We know that linear speed of object going around a circle is $\omega * r $ Now let us take an elastic string and rotate a body of negligible mass with $\omega = 500  rad/s$ It is possible to further stretch this string while maintaining $\omega$ constant using a super powerful motor.  If we extend the chord length to say $1,000,000 m$ then the linear speed of the body should come out to be equal to $500,000,000 m/s $ which is greater than the speed of the light.   Where is the fallacy in the above argument?
358
Imagine a bar   spinning like a helicopter propeller,  At $\omega$ rad/s because the extremes of the bar goes at speed  $$V = \omega * r$$  then we can reach near $c$ (speed of light) applying some finite amount of energy just doing   $$\omega = V / r$$  The bar should be long, low density, strong to minimize the amount of energy needed  For example a $2000\,\mathrm{m}$ bar  $$\omega = 300 000 \frac{\mathrm{rad}}{\mathrm{s}} = 2864789\,\mathrm{rpm}$$  (a dental drill can commonly rotate at $400000\,\mathrm{rpm}$)  $V$ (with dental drill) = 14% of speed of light.  Then I say this experiment can be really made and bar extremes could approach $c$.  What do you say?   EDIT:  Our planet is orbiting at sun and it's orbiting milky way, and who knows what else, then any Earth point have a speed of 500 km/s or more agains CMB.  I wonder if we are orbiting something at that speed then there would be detectable relativist effect in different direction of measurements, simply extending a long bar or any directional mass in different galactic directions we should measure mass change due to relativity, simply because $V = \omega * r$  What do you think?
75618
You swing a ball on the end of a lightweight string in a horizontal circle at constant speed.  Can the string ever by truly horizontal?  If not, would it slope above the horizontal or below the horizontal?  Why?
61386
Imagine a massless and frictionless pulley with two weights hanging either side of the pulley by a massless string.    Rather than being fixed to a ceiling, the pulley is being pulled upward by an external force F, with the weights and string still attached.  Due to Newton's 2nd Law,  $\Sigma F_y=F-2T=ma$,  where $T$ is the tension in the string on either side of the pulley and $a$ is the vertical acceleration of the pulley.  Clearly, since there is a net upward force, the pulley itself will accelerate upwards.  But because the $m=0$,   $F-2T=0$.  Does this not then suggest that the pulley has a constant velocity?
101700
Lets say an object is tied to a string and it is being swung in a circle:    In this case, the tension of the string acts as a centrifugal force that keeps the object moving in a circle. However, if centrifugal force is the only force acting on the object, why doesn't it collapse into the center? I am pretty sure this has to do with the object's inertia (or the centrifugal pseudo-force), where it moves slightly forward in a tangent before falling, restoring the orbit.  Also, what keeps the string taut during the swinging? There has to be some force countering the centripetal force that keeps the string tight. I think this might again have to do with the intertia/centrifugal pseudo-force but can a pseudo-force balance a real force? Also, if centripetal were to be balanced by the pseudo-force, wouldn't the object not move at all (since the net force would be 0)?
82686
$$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center?
56665
My question is not about why speed of light has a particular constant value which has been sufficiently addressed in other questions on SE already. I want to know whether the fact that speed of light has to be constant is a consequence of other fundamental laws of Universe, or is it just a fundamental law itself.  My motivation for asking this question is thinking about Special Theory of Relativity. One way of understanding why time dilation and length contraction happen is to look at it from the perspective of constancy of speed of light. Alice who is on the moving source of light and Bob, a fixed observer see the speed of light as constant. In order for Alice to see the speed of light as a constant value, time slows down and length expands (in forward direction) thus keeping the speed constant.   In the  I'm doing, the Professor tells that this constancy of speed of light could be more generally referred to as, "Principle of wave constancy". That is, the speed of any wave is a property of the medium (and thus it remains constant). So, the speed of a wave in water, for example, remains constant even if the source of the waves (say a paddle keeps moving). Frequency and wavelength may change but the speed remains constant. In a similar fashion, light is an electromagnetic wave and thus its speed is constant. And the constant speed is defined by the medium (which happens to be vaccum i.e., no medium for light).   My questions are twofold now -    Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general) Theoretically, could there be other waves that could travel through vaccum at a different speed? If yes, time dilation and length contraction will have to be different.   I'd appreciate it if the answer is kept non-technical (essential math is okay). More importantly, I request you to try and explain without using General Theory of Relativity. Because I'm thinking how the scientific understanding by 1905 had already knew that speed of light has to be constant.
