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Closest distance of approach

Rewrite the total energy as: $$ H = C + \frac{1}{2}\mu v^2 + \frac{kq_1q_0}{r}\;, \qquad(1) $$ where $r$ is the relative distance, $v$ is the relative velocity, $\mu$ is the reduced mass, and $C$ is a ...
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Closest distance of approach

Identical velocities implies zero relative velocity, meaning the objects are not moving with respect to each other - they are not getting any closer or farther from one another. Since the objects were ...
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Closest distance of approach

Say the two objects are moving to the right. As long as the one on the left is moving faster it gaining on the other and the the distance between them is decreasing. Once the one on the right is ...
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Total rotational energy of oval-shaped object

As requested, I’ll try to make my comment more accessible. For a rigid body rotating about a point, you can associate two vector quantities: angular velocity $\vec \Omega$ and angular momentum $\vec L$...
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How did Noether use the total time derivation to get her conservation of energy?

You have to use the Euler -Lagrange equation also $\left( \frac{\partial L}{\partial x}-\frac{d}{dt}\frac {\partial L}{\partial \dot x}= 0 \right)$
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1 vote

How did Noether use the total time derivation to get her conservation of energy?

There are two ways of seeing that the two equations can be combined to yield the desired result... The first one is simply to substitute $$\frac{d}{dt}\bigg(\frac{\partial L}{\partial \dot{x}}\dot{x}\...
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1 vote

Understanding conservation of energy in a pulley problem

I think you can argue as follows. We have a tension T acting on both masses. The variation of kinetic energy for $m_1$ is $$ KE^1_{f} - KE^1_{i} = \int_0^h (- m_1 g + T) \ dx, $$ while for $m_2$ we ...
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2 votes
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Yet another perpetual motion machine: does this imply large selective membranes are not possible?

Many devices that seemingly constitute perpetual motion machines contain two parallel paths that clearly provide excess energy each way. One example is a capillary wick that lifts water to be ...
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1 vote

Does a water heater consume more energy if constantly kept on than if it is made to heat the water on demand?

Yes, your reasoning is correct. Let's consider a cylinder: Diameter 0.5m Length 0.8m Internal volume 157l It has a total area of $ 1.65 m^2 $. Suppose it is insulated with PIR foam, thickness 5cm, ...
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6 votes
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Where does the excess energy go in this problem?

By postulating the collision is an inelastic collision, you postulate it to not conserve mechanical energy (that's what inelastic means). As this is your postulate and you haven't set a place for the ...
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-1 votes

How conservation of energy looks like in the moving frame?

Let' say there are two earths, on one of those earths a car is accelerating. Now the car says that the kinetic energy of the earth that the car is not pushing increases by a huge amount, and the ...
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How conservation of energy looks like in the moving frame?

Energy is not invariant across reference frames. From the car's frame you would correctly calculate the car's kinetic energy to be 0. The earth, on the other hand, would have massive kinetic energy. ...
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2 votes

How conservation of energy looks like in the moving frame?

From the car's point of view, there is a fictitious force, which does work. Therefore, the car sees extra potential energy. You can think of the fictitious force as weak constant gravity. Similar to ...
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Why did Noether use the Lagrangian for her conservation of energy theorem?

The very starting point of Noether's theorem (NT) is an action formulation (and hence a Lagrangian), and quasisymmetries thereof. Noether did not formulate her theorem with only energy conservation ...
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2 votes

Why did Noether use the Lagrangian for her conservation of energy theorem?

However, I know that the Lagrangian doesn't always equal energy. The Lagrangian never equals the energy, unless there is no potential $U$ so the problem is trivial. In many cases, we can show that ...
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Microscopically, are all collisions really elastic collisions?

Even at a grade-12 level, the answer is "no." It's perfectly reasonable to construct a collision where energy is conserved, but kinetic energy is not conserved. For an example you could ...
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Microscopically, are all collisions really elastic collisions?

In practice, ordinary materials have a mesoscopic structure (fibers, connections between supra-atomic structures, arrangement of atomic planes). The inelasticity observed in macroscopic bodies, like ...
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How is a motorcycle whip possible?

To me, this looks like it can be explained by the classic experiment of standing on a round table and holding a spinning bicycle wheel. If you stand on a turntable, take a bicycle wheel, hold it ...
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2 votes
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Why do we give so much importance to energy, i.e., the conserved quantity under time symmetry?

