newtonian mechanics - Moment of Inertia, why $r^2$and not $r$?

So my engineering mechanics book includes a brief discussion on area moments of inertia.

Unfortunately, the ensuing chapter is predominately computational in nature. I don't have a thorough grasp of where the equations for this come from. The equation in question is expressed as:

$$I_x = \int_A y^2dA \space \mathrm{ and } \space I_y = \int_A x^2dA$$

Because of the simple equation for torque $\tau=Fd$, it is easy to imagine that the angular momentum of an object being rotated about an axis is greater the further it's mass is from the axis.

Simply divide any continous object up into a set of differential elements and violá, you have an integral that can be vaguely matched to the angular momentum.

The problem, however, is I can't possibly derive this. In fact, it makes more sense to me to integrate over simply $x$ or $y$ rather than $x^2$ or $y^2$.

Due to a logistical nightmare, I have no access to a physics book to look this up. I would be grateful to anyone who could explain (ie, explain the motivation of) this to me!

quantum mechanics - Long time deviations from exponential decay in radioactivity

Are there any examples of common substances whose decay is not exponential?

We're used to thinking about radioactivity in terms of half-lives. This is a concept that makes sense only for a decay that is exponential. However, there are plenty of physics articles on the subject of non exponential decay. It seems to be theoretically ubiquitous. For example:

The decay of unstable quantum states is an ubiquitous process in virtually all fields of physics and energy ranges, from particle and nuclear physics to condensed matter, or atomic and molecular science. The exponential decay, by far the most common type, is surrounded by deviations at short and long times$^{1,2}$. The short-time deviations have been much discussed, in particular in connection with the Zeno effect$^{3,4,5}$ and the anti-Zeno effect$^{6,7,8,9}$. Experimental observations of short$^{10,11}$ and long$^{12}$ time deviations are very recent. A difficulty for the experimental verification of long-time deviations has been the weakness of the decaying signal$^{13}$, but also the measurement itself may be responsible, because of the suppression of the initial state reconstruction$^{2,14}$.

1) L. A. Khalfin, Zurn. Eksp. Teor. Fiz. 33, 1371 (1957), English translation: Sov. Phys. JETP 6 1053 (1958).
2) L. Fonda and G. C. Ghirardi, Il Nuovo Cimento 7A, 180 (1972).

10.1103/PhysRevA.74.062102, F. Delgado, J. G. Muga, G. Garcia-Calderon
Suppression of Zeno effect for distant detectors

So are there any examples of deviations from long time decay? If not, then why not? Is the theory wrong or simply impractical? And is there a simple, intuitive explanation for why long decays should not be exponential?

Answer

To be honest, the paper listed as reference 12 in the OP's quote (1) gives as good an answer to most of these questions as you can hope to get:

For example, taking $\omega(E)$ as a Lorentzian function for all E yields the well-known exponential decay at all times. However, in real physical systems, $\omega(E)$ must always have a lower limit, which is, for example, associated with the rest mass of scattering particles.

In other words: a system with a perfectly exponentially decaying probability of remaining in the initial state, of the form

$P(t)=e^{-t/\tau}$,

must necessarily have an energy distribution of that initial state that is exactly a Lorentzian:

$\omega(E)=\frac{1}{\sqrt{2 \pi}}\frac{2/\tau}{(1/\tau)^2+(E-E_0)^2}$

But this is a function that is nonzero on the entire interval $(-\infty,\infty)$, while any real system necessarily has a minimum energy. Using some fancier Fourier analysis (2), one can prove that this means that in the $t \rightarrow \infty$ limit, any decay must go slower than exponential, and it generally turns out to be a power law. As rob mentions, this non-exponential behavior can also be seen as the result of the inverse decay process becoming non-negligible, and that point of view is also discussed in (2).

Most of the theory and review papers on this stuff that I've seen dates back to the 50s-80s, and don't even speculate on experimental prospects. However, (1) from above claims to give the first observation (in 2006) of this power-law decay. The authors note that in searches for this behavior using radioactive decay, one must look for deviations at the level of $10^{-60}$ of the initial signal (!), so it's not surprising that no one has been successful. Instead, they use the decay of the excited state of some organic molecule instead, which is very broad compared to the resonant energy and as a result crosses into this nonexponential regime much sooner.

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As a final side note, the OP doesn't ask about short time deviations from exponential decay, but it's easy to see where those come from too. Using

$P(t)=|\langle \psi_0 | e^{-i H t} | \psi_0 \rangle|^2$, and Taylor expanding the time evolution exponential, one finds

$P(t)\approx 1-\sigma_E^2 t^2$,

where $\sigma_E$ is the energy spread that $\psi_0$ has among the states of $H$. This is a very general consideration that applies to pretty much any time evolution.

general relativity - Does Hawking radiation need an apparent horizon and when does it switch on during stellar collapse?

I've read that Hawking radiation is implicitly linked with the existence of an apparent horizon (1). This seems a slightly less onerous than linking Hawking radiation with a genuine bona fide event horizon and its dependence on the global spacetime properties.