552
I don't have a lot of knowledge of special relativity and associated topics; some of the few things I know are that "all motion is relative" (that is, there is no 'stationary reference frame'), and the speed of light in vacuum ($c \simeq 3 \cdot 10^8~\mathrm{m~s^{-1}}$) is the absolute asymptotic speed limit (asymptotic meaning that you can never equal it, only get arbitrarily close). What escapes me is how those concepts work together - to my naive understanding, an object would never move in its own reference frame (and so would never reach $c$). What reference frame is $c$ measured against? (Is it measured against a reference frame?) Or am I looking at this the wrong way?
16430
There's another question on physics.SE whose , if I have understood it correctly, explains that the farther the points are in space the faster they are moving away from each other.     Actually, the apparent speed with which two of the points on the   circle in a distance $D$ of each other would move relative to each   other will be $v = H_0 D$ where $H_0$ is the speed the balloon itself   is expanding.   This means that even the ruler sitting on my desk is expanding, just that the expansion is very very slow owing to the ruler's small size relative to the cosmic scales. However, if everything is scaling then everything should apparently be of the same size to an observer in such an expanding space. The  quote from the show "The Big Bang Theory" comes to mind.   To be able to counter that, there're some phenomena that we can observe which are scale invariant. Red-shift would be one such phenomenon. But is red-shift the only way we can tell that the space is expanding? Is there a logical flaw in my thinking? I know that red-shift is a result of the Doppler effect - there's bound to be a decrease in the frequency of a signal, as perceived by an observer, emitted by a source which is moving away relative to the observer.
107143
Under Newton, the escape velocity is  $$v_{esc} = \rm c \ \sqrt{r_s/r}$$  where $\rm r_s=2 \ GM/c^2$. In the nonrotating relativistic case (the Schwarzschild case) the radial escape velocity is the same:  $$v^{\parallel}_{esc} = \rm c \ \sqrt{r_s/r}$$  which is the speed of light, $v={\rm c}$, at the event horizon at $\rm r=r_s$. Since the photon sphere, where an object with $v={\rm c}$ would not escape, but be captured into orbit, is at $r=3 \ {\rm GM/c^2}$, the transverse escape velocity is  $$v^{\perp}_{esc} = \rm c \ \sqrt{r_s/r}\div \sqrt{1-r_s/r}$$  and for the observed (shapiro delayed) velocity ${\rm v}$ the transformation is  $${\rm v}^{\perp}=v^{\perp} \ \sqrt{\rm 1-r_s/r} \ , \ \ {\rm v}^{\parallel}=v^{\parallel} \ (\rm 1-r_s/r)$$  so the observed angular motion is the same as with Newton, yet locally higher, while the observed radial motion is slowed down because of the radial length contraction and time dilation.  But what about the Kerr case where the rotation is significant and the event horizon is at $\rm r=(1+\sqrt{1-a^2}) \ {\rm GM/c^2}$ instead of $\rm r=r_s$?  A testparticle which is locally at rest and therefore corotating with the angular velocity of the frame dragging  $${\Omega = \rm d \phi/ {\rm d t} = {\rm 2 \ a \ r/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  should experience the time dilation factor  $$\rm \dot t = {\rm d t}/ {\rm d}\tau = \sqrt{{\rm ((a^2+(r-2) \ r) (a^2 \cos ^2(\theta )+r^2))/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  relatively to the far away observer, so multiplying $\Omega$ with $\rm \dot t$ should give the local frame dragging angular velocity, which, multiplied with the $\rm \sqrt{x^2+y^2}$ value of the Boyer-Lindquist to cartesian transformation should give the transverse velocity - this is slightly below $\rm c$ for a maximaly rotating black hole (with $\rm a = 1$ - at the edge of the outer equatorial ergosphere where $\rm r=1$ and $\rm R=\sqrt{r^2+a^2}$).  But how do I find the radial and total escape velocity, and what are the rules to convert the shapiro-delayed coordinate velocities $\rm d r/d t \ , \ d \phi/d t \ , d \theta/d t$ to the local 3-velocity components $v^{\parallel}$ and $v^{\perp}$ of a test particle (local meaning relatively to a frame dragged probe with contant $\rm r$)? The radial length contraction seems a bit different than in the Schwarzschild-case, where it is the same as the gravitational time dilation factor...
