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28

What keeps a bicycle up is a variety of things, but it all comes down to the front wheel, which can move left/right. The bike is always out of balance, and if it starts to fall to the left you unconsciously turn to the left, which moves the point of support (the wheel on the surface) to the left, which arrests the fall and may start the bike falling to the ...


19

By the principle of relativity, you will not fall over – assuming that you know how to use the bike and you won't be deliberately "confused". The principle says that the laws of physics have the same form in all inertial frames that are moving by a constant velocity relatively to each other. The reference frame associated with the moving sidewalk is as good ...


16

In an ideal situation (no air resistance) there will be absolutely no difference in the place where the coin lands! Whether you toss the coin up from inside the train or while standing on the roof, the coin will land back in your hand (provided you've tossed it perfectly vertically). However, in practice, while standing on a fairly fast train's roof, there'...


16

The answer is simple: Maxwell's equations. Maxwell published his electromagnetic theory in the 1860s. This generated a huge schism in physics. Maxwell's electromagnetism was in direct conflict with Newtonian mechanics. There is no allowance in Maxwell's electrodynamics for the speed of the emitter or the speed of the receiver. The speed of light is constant ...


6

We have to be careful in stating exactly what we're going to allow ourselves to assume here. We need some sort of principle of relativity -- that the laws are the same for both observers. But we don't want to assume anything else a priori, right? For instance, we don't want to assume at first that rulers have the same length for both observers -- we need to ...


6

I suspect your confusion is because you're holding two conflicting notions in your head, something like: "Motion is relative, so physics works exactly the same inside a moving train car as inside a stationary one." and: "Directions are absolute, so if I throw the coin straight up from the roof of the train, it goes the same way whether the train is ...


6

Your prof is using slightly wrong words: the group is a Lie group of dimension 10, not order 10. A group's order is the number of its elements, which is here uncountably infinite. A group of order 10 is a finite group (and indeed there are only two possible groups with 10 elements). I'm not sure how much continuous group theory (Lie group) theory you have ...


5

If I remember correctly, assuming only a homogeneous and isotropic spacetime, on top of an arbitrary group structure, the only 4D spacetime symmetries that are allowed are either galileo group SO(3,1) (that is Lorentz group) SO(4) (that is euclidean 4D rotations). The relevant references where this was shown are (according to link below) J-M. Levy-...


5

how Galilean transformations which are wrong (are approximately correct) give the correct answer for k? The Lorentz prediction and the Galilean prediction must agree in the limit that $v \to 0$ (or in the limit that $c \to \infty$). This is because $v=0$ corresponds to no transformation at all, so they had better both agree there. So if you take the ...


5

In special relativity, proper acceleration is defined as $$ a = \frac{du}{dt}, $$ where $$ u = \frac{dx}{d\tau} = v\frac{dt}{d\tau} $$ is the proper velocity, and $$ d\tau = dt\sqrt{1-v^2/c^2} $$ is the proper time. So $$ \frac{d}{dt}\left(\frac{v}{\sqrt{1-v^2/c^2}}\right) = a. $$ If we integrate this over a time interval $[0,t]$, we get, if $a$ is constant, ...


5

Take the unprimed frame to be your and my rest frame. For some body we measure $v(t)$ and by differentiating it we get $a(t)$. Now consider another observer in the primed frame moving at constant velocity $V$ relative to us. because of the law of addition of velocities we know that the other observer measures the velocity of body to be: $$ v' = v(t) - V $$ ...


5

I'll point out the more detailed differences below, but a nice rule of thumb to follow for these is that since the Galilean transformation gets it's name from a man who lived several centuries ago, the physics formulation for them is more basic than the Lorentz transformation, which is a more modern interpretation of physics. That way you can remember that ...


5

Actually given that the first postulate says that all physical laws are the same in all inertial frames, you could replace the second postulate by the postulate: "Maxwell's equations are the physical laws for electromagnetism". From Maxwell's laws you can derive that the speed of light in vacuum has a specific, constant value, in SI units $c=1/\sqrt{\...


5

You're not correct that the stone will appear to fall straight down if the train's acceleration is $\neq 0$. It would appear to fall in a (straight) slanted line (with angle $\arctan\left(\frac{a}{g}\right)$) because the stone is accelerating in the $y$ axis (due to gravity) and the observer in the train is accelerating in the $x$ axis. From outside the ...


