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9

The energy of an element of a traveling wave is not constant. Halliday-Resnick-Krane is right. For a string of density $\mu$ and tension $T$ the kinetic energy of an element $dx$ is $$dK=\frac 12\mu dx\left(\frac{\partial \xi}{\partial t}\right)^2.$$ For the potential energy we have $$dU=Tdl,$$ where $dl$ is the stretched amount of the string. A small ...


5

The second derivation is correct, as explained by Diracology. However, the first derivation is 'sort of' correct, in the sense that the location of potential energy can be ambiguous. For example, consider the three following systems. A mass on a stretched spring. A mass sitting on a table. A charged mass next to another charge. These three systems have ...


4

Notice that in the finite approximation the vector PR is not perpendicular to the radius, so work is done on it. By the drawing, this work is of the opposite sign that that in QR, so they both compensate to zero. I'll leave to you to compute both if you are really interested. Another way to see it, the gravity force is derived from a conservative field, so ...


4

There are already good answers here, but I'm afraid that to the best of my knowledge, Diracology's (and indeed Halliday-Resnik-Krane's) expression of the potential energy is not correct. I would like to point to this paper by Lior M. Burko which focusses on the subtleties of the derivation of the kinetic and potential energy of the string as a whole and ...


2

I think your problem is that you didn't change the units in the constant g. It has a value of approximately $9.8ms^{-2}$. Notice that it depends on meters. To obtain the correct result, you should use $980cms^{-2}$. Notice that this constant is off by a factor of 100, so that the result (after the square root) is off by a factor of $\sqrt{100}=10$.


2

There are different ways of stating conservation of energy and accounting for energy, which can make the issue confusing. One such statement is "the total energy of an isolated system is constant". This is true, and is the simplest way to state conservation of energy. This form of conservation of energy is the earliest taught. There's another way of stating ...


2

This question is easilly answered by considering the gravitational potential of earth, and invoking conservation of energy. The potential is $V(r) = -G\frac{m_1 m_2}{r}$ and the kinetic energy of the moon will be given by $$ K = V(r_{moon}) - V(r_{impact}) $$ where $r_{moon}$ is the current distance from the center of the earth to the center of the moon, ...


2

Short answer: no. I'll give some context with the details of the simplest examples. In the context of conservation laws, "energy" refers to the Hamiltonian. In classical mechanics, a quantity without explicit time dependence is conserved iff its Poisson bracket with the Hamiltonian is 0. In quantum mechanics, quantities are promoted to operators on a ...


2

I assume you are asking about a light sphere falling from infinity towards a heavy one. Well, the potential energy at $r$ is $$P = - \frac{GMm}{r}$$ And kinetic energy is $$K = \frac{mv^2}{2}$$ Total energy is zero, so $P=-K$ or $$v^2 = \frac{2GM}{r}$$


1

The simple answer is that not only momentum but also energy needs to be conserved. This puts constraints on the number of balls that can be activated in the cradle. Note that this does not always give a unique solution either. But it enforces that $ n $ balls to $ n $ balls is a "stable" solution.


1

Work done by a central force is Zero. At every moment the force is perpendicular to the displacement of the test particle. If you see your diagram it's very easy to see that at the final and initial position of the particle the force is not in the same direction. What he actually does is to assume that P and Q are actually arbitrarily close. So now the ...


1

Einstein's equation does not indicate how mass can be converted into energy. It only says that mass and energy are equivalent. Nobody has as yet found a way to convert mass directly and completely into energy. The best that can be done at present is nuclear fission or fusion reactions. Fission is already well established in the nuclear energy industry. ...


1

Energy is conserved in an isolated system. The energy of the system is constant. But it can flow from one part of the system to another. Energy is not conserved in the part. The energy in the part can increase or decrease. No energry is created or destroyed. It just moves to a place where you stop counting it.


1

Instead of forces, you work with conservation of potential and kinetic energy. In this case, the kinetic energy that gives this force is $-\frac{Gm_1 m_2}{r}$. The difference in potential energy is transformed into kinetic energy. With forces, you'd have to go the long way around (integrate the acceleration over time to get the velocity and figure out the ...


1

Yes you are. If a force is conservative, its work does not depend of any path between any points $A$ and $B$. Since the work integral can depend only on the initial and final points themselves, we define $$W_{A\rightarrow B}=\int_A^B\vec F\cdot d\vec r\equiv U(A)-U(B).$$ Now define the mechanical energy as $E=K+U$ so that $$dE=dK+dU.$$ Suppose there are two ...


1

Energy is proportional to amplitude squared. The energy in the wave is spread out over the surface of a sphere. The area of this surface increases as the wave propagates outwards from the source and is proportional to $r^2$. So the intensity of the wave (power/area) decreases in proportion to $1/r^2$.


1

Yes, at the fundamental level all energy terms are normally either kinetic or potential energy. The only demonstration of this that I know of requires a tool called the Lagrangian, which you might not be familiar with. But maybe you can at least get a flavor of how it goes. The Lagrangian, very briefly, is a particularly useful way to represent all the ...



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