13417
Refer, "The classical theory of Fields" by Landau&amp;Lifshitz (Chap 3). Consider a disk of radius R, then circumference is $2 \pi R$. Now, make this disk rotate at velocity of the order of c(speed of light). Since velocity is perpendicular to radius vector, Radius does not change according to the observer at rest. But the length vector at boundary of disk, parallel to velocity vector will experience length contraction . Thus, $\dfrac{\text{radius}}{\text{circumference}}&gt;\dfrac{1}{2\pi}$ , when disc is rotating. But this violates rules of Euclidean geometry.   What is wrong here?
4034
As  says:     [...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $\frac{1}{2}mv^2$.   Why does this not increase linearly with speed? Why does it take so much more energy to go from $1\ \mathrm{m/s}$ to $2\ \mathrm{m/s}$ than it does to go from $0\ \mathrm{m/s}$ to $1\ \mathrm{m/s}$?  My intuition is wrong here, please help it out!
65200
Wikipedia has this page on  using planets. In some cases this effect was used to accelerate the spacecraft to a higher velocity. This diagram shows this in a very oversimplified manner.   That got me thinking that if light is affected by gravity, and if it slingshots around the black hole/massive object, can't it gain a higher speed than $c$? What limitations are stopping it from doing this?  Forgive me if the answer to this question is pretty straight-forward or staring-you-in-the-face kind. I haven't fully understood the mechanics of the gravitational slingshot fully yet, but I couldn't wait to ask this. Note: If it is possible (I highly doubt that!), could you provide an explanation using Newtonian mechanics, I'm not very familiar with general relativity because I'm in high school.
108663
Is there an upper limit for  jerk in physics? What about higher derivatives?   A consequence of  special relativity is that no material body can reach or exceed the speed of light in vacuum (due to the relativistic mass increase, or the Lorentz contraction).    I am not interested in obtaining these upper limits  based on postulates (and following mathematical implications), like the postulate  of the constancy of speed of light in special relativity.    I would like to know whether there are experiments that would lead to these upper limits, as a matter of consistency with empirical evidence. For the speed of light in vacuum we have Maxwell's equations. As far as I know, it was established experimentally that the constant that appears in Maxwell's equations is the speed of light in vacuum.  For acceleration,  there is a previous question on StackExchange, , where in an answer it is discussed Caianiello's maximum acceleration (the result is linked to Heisenberg's uncertainty principle, that's the type of answer that I am interested in, as an example).   So for speed and acceleration,  I understand that there is  experimental evidence that tells us that in order to have consistency (with the empirical evidence),  then we must accept that there is a maximum speed and acceleration in physics.  I am not interested in arguments involving the Planck system of units, since these arguments would be sensitive to the choice of normalization.   What about jerk in physics, and higher order derivatives. Is there any experimental data that would force us to accept upper limits for these quantities? Would it be useful to imagine experimental scenarios that would force us to accept these upper limits?  I will reformulate the question,  for clarity. I will not focus on the chronological emergence of the mathematical models that explain the experimental data.  In physics, maximum speed can be linked to Maxwell's equations. Maximum acceleration can be linked to Heisenberg's uncertainty principle. Does this trend continue,  for jerk and higher derivatives? Because  intuition tells me that in physics there must be upper limits/bounds for all of them, and there is a countable infinity of these quantities.
