Is there a geometrical way to obtain a relationship between these vectors?
Suppose we have a setup like this. Here $a_1,a_2,b_1,b_2$ are acceleration magnitudes($b_1,b_2$ being relative) and $P,Q,R,S$ are not pulley/blocks but are points on the rope. If I use a geometrical ...
3
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1answer
227 views
Intuition behind classical virial theorem
I am continuing to brush up my statistical physics. I just want to gain a better understanding. I have gone through the derivation of the classical virial theorem once more. I have thought about it, ...
3
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0answers
53 views
Reference Request: Classical Mechanics with Symplectic Reduction [closed]
I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
3
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4answers
289 views
Why does higher acceleration minimize a car's fuel consumption?
I generally try to optimize my car's fuel consumption when driving, using my car's real-time MPG gauge and average-trip MPG indicator.
Until recently, I believed the slower the acceleration, the ...
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3answers
6k views
Has any permanent magnet motor been proven to run?
I have read lots of articles about permanent magnet motors, some of which claim the possibility and other which refute it. Is it possible to have a permanent magnet motor that runs on the magnetic ...
2
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3answers
105 views
Relating generalized momentum, generalized velocity, and kinetic energy: $2T_{i}=p_{i}\dot{q_{i}}$
According to equation (6) on the first page of some lecture notes online, the above equation is used to prove the virial theorem. For rectangular coordinates, the relation
$$
2T_{i}=p_{i}\dot{q_{i}}
...
1
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1answer
55 views
Deriving equations of motion of polymer chain with Hamilton's equations
This is related to a question about a simple model of a polymer chain that I have asked yesterday. I have a Hamiltonian that is given as:
$H = \sum\limits_{i=1}^N \frac{p_{\alpha_i}^2}{2m} + ...
2
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1answer
53 views
Hamiltonian of polymer chain
I'm reading up on classical mechanics. In my book there is an example of a simple classical polymer model, which consists of N point particles that are connected by nearest neighbor harmonic ...
1
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1answer
46 views
kinetic energy of the stone
Suppose we have a man traveling in an open car (roof open) with speed $v$ towards right (man faces right). He throws a stone (mass $m$) towards right, in his frame-forward with speed $V$.
In the ...
2
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0answers
67 views
Classical scattering of two particles by a Yukawa potential
A point-like particle $A$, coming from minus spatial infinity, heads at another one, $B$, with an impact parameter of $b$.
Initial momenta are $p_A$ and $p_B=0$.
They repel each other via a Yukawa ...
3
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2answers
128 views
A partial differential equation for kinetic energy
The kinetic energy of a point particle of mass $m$ and speed $v$ is $K = \frac{1}{2}mv^2$. An elementary mathematics textbook I saw asked one to show that
$$ \frac{\partial K}{\partial ...
-1
votes
1answer
109 views
Lagrangian formulation for relativistic case
Lagrangian for a real scalar field:
$$\mathcal{L}=\frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^2\phi^2 $$
Can someone simply drive me how can I write it from ...
1
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4answers
202 views
Bat hitting a ball
When a bat hits a ball, consider two cases:
1) The batsman goes for a defense, and stonewalls it, to reduce its speed.
2) the batsman goes for a shot, e.g. a home-run, etc.
in which case will the ...
5
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1answer
215 views
Meaning of negative frequency of sound wave
Suppose that Alice and Bob are both holding speakers emitting sound at a frequency $f$. Alice is stationary while Bob is moving towards Alice at twice the speed of sound.
In the case of Alice, if I ...
1
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3answers
159 views
Must the Lagrangian always be known for the Euler-Lagrange equations to be of any use?
When studying classical mechanics using the Euler-Lagrange equations for the first time, my initial impression was that the Lagrangian was something that needed to be determined through integration of ...
8
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4answers
1k views
Are water waves (i.e. on the surface of the ocean) longitudinal or transverse?
I'm convinced that water waves for example:
are a combination of longitudinal and transverse. Any references or proofs of this or otherwise?
1
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1answer
47 views
Measure density with the help of buoyancy
I am trying to derive a formula to calculate the density of a irregulary shaped object.
I can measure the (false) weight of the object in pure air (of known density), and the (false) weight of the ...
1
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3answers
174 views
Is a quantum system mandatory for generating true random sequence?
Is a quantum system necessary if we want to generate true random sequence? The mathematical framework used for classical mechanics doesn't involve any random value. But the mathematical framework of ...