However I've also read a number of papers that seem to me to suggest than Hawking radiation occurs during stellar collapse and can occur in, albeit physically implausible models, before an apparent horizon forms or even when an apparent horizon is miraculously avoided (2).

So my question is: what is the connection between Hawking radiation and the apparent horizon and for a far-off observer watching a star collapse, when would they observe Hawking radiation to "switch on"?

gravity - Event Horizon violability?

Is the "event horizon" of a black hole potentially violable? Black holes are commonly described as being unidirectional (matter / energy goes in, but doesn't come out), but is the event horizon of a black hole potentially deformable? That is, in the case of two black holes colliding (let's say), is there not a point when the effective gravitational field between the two black holes "cancel out", thereby causing a potential "escape path" for light traveling perpendicular to the axis the two black holes are on?

In a more complicated case, consider two black holes of similar mass orbiting each other in a stable configuration; since no matter or energy actually pass the event horizon of a black hole, would the area between the black holes be a potential zone for escape of energy? (I'm assuming minimal frame dragging effects here.) And since the black holes would need to rotate at a very high rate of speed to avoid actually colliding, would that not restrict the emission of energy to be along the axis of rotation?

A bit of clarification here; the primary question here is whether or not the gravitational force vectors of the black holes can properly cancel each other out at an appropriate point between them. If that's the case, then (in a static system) it can be easily determined that there is a plane through that point perpendicular to the axis along which the two black holes exist where the gravitational force vectors are deeply reduced (not zero, of course, except at the precise point where the masses perfectly balance; however, reduced significantly based upon the inverse distance from said axis).

In an (again static) two black hole system, orbiting material could enter a "figure eight" orbit; much of this material would pass through this "nodal point" (or very nearby it). The high speeds and volumes of the material would indicate a large number of collisions; some resultant (high-energy) ejecta from these collisions would result in some (small) quantity of ejecta traveling along the plane of reduced gravitational force. Now consider the (more dynamically stable) case of stably orbiting black holes; the plane of reduced gravitational force is itself rotating (about the axis of rotation of the black holes); this turns the "escape plane" (not entirely just a plane, it having some (small) quantity of thickness) into a line (again, not entirely just a line, but a very thin cone).

The primary point is that that the cancellation of force vectors effectively makes for a "dimple" in the event horizon of each of the black holes; that is, the existence of another strong gravitational attractor in close proximity deforms the event horizon of a black hole.

The question is, is there anything which invalidates this analysis? As mentioned before, frame dragging is a concern; is there anything else which would be a concern? (I suspect the frame dragging concern would actually be resolved by the mutual forced frame dragging, i.e. the holes' spins would "synchronize" because of the mutual frame drag effect.)

If this is a possibility, then does this not imply another means of black hole mass "evaporation" other than Hawking radiation? (And, in fact, wouldn't it imply the type of "polar jet" that has been observed?)

homework and exercises - Net flux through insulating cylinder

In a non-uniform electric field, say with field lines curving away from each other along x, will the net flux on a cylinder be negative or zero?

thermodynamics - Is gravity Maxwell's demon in disguise?

Maxwell's demon is a thought experiment in which physicist James Clerk Maxwell suggested how the Second Law of Thermodynamics might hypothetically be violated. Basically, a demon controls a small door between two chambers of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that fast molecules pass into the other chamber, while slow molecules remain in the first chamber. Because faster molecules are hotter, the demon's behavior causes one chamber to warm up as the other cools, thus decreasing entropy and violating the Second Law of Thermodynamics.

So what if you flip the image above 90% clockwise and assume the existence of gravity in the direction of that yellow arrow?

Inside a vertical cylinder full of gas, the pressure should be lower at greater heights (e.g., greater distance from the gravity exerting center) --akin to the atmosphere above our heads-- considering that the pressure p exerted by a column of fluid of height h and density ρ is given by the hydrostatic pressure equation p = ρgh, where g is the gravitational acceleration.

In our flipped image, A is the top half of the column, and B the bottom half. The column height above molecules in A is then of course less than that of molecules in B. Finally, due to the ideal gas law a difference in pressure between A and B corresponds to an equivalent difference in temperature between A and B.

Unless gravity indeed manifests itself as Maxwell's demon, thereby violating the 2nd law of thermodynamics; I must have missed something. What is that something?

nuclear physics - Why doesn't orbital electron fall into the nucleus of Rb85, but falls into the nucleus of Rb83?

Rb83 is unstable and decays to Kr-83. Mode of decay is electron capture. Rb85 is stable.

The nuclei Rb83 and Rb85 have the same charge, but Rb85 is heavier than Rb83. While gravity acts more strongly on Rb85, this is probably not the factor producing the stability of Rb85. So, why does the orbital electron fall into the nucleus of Rb83, and Rb85 is stable?

Answer

It is not a matter of "falling in": all s orbitals have non-trivial probability densities at the center.

It is about energy balance in the nucleus.

Kr-83 is a lower energy configuration than Rb-83 by enough to make up for the neutrino and the gamma(s). Evidently Kr-85 is not a sufficiently lower energy state than Rb-85.