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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

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 ...


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 ...


2

Your scenario is actually a classic transient conduction problem tackled in undergraduate engineering heat transfer, so we can handle this scenario easily. I took the figure below and adapted a derivation from a popular heat transfer textbook by Incropera and DeWitt: In this figure a warm object is placed in a tank filled with a known liquid (the ...


1

All three are "correct", and all three refer to mass-energy equivalence discovered by Einstein. Equations (2) and (3) are algebraically identical, and are generalizations of (1). Equation (1) only takes into account an object's rest mass, whereas equation (2) also takes into account the momentum $p$ of the object, and (3) takes into account velocity $v$. ...


1

The photons released are individually massless, but all of them together have an effective mass equal to the original masses of the particle and antiparticle; see my answer here. This isn't some mathematical abstraction either -- you can put the photons in a reflective box and weigh it, and it'll have extra weight. It's safe to say that the phrase ...


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

But the acceleration is not a partial derivative! Its a total derivative, $\frac{\mathrm dp}{\mathrm dt}$, with a $\mathrm d$ instead of a $\partial$. Anyway, I guess you might want to read about the Hamilton-Jacobi equation.


1

This is an every day term, with no meaning for physics. In the Oxford dictionary for "energy" one gets: Physics The property of matter and radiation which is manifest as a capacity to perform work (such as causing motion or the interaction of molecules) In science fiction one might separate zero mass particles, like the photon and the graviton ( ...


1

Your question seems to be about heat transfer through convection. The formula that describes this phenomenon is: $dQ/dt=h*A*(To-Tenv)$ where $dQ/dt$ is the heat transferred per unit time, $A$ is the area of the object, $h$ is the heat transfer coefficient, $Ta$ is the object's surface temperature and $Tenv$ is the fluid temperature (temperature of air ...


1

Almost everything from the wikipedia page you link is just false, or at best very misleading. IMHO, that page was written by someone that doesn't know anything about quantum mechanics beyond what one could find in TV documentaries. "Not even wrong" came into my mind many times as I was reading the article. In quantum physics, a quantum fluctuation (or ...



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