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How can I interpret the normal modes of this mechanical system?

In this example all the linkage does is convert a displacement of $u_3$ at the top to a displacement of $-\frac ba u_3$ at the bottom instantaneously so you should not expect there to be any ...
Farcher's user avatar
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2 votes

How can I interpret the normal modes of this mechanical system?

The system you wrote is a system of DAEs (differential-algebraic equations) since the third equation is a algebraic equation representing an algebraic constraints. As the rod is massless, its "...
basics's user avatar
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1 vote
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Work Done by kinetic friction in Circular Motion

In circular motion, your total displacement $$\oint \mathrm{d}\vec{s}$$ is zero, correct. But for work, you are not integrating just the displacement $\mathrm{d}\vec{s}$, but the quantity $\vec{F}\...
CompassBearer's user avatar
3 votes

How can I interpret the normal modes of this mechanical system?

I have not attempted to do the algebra, but I presume that you have found that the coeffeicient of $(\omega^2)^3$ in your characteristic equation is zero. To understand what this means consider the ...
mike stone's user avatar
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-1 votes

Does a rocket moving in a circle expel exhaust at a greater velocity?

Accordingly, must not the exhaust now obtain the entirety of the spent energy? Yes. From the rocket's frame of reference, would the exhaust gases be perceived as exiting at a greater velocity than ...
BowlOfRed's user avatar
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Does a rocket moving in a circle expel exhaust at a greater velocity?

Energy is frame dependent. In a inertial frame that is momentarily comoving with the rocket, its kinetic energy is zero, and the kinetic energy of the gas is the same for both cases, if we suppose the ...
Claudio Saspinski's user avatar
7 votes

Does a rocket moving in a circle expel exhaust at a greater velocity?

This is an interesting question, because when it is moving in a circle, the magnitude of its tangential velocity is constant and its angular velocity is also constant. Therefore the total kinetic ...
KDP's user avatar
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4 votes

Does a rocket moving in a circle expel exhaust at a greater velocity?

The energy of the fuel does become the kinetic energy of the exhaust. Let us ignore the fact that the rocket becomes lighter as it burns fuel, and that it will eventually run out of fuel. The rocket ...
mmesser314's user avatar
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Doppler shift from a moving reflector and source if only the relative velocity is known?

can we determine the relative velocity with doppler effect? Yes we can. This is similar to the principle that that police speed radar works on, except that the police radar uses electromagnetic waves....
KDP's user avatar
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Doppler shift from a moving reflector and source if only the relative velocity is known?

In a manner it is possible as Doppler effect dictates that the received frequency of a wave changes based on the relative motion between the source and observer. In this case, the sonar acts as both ...
Joshua's user avatar
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Question regarding gravitational force as external force

I will use Lagrangian method for the simple reason that the system is complex so, and we need to understand better. The small ones stand for the object and the capitals for the ramp. The Lagrangian is ...
Dionysis Balasis's user avatar
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With two balls connected to a string find minimum upward velocity that can be given to one of the balls such that the other leaves the ground

This statement that the both balls have equal horizontal velocity is not correct. When string is vertical let velocity of A be v(A) and velocity of B be v(B). Then, mv(B)²/(L/2)=T+mg mv(A)²/(L/2)=T-mg(...
OP BOSS's user avatar
1 vote

In the flow equation in Lecture 9 from Susskind’s ‘Classical Mechanics’, why is there a negative sign? Why is there partial and regular derivatives?

He says that if the flow velocity $v_x$ in the $x$ direction varies across the box, then the net flow into the box in the $x$ direction (across two faces) will be proportional to $-\frac{\partial v_x}{...
hft's user avatar
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16 votes
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Areas with anti-parallel gravity in classical physics

If we let the $y$-axis point upwards then OP's 2D gravitational field is $$\vec{g}~=~ \begin{pmatrix} 0 \cr g~{\rm sgn}(x)\end{pmatrix}.$$ It has a non-zero curl $$(\vec{\nabla}\times \vec{g})_z~=~2g\...
Qmechanic's user avatar
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5 votes

Areas with anti-parallel gravity in classical physics

The potential that yields your provided force field is discontinuous. One possible choice would be $$V(x,y) = \begin{cases} gy & x\leq 0 \\ -gy & x>0 \end{cases},$$ which has a dicontinuity ...
Refik Mansuroglu's user avatar
0 votes

Equilibrium in physics

I think there are several possible answers for this problem. It is definitely true that the system will converge to its equilibrium state if you properly consider dissipation, as JalfredP answered. ...
Jun_Gitef17's user avatar
1 vote