4

In the context of Special Relativity, you do need to be careful about constraints such as "assume constant acceleration" without further qualification because, just as there is a need to distinguish between proper time and coordinate time, one must distinguish between proper acceleration (acceleration measured by an accelerometer) and coordinate acceleration,...


4

Either I should turn in my medical marijuana card, or the author of your textbook should. The exercise doesn't make any sense. Since we only have one particle, whose mass is fixed, we can set $m=1$. Also, the factor of 1/2 in the equation for KE is purely conventional, so let's drop that as well. We then have the following: Given the definitions $p=v$ and $...


4

The answer is negative. There is no action of the free particle invariant under the Galilean group. In the following, a heuristic explanation will be given and in addition a reference where a more detailed proof is provided. The basic reason is that the Galilean group cannot be realized on the Poisson algebra of functions on the phase space of the free ...


4

Here I would like to expand some of the arguments given in Ron Maimon's inspiring answer. Consider $N$ point particles with positions ${\bf r}_1, \ldots, {\bf r}_N$. The Galilean transformation group is, e.g., explained here. The only transformation, which we will bother to mention explicitly from now on, is the shear transformation $$ t \longrightarrow t, ...


4

Yes: I've done that. I used to have a device for the purpose, commonly called "rollers". It's like a treadmill for bicycles.


3

In general, it won't drop to the place right above where you dropped it in the roof-of-the-train case, but its motion will be different. The situation is different in that generally there are other forces on the coin: the air is moving relative to the coin not only vertically, but also along the direction of the train's motion. In other words, when on the ...


3

I can not answer you question mathematically, but my experience with Burgers equation tells me that there is no such transformation. If you think of Euler equation as Navier-Stokes in the limit where the viscosity vanishes $\nu \to 0$, then time reversal symmetry is simply spontaneously broken. As long as you have a finite viscosity the system is ...


3

You seem to be totally skimming by conservation of momentum. Let's look at your situation initially, if you look from there frame it does not matter whether they are at rest or moving with constant velocity. Let's say the mass of small robot(blue circle) is $m$ and mass of big spaceship(large rectangle) is $M$ ($m \lt M$) If you just see in there frame, ...


3

The basic assumptions on the space-time structure in classical mechanics are: (1) Time intervals beetween events are absolute. (2) Space intervals beetween contemporary events are absolute. We may refere to this two properties as to the "galilean space-time structure". In the first chapter of Arnold's "Mathematical Methods of Classical Mechanics" you can ...


3

An example of a law that is not invariant: $F = -\mu mv$. That is, some kind of universal friction slows down all moving objects. This requires a point of reference they are slowing down compared to, so it is not invariant. Any law that can be written in the form of a tensor is invariant, but this law cannot be written in that form. Not unless you have some ...


3

The Galilean spacetime is indeed the affine space $\mathbb{A}^4$. Affine space can be considered as a 'space with no origin', which makes intuitively sense because why would some point (the origin) be special. For example a trivial Galilean space is $\mathbb{E}\times \mathbb{E}^3$ where $\mathbb{E}$ is Euclidean space. The $\mathbb{R}\times \mathbb{R}^3$ ...


3

Unlike light, sound can only travel through a medium - in most situations, air. The velocities in your equations are relative to a fixed reference frame - that of the body of air in which the sound is travelling (which in a typical physics problem is the same as the ground's reference frame). So there really is a tangible difference between the case where ...


3

As wikipedia says about the second law, As Newton's second law is only valid for constant-mass systems, mass can be taken outside the differentiation operator by the constant factor rule in differentiation This is expanded in the article about variable mass systems. You derived the formula $$ F = m\dot v + \dot m v. $$ where $\dot v \equiv {\text{...


3

The action of the boost on momentum states specifies the matrix elements of $T_V$ in momentum representation, it does not "change the momentum representation of the state". In other words, from $$ T_V\left|p_1,p_2,..,p_N\right\rangle=\left|p_1+m_1V,p_2+m_2V,..,p_N+m_NV\right\rangle $$ it just follows that $$ \langle p'_1,p'_2,..,p'_N | T_V |p_1,p_2,..,p_N\...



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