A U-shaped structure made of the strongest steel contains a detonator switch con

Question

A U-shaped structure made of the strongest steel contains a detonator switch connected by wire to one metric ton (1000 kg) of the explosive TNT as shown in the figure below. A T-shaped structure made of the same strong steel fits inside the U, with the long arm of the T not quite long enough to reach the detonator switch when both structures are at rest in the laboratory. Now the T structure is removed far to the left and accelerated to high speed. It is Lorentz-contracted along its direction of motion. As a result, its long arm is not long enough to reach the detonator switch when the two collide. Therefore there will be no explosion TNT switch BOTH AT REST However, look at the same situation in the rest frame of the T structure, In this frame the arm of the T has its rest length, while the two arms of the U structure are Lorentz-contracted. Therefore the arm of the T will certainly strike the detonator switch and there will be a terrible explosion TNT REST FRAME OF U STRUCTURE a) Is there an explosion or isn't there? Explain clearly your reasoning b) The wire from the detonator switch to the TNT is restrung through point B on the U structure when both structures are at rest and a laser is installed at point A on the T structure. Later, when the two structures collide at A, the laser files a pulse at B that cuts the detonator wire. Does this new apparatus change your prediction about detonation of the TNT? TNT REST FRAME OF T STRUCTURE

Explanation / Answer

a.) There will and will not be an explosion. How? lets break it into two parts:

  i.) No explosion - We know that both the frames are made of very strong steel and strenght is based on the bonds between the atoms and total mass necessarily does not play a role in it. Accelerating U-Frame is lorentz contracted and upon collision will detonate the TNT BUT there is one important thing to take into consideration i.e the TNT, more precisely its mass. The frame is a sturdy one but is not as heavy as the explosive. So while accelerating towards the T-Frame(at rest), Lorentz contraction is observed but due to the presence of the TNT the U-Frame starts to decelerate and the contarction vanishes. Hence leading to the First image given meaning once the U-Frame is back to its original length the arm of T-Frame whose length as we know will be unable to trigger the detonation.( might lead to stopping the U-frame before it reaches the other frame, this can just be assumed for the sake of the arguement)

ii.) Explosion : Explosion can occur IF the U-frame is accelerated at such a rate where it can overcome the weight of the TNT. in such a case, the frame is lorentz contracted and the arm of the T-Frame at rest can trigger an explosion.

b.) With new changes made for the detonation, in case of the Rest T and U- Frames and accelerating T-Frame there will be no explosion at all beacuse in both cases the lenght of T-Frame arm is small. Hence we shall consider the following cases: Rest T and accelerating U & Accelerating T and U.

Rest T and acceleration U (we shall use the above a.ii.) case)

in this case the explosion will definitely occur as the U frame owing to Lorentz contraction because the Laser can only file a pulse when the collision is with point A, in this case which is not. Hence this arrangement will fail in this situation.

Accelerating U and T

In both accelerating frames, Lorentz contraction is observed. Upon collision, there will be no explosion not because the laser filed a pulse but the situation will be similar to the first image reason being is even though the arm of T is small but the vertical arm will prevent the contracted arm from triggering the switch.

Hence it can be concluded that this new apparatus change does not contradict the detonation predicton of the TNT.

  

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