15718
In replying to a  I stated:     Gravitational waves have not been yet experimentally observed so as to have their velocity measured.   Which after the fact prompted me to try and verify it. I found a "speed of gravity" claimed measurement, which has been in a controversial discussion. In the last paragraph of  I found:     In September 2002, Sergei Kopeikin and Edward Fomalont announced that they had made an indirect measurement of the speed of gravity, using their data from VLBI measurement of the retarded position of Jupiter on its orbit during Jupiter's transit across the line-of-sight of the bright radio source quasar QSO J0842+1835. Kopeikin and Fomalont concluded that the speed of gravity is between 0.8 and 1.2 times the speed of light, which would be fully consistent with the theoretical prediction of general relativity that the speed of gravity is exactly the same as the speed of light.[18]   There has been criticism, described in the paragraph.   Has this matter been resolved ? is the measurement valid?  The speed of gravity has also been  calculated from observations of the orbital decay rate of binary pulsars and found to be consistent to the speed of light within 1% ( same link).  This is the speed that gravitational fields transfer information, and though the number itself depends on the theoretical framework solving for the system parameters it is true that just the existence  of gravitational damping in the binary system would imply that the speed cannot be infinite.  Along these lines I was wondering whether studying the second star in a binary system where the first has gone nova would not give a cleaner and model independent measurement of the speed of transfer of information gravitationally. In this day and age where everything is digital one should be able to have data on this from now on.     (Feb 11,2016) announced the observation of gravitational waves , at the same time  consistent with the speed of light being c, and thus measuring it. See the  for a summary.
17897
There is always one point in introductory mechanics that has been continuously bothering me ever since I had taken my freshman physics course - Why does a massless rope have its tension equal on both ends?  I have been searching for a good explanation on various textbooks at different levels, but none of them provided me a satisfactory explanation. They either   State directly "we assume the string is massless and the pulley has no friction, therefore the tension must be the same throughout the string" without further any elaboration and continue the calculation, Claim without any reasons that "whenever the rope and the pulley are massless and frictionless, the tension will be the same on both sides", or Explain using a lot of hand-waving arguments regarding infinite acceleration, non-zero moment of inertia, etc. , in one line or two.   Although those hand-waving arguments of type $(3)$ do sound quite reasonable to me and will convince me for a while every time I review them. But after a while, I will start feeling not well about this whole tension concept again.  Therefore, I am looking for a rigorous (in the physicist's sense) proof of exactly why and when do the tension in rope will be uniform, and hopefully from that I can clean up all of my conceptual difficulties.
103576
It makes intuitive sense to me for the apparent frequency of a sound as modified by the Doppler effect to be based entirely on the speed at which the observer and the source get closer or farther apart, regardless of whether the source or the observer is moving, based on the principle of relativity. However, the general equation I receive from every source $$f'=f((v±v_o)/(v∓v_s))$$ does not have this property. For example, if the speed of sound in the medium is 2 units and the observer and source are coming closer at a rate of 1 unit, then if we take the observer to be moving then we get $$f'=f(3/2)$$ but if we take the source to be moving then we get $$f'=f(2/1)$$ What is the difference between these two cases?
54328
Newton's first law states that in an inertial frame, a body at rest continues to be at rest, and a body in constant rectilinear motion continues its motion, unless an external force is applied upon the bodies. My question is how far have these assertions been tested experimentally? More precisely, is there any need to test this statement, or is it taken as a "definition" of sorts?  I assume the body at rest is a trivial case. On the other hand the case of the body in constant motion would be considerably harder to test, since no external force on a body is an idealization. But I am still curious whether there have been any attempts to test the case of the moving body.