As a complement to Nickolas's great answer, note that throughout physics, conservation laws are so useful that even approximate conservation laws can be of enormous interest. Conservation laws let you ...
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Is the total gravitational energy in the sun greater than the energy that is produced by the sum total of the nuclear fusion contained therein?

I will address the title: Is the total gravitational energy in the sun greater than the energy that is produced by the sum total of the nuclear fusion contained therein? Two concepts are confused in ...
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Is the total gravitational energy in the sun greater than the energy that is produced by the sum total of the nuclear fusion contained therein?

Unlike a chemical chain reaction or a fission chain reaction I believe fusion cannot sustain inself. Fusion reactions in the sun proceed very similarly to how regular exothermic chemical reactions ...
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4 votes

Is the total gravitational energy in the sun greater than the energy that is produced by the sum total of the nuclear fusion contained therein?

To sum up: energy is conserved, fusion converts some mass into energy, this is exothermic, and the gravitational binding energy doesn't contribute much to the heat of the sun. The total gravitational ...
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1 vote

Is the total gravitational energy in the sun greater than the energy that is produced by the sum total of the nuclear fusion contained therein?

Stars sustain fusion for millions or billions of years. The density and temperature in the core of a star is high enough to sustain fusion at a rate that replenishes the outgoing energy. The energy to ...
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Statistical Analysis of motion under central force

i'd recommend reading this, the motion really depneds on the initial conditions of the particle: https://en.wikipedia.org/wiki/Classical_central-force_problem
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10 votes

Why do we give so much importance to energy, i.e., the conserved quantity under time symmetry?

At the end of the day, physics is an experimental science. If your experiment can't measure some effects, you might just as well ignore them. That is quite often the case with cosmological effects. If ...
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1 vote
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Are one-dimensional bifurcations dissipative or conservative?

In general, a dynamical system $\dot{x} = f(x)$ is conservative if and only if $∇·f = \mathrm{div} f = 0$ (in the entire phase-space volume of interest). This works irrespective of the dimension of ...
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Energy conservation in a system of two bodies

You choose the system. Once this is done then internal and external forces need to be identified. An internal force will always have a Newton's third law pair and an external force will not have a ...
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2 votes

Piano on the moon

both the strings on the piano and the structure of the piano itself will vibrate without radiating away any sound energy, but since they both possess internal friction, the vibrational energy will ...
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4 votes

Piano on the moon

The strings making up the chord and the body of piano will vibrate for a longer period of time since none of that energy will be used to create pressure waves in 1 atm of air.
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Conserving energy in newtonian mechanics

But we know energy is always conserved For mechanical systems, a more general statement is that the work of the net force is equal to the change of the kinetic energy. For example, if the object is ...
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1 vote

Energy conservation in a system of two bodies

Total mechanical energy (KE+PE) of a system is conserved if the sum of external forces acting on the system is zero. Consider the following systems: $m_1$ plus Earth $m_2$ plus Earth $m_1$ + $m_2$ + ...
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Energy conservation in a system of two bodies

Generally, only the total energy of an isolated system is conserved. You write No external force us acting on that system. meaning the system of earth and one of the masses... But it is not true: ...
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2 votes
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Where is the excess energy going to when a solar inverter is operating a photovoltaic (PV) panel away from the maximum power point (MPP)?

The answer is easier if you only consider a single cell, and apart from some details the same answer holds for an entire string of cells and then for an entire panel that consists of several strings. ...
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6 votes

Is David Tong incorrect in this remark about classical mechanics in his QM lectures?

Well, in Tong's defense he did insert the phrase roughly speaking to indicate that the remark is not precise. Concerning that the Lorentz force is not a conservative force according to the ...
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14 votes
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Is David Tong incorrect in this remark about classical mechanics in his QM lectures?

Indeed, I think that the statement in Tong's book is quite ambiguous (though it is not definitely false as I discuss below). In principle there is no relation between the possibility of a Hamiltonian ...
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1 vote

Conserving energy in newtonian mechanics

Energy is NOT always conserved. Energy is conserved ONLY in closed systems. If you take up a stone in your hand, the stone is not a closed system. If you put energy into a stone by accelerating it, it ...
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Conserving energy in newtonian mechanics

So, right now, you are considering the ball as a part of the system, which obviously will lead to contradictions in the conservation of energy in a similar, but not the same way as you have told. See, ...
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