Equilibrium in physics

In short, it has to do with dissipation. I will start with a very qualitative answer. A classical pendulum (or oscillator) does not exchange energy/heat with the environment so it is "stuck" ...
JalfredP's user avatar
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2 votes
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Question regarding gravitational force as external force

My doubt is if gravitational force is considered as an external force Yes, it is considered as an external force, an external conservative force. ...so how come can we apply Total mechanical energy ...
hft's user avatar
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0 votes

Question regarding gravitational force as external force

If you chose the system to be the mass and the wedge then the gravitational force due to the attraction of the Earth is an external force to your system and you should equate the work done on the ...
Farcher's user avatar
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6 votes
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Is it possible to understand in simple terms what a Symplectic Structure is?

At the most rough level possible, a symplectic structure (geometrically) is an even-dimensional manifold together with a preferred choice of two-dimensional planes which, taken together, span the ...
11zaq's user avatar
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Massless String Having Different Tensions

The reason that an isolated massless string cannot have a net force on it is that if there was a net force the string would suffer an infinite acceleration. System: string and pulley In the case you ...
Farcher's user avatar
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Massless String Having Different Tensions

We often assume the tension in a string, particularly a massless string, is the same throughout but we have to be careful that we are considering one string, not two. If you think of the classic toy ...
M. Enns's user avatar
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2 votes

Understanding the “source” of magnetic energy in a bar magnet

In addition to the good answer already provided I want to point out that simply exerting a force does not require any energy if the force does not act through any displacement. So a magnet could ...
M. Enns's user avatar
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3 votes

Understanding the “source” of magnetic energy in a bar magnet

Energy is the ability of a system to perform work on another system. One can indeed convert the energy of a permanent magnetic field into useful work by demagnetizing the magnet, but then the useful ...
FlatterMann's user avatar
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6 votes

Noether's theorem by a taste of logic

Noether’s theorem is a mathematical theorem. It connects conservation laws to the properties of Lagrangian functionals. If these properties are contradicted by experiment then this means that a ...
my2cts's user avatar
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5 votes

How much time does it take for an object to fall from space?

If the body has no angular momentum (no tangential speed with respect to Earth), what you want to solve is Newton's second law in this form: $$m \frac{\mathrm d^2h(t)}{\mathrm dt^2}=-\frac{GMm}{(R_{\...
Mauricio's user avatar
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2 votes

How much time does it take for an object to fall from space?

Assuming the fall is purely radial and neglecting any fancy rotation and coriolis effects, you would obtain a single differential equation: $$\frac{GM_e}{(R+h)^2} = -\frac{\mathrm{d}^2 h}{\mathrm{d}t^...
CompassBearer's user avatar
3 votes

How much time does it take for an object to fall from space?

The solution can be found on Wikipedia, the time $\rm t$ to fall from $\rm r_0$ to $\rm r_1$ is $$\rm t=\left(\sqrt{\frac{{r_1} \left(1-\frac{{r_1}}{{r_0}}\right)}{{r_0}}}+arccos\left(\sqrt{\frac{{r_1}...
Yukterez's user avatar
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4 votes
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QFT introduction: From point mechanics to the continuum

What do Peskin & Schroeder mean by $\dot{\phi}(\vec{x})$? How can we define a time derivative of a function, which depends on position only? Notation $\dot \phi$, when writing down Lagrangian/...
Ján Lalinský's user avatar
0 votes

QFT introduction: From point mechanics to the continuum

I know that the four derivative (and the four vector) inside the Lagrangian of (Eq. 2) is crucial to maintain Lorentz invariance, but that is the only thing that is not predicted by the four ...
hft's user avatar
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3 votes

Designing a thought experiment on Noether's Theorem

Experiments on crystals are translationally variant because the crystal structure is only the same up to translations that reproduce the same structure, in such cases there is "crystal momentum&...
mike1994's user avatar
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1 vote
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Non-inertial frames in quantum mechanics

Inertial forces are completely described, in Hamiltonian mechanics, in terms of an electric-like potential plus a magnetic-like vector potential. The Hamiltonian of a free particle in a non-inertial ...
Valter Moretti's user avatar
16 votes

In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?

The notation is a little sloppy from a purely mathematical point of view (although common in physics) so it might be causing a little confusion. To help clarify, it might help to use different letters ...
Andrew's user avatar
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3 votes

In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?

They are partial derivatives. From the chain rule, we have $$ \frac{\partial V(aq_1-bq_2)}{\partial q_1}= a V'(aq_1-bq_2),\\ \frac{\partial V(aq_1-bq_2)}{\partial q_2}= -b V'(aq_1-bq_2). $$ For ...
mike stone's user avatar
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2 votes

In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?

