special relativity - Which clock is the fastest inside an accelerating body?

The picture shows an accelerating spaceship with two clocks inside it. It is so far away from all other bodys that gravity is of no importance.

Will the bottommost clock be slower than the topmost one? Or will both clocks have the same speed?

general relativity - How to determine "timelike"-ness without using a coordinate system?

It has been stated here that:

we can say, without introducing a coordinate system, that the interval associated with two events is timelike, lightlike, or spacelike

.

This assertion appears at variance with

Therefore I'd like to know:
How can be determined whether the interval associated with two events (which are given or characterized and distinguished by naming, for either event, the distinct individual participants which had been coincident at that event) is for instance "timelike", without using any coordinates and coordinate system?

Is it correct that the interval associated with two events (given as described above) is "timelike" if and only if there exists at least one participant who took part in both of these events?

Is conservation of energy only for isolated systems?

I am thinking in the mechanical context.

Everywhere I research (e.g. Wikipedia) the law of conservation of energy is stated only for the special case of an isolated system. I am wondering if conservation of energy holds beyond that special case; it seems that it should. After all, if a property holds only under special cases then the property is not a law.

Reading Feynman's lecture 14 volume I, I understand that if only conservative forces act on an object its total energy remains unchanged. For example, a falling object subject only to gravity has a constant sum of kinetic and potential energies. However, the system consisting of just this object is not an isolated system because it is subject to the external force of gravity. It seems this is an instance of conservation of energy holding outside the special case of an isolated system.

Feynman argues that at the fundamental level all forces are in fact conservative. This implies that at the fundamental level conservation of energy applies to all systems. Is this true? If so, why is conservation of energy continually stated under the special case of an isolated system?

(this site's "energy-conservation" tag says "the amount of energy in a system is constant" implying the system need not be isolated, further confusing me)

Answer

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 conservation of energy, "the energy in a region changes by the amount of energy flowing into or out of a region, and energy in adjacent regions changes by the same amount". You could call this local conservation of energy, and is a much stronger statement. It not only tells us that energy is conserved, but it also tells us that energy can't disappear from a region and reappear far away. This is the kind of conservation of energy that Feynman is considering, so he can apply it to systems that aren't isolated.

electricity - How is current used, say, in a light bulb?

I understand that current is the flow of charge, not necessarily of electrons (which drift slowly). The way I imagine this is a long tube filled with balls that just fit within the tube - when you push a ball at one end of the tube, the ball at the other end is pushed basically instantly, even though none of the balls move particularly much.

A filament light bulb works because a piece of wire with high resistance heats up and causes light to be seen. In the ball-in-tube analogy, I guess this would be like having three tubes connected to each other - one full of air, one full of oil and another full of air. All would hypothetically be connected to each other (pretend the water had no way of filling into the air tubes).

Essentially:

[air][water][air] = [wire][light bulb][wire]

I don't understand how moving one electron and essentially causing charge to be propagated throughout the circuit can cause a filament light bulb to work.

Can someone explain this using the ball-in-tube analogy, or if I have some conceptual misunderstanding in the analogy, correct me?

electricity - What exactly is resistance and Ohm?

Ohm is defined as

"a resistance between two points of a conductor when a constant potential difference of 1.0 volt, applied to these points, produces in the conductor a current of 1.0 ampere, the conductor not being the seat of any electromotive force"

according to this wikipedia page. However, I still don't understand what exactly is resistance and ohm. Does resistance decrease the amount of electrons passing through a conductor per unit time or does it decrease the energy that a single electron carries or none of them? I know what Ohm law is and I also know the mathematical background of resistance but I can not understand the logic behind it.

For example, when we talk about a conductor that should be insulated by an electrical insulator we have to know how many ohms that is needed to determine the size of the insulator for a good insulation. But I would like to know how it is determined. Thus, I would be pleased if someone could explain the logical background of resistance and ohm. Thanks.

newtonian mechanics - Intuitive Explanation of Tippe Top Effect?

A friend showed me a tippe top (a special kind of spinning top) lately and asked me about the physics behind it. I thought about it for a while but cannot quite figure it out. So I will throw the question to SE. Below is a picture of tippe top:

                                       

So basically the strange effect is that when you spin such a top fast enough with its round surface touching the table, the top will wobble and eventually invert itself. There is a video of this effect at wiki commons and here is an illustrative picture:

                        

So my questions are:

1. Why is the top so unstable? Why does it invert?


2. Why does the initial angular velocity of the top matter? (Why do we have to spin faster than a certain velocity in order for the top to invert?)



3. What other shapes will lead to inversion?

I am looking for some intuitive explanations that do not involve too much mathematics. Heuristic approaches are welcomed!

special relativity - Does this count as moving faster than light?

I'm not familiar with any complicated physics equation, however I do understand some basics. Suppose there is two objects, both of them are moving away from each other in a 3-dimensional space, which they both have the speed of half the speed of light ($c/2$). Relative to the reference frame of object A, object B would be moving away from itself equal to the speed of light.

   A          B
<-----      ----->
  -c/2       c/2   to a 3rd observer
   0          c    to A
  -c          0    to B

This seems to break the law of "nothing can move faster than the speed of light". I believe that there is some other explanation for this case. I have done some research, and there are quite a few questions on this topic. Take this one and this one for example. However, they are about adding velocities together, which isn't quite the same as in the case I was describing.

This question is also similar to my case, however the two objects described are both moving toward the same direction, and I'm not shooting any beam or laser toward the other object.

Most probably in reality there are some extremely complex laws and equations which makes this question more complicated. However, the two objects are not interacting with each other and I'm not trying to "detect" the speed of the other object in the reference frame of A or B. Also none of the objects are moving at a significant fraction of the speed of light relative to the third observer, so special relativity doesn't apply(?) I just could not think of any reason why wouldn't B be moving away from A in the speed faster than light.

A simple explanation would be great.

Answer

In your frame of reference, it does indeed look as though the difference in speed between A and B is greater than $c$. But the question is - does A think that B is moving away at that speed? And the answer is "no".

There is a thing called the Lorentz transformation which describes how the observed speed of an object is a function of the speed of the observer; conveniently, this prevents the breaking of the speed limit of special relativity.

Without giving you the math (you asked for a "simple explanation"), there are two things that happen when you are in a moving inertial frame of reference: length contraction, and time dilation. Clocks moving relative to you seem to go slower, and distances become shorter. These changes are described in the Lorentz transform. Net result is that velocity is also changed - this is described in the Einstein velocity addition equation which states

$$u' = \frac{u+v}{1+\frac{uv}{c^2}}$$

When we put $u=v=c/2$, we get $u' = 0.8 c$, so no speed limit is broken.

The key to understand here (after I re-read your question I realized I needed to add this): it is OK for two things to appear to move faster than the speed of light relative to each other - for example, you can see two beams of light, one traveling to the left, and the other traveling to the right, and say "the difference in speed between these photons is 2c". However, there is no frame of reference in which you can observe anything moving faster than the speed of light - and THAT is the condition that special relativity imposes. Does that difference make sense?