Recent content by TSny

For online calculation of the incomplete elliptic integral ##F[\phi, m]##, where ##m = k^2##, see this site at WolframAlpha. The link shows the calculation of ##F[0.3, 0.8]##.

I don't see much difference between the MTW (Darwin) calculations and your calculations using ##r_1, r_2, ## and ##r_3##. Darwin shows how the three roots can be expressed in terms of just ##R##, the distance of closest approach. (See post #2). The expressions used in MTW are easily shown to...

The relations between the three root ##(r_1, r_2, r_3)## and the quantities ##R## and ##Q## are given in the paper by C. Darwin referenced in Fig. 25.7 of MTW. You can read this paper online here if you register for a free account. Section 8 of the paper deals with the orbits of light rays...

From ##t = \alpha \sinh(\tau/\alpha)##, we have ##z = (t^2 + \alpha^2)^{1/2} = \alpha \cosh(\tau/\alpha)##. Use these to express the denominator of (3.59) in terms of ##\tau## and ##\tau'##. Note ##\mathbf x = z## and ##\mathbf x' = z'##. Using identities for the hyperbolic sine and cosine...

Yes, the Coriolis force is toward the right. In the video below, you can see that when the rotating frame rotates counterclockwise relative to the inertial frame of the room, the ball deflects "toward the right" relative to the person tossing the ball. So, the direction of the Coriolis force is...

Yes, I agree that the x-axis as drawn is bad. The x-axis (toward the east) should be into the page (perpendicular to the y-z plane). The angle ##\lambda## should be the angle between the equatorial plane of the earth and the z-axis, not the angle between the x and z axes. But the mathematical...

Your work looks correct to me (except, as pointed out by @kuruman, ##m## should not be in your expression for the acceleration). I agree with your answer of west.

Looks good. Recall @haruspex 's comment: If the charge, ##-q##, on the inner surface were uniformly spread, what would the potential at ##A## be due to this charge? Does it matter that the charge is actually not uniformly spread?

I'm not familiar with any treatments of thermodynamics that prove differentiability of ##U##. I think physicists tend not to worry too much about it. They just assume differentiability. For example, regarding entropy, Callen gives on page 28: So, Callen postulates differentiability of ##S##.

Even though we need quantum mechanics to describe atoms, we still assume that systems of atoms have well-defined energies and that the law of conservation of energy is valid. On page 17 of Callen, we read When work is done on an adiabatically enclosed system, we can imagine it being performed...

I think Callen’s conclusion follows from his discussions earlier in Chapter 1. The change in the internal energy of a system during a process equals the net amount of energy transferred to the system during the process. This follows from the principle of conservation of energy. For simple...

You can read a discription for the chart here. Click on the picture and zoom in.

I don't know. A highly reflective metal surface or mirror would not absorb very much. Each ray of light in the beam obeys the law of reflection at the curved surface.

You might experiment with fanning out the beam by reflecting off a smooth cylindrical surface. Here, I used a ceramic mug.

Very good. I'm glad I could help.