No. The notation means that the V on the RHS is a function only of one variable, and so its derivative is the simplest, one-variable derivative.
naturallyInconsistent's user avatar
1 vote
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Invertibility between generalized and actual coordinates

While it is true that you use 3 numbers for Cartesian coordinates, you do not use the full set of triplets. You use the subset that satisfies $x_1^2 +x_2^2 + x_3^2 = r^2$. This subset has two degrees ...
mmesser314's user avatar
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2 votes
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Meaning of $d\mathcal{L}=-H$ in analytical mechanics?

I think it is gibberish in your own words. $\frac{\partial}{\partial \dot{q}}$ cannot be replaced by $dt \frac{\partial}{\partial {q}}$ even in the most cavalier approach because $\dot{q} = \frac{dq}{...
John's user avatar
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1 vote

Meaning of $d\mathcal{L}=-H$ in analytical mechanics?

Apart from the fact that I am really skeptical about its mathematical validity, your replacement $\frac{\partial}{\partial\dot{q}}\rightarrow\frac{dt}{\partial q}$ makes little sense in the context of ...
paulina's user avatar
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0 votes

Why aren't all objects and their images same in size?

Dicing with infinities is usually dangerous. Start with the idea that a small part of the object, size o$\times$o, produces an image of size i$\times$i. Now let o become smaller and smaller. What ...
Farcher's user avatar
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1 vote
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Work performed by hydrostatic pressure

With a little less math, the power of a stress distribution $\mathbf{t}_n$ over the boundary of a volume $V$ is $$P(t) = \oint_{\partial V} \mathbf{t}_n \cdot \mathbf{u} \ ,$$ being $\mathbf{u}$ the ...
basics's user avatar
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0 votes

Is there an error in Susskinds' derivation of Euler-Lagrange equations?

Here are my thoughts (I may be wrong). The first equation states the action which is the integral of the Lagrangian with respect to time. Integrating is the equivalent of finding the areas of strips, ...
Bradley Peacock's user avatar
2 votes

Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?

OP asks an interesting question, which we rephrase as follows: Given an action $$S[q]~=~\int_{t_1}^{t_2} \!dt~L(q,\frac{dq}{dt},t),$$ are the stationary action principle (SAP) and Euler-Lagrange (EL) ...
Qmechanic's user avatar
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1 vote
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How to compute the vector field from a potential in the complex plane?

What did you plot exactly, I rather obtained something like this: which is compatible with the lecture. The velocity field is: $$ \dot z = w := \sqrt{E-V(z)} $$ with $\min V<E<0$. Since it is ...
LPZ's user avatar
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1 vote
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Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?

As I understand it you are wondering about the following: What will be the implications when Hamilton's action is evaluated over a time interval where the end point is earlier than the start point? I ...
Cleonis's user avatar
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What happens to the coefficient of friction as the normal force increases?

Well an old kemps engineers year book I took notes from, gave static coefficient values for cast iron on steel for a number of different pressures, which showed a rise in value with pressure increase. ...
John Grimes's user avatar
1 vote

Why the interaction between system and thermal bath does not affect the energy levels of the system?

First, let's recall that already in classical theory, Hamiltonian value is not necessarily energy value. Energy is usually defined by some definite expression motivated by a conservation law, which is ...
Ján Lalinský's user avatar
0 votes

Work performed by hydrostatic pressure

My intuition is that we are looking at a special case of Stokes' theorem, since we are relating surface integrals to volume changes. To this end, it seems clear enough to me that the differential ...
creillyucla's user avatar
3 votes

Would a nearby electron be attracted/repulsed due to the oscillating $\vec E$ and $\vec B$ field of a passing electromagnetic wave?

Suppose $$\vec E (x, t) = \begin{bmatrix}0 \\ E_{max} \cos (kx-\omega t) \\ 0 \end{bmatrix} $$ $$\vec B(x, t) = \begin{bmatrix}0 \\ 0 \\ B_{max} \cos (kx-\omega t) \end{bmatrix} $$ a moving electron ...
hft's user avatar
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Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates

There are some issues in your computation. It seems that you assume that you will be able to use identities from the 1D harmonic oscillator to solve for AA coordinates in this more general case. This ...
Void's user avatar
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1 vote

The conservative force

While @basics has the math right, I understand your question is about the physical interptetation. To answer this, we must understand what a force field is, since the definition of the rotation ...
paulina's user avatar
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