Limit to length of expanding spring?

[johne1618]

Imagine a massless compressed spring with its left-hand end anchored to x=0.

Now I release the right-hand end of the spring.

Within an interval of time each part of the spring will expand a certain amount.

Thus the velocity of the spring at distance x should be proportional to x.

Is it true that the length of the spring must be limited by the fact that its right-hand end cannot travel faster than the speed of light?

 

[Ich]

Imagine a massless compressed spring with its left-hand end anchored to x=0.

Massless springs are evil. They don't even work in Newtonian mechanics, and they create all sorts of paradoxes in SR.
The speed of sound in any material must be smaller than c, that means [tex]c^2 \gt \frac{E}{\rho},[/tex] which is why you need massive springs.

Within an interval of time each part of the spring will expand a certain amount.

No. For example, the left hand side of the spring wouldn't know that you released the other side until the signal of that disturbation has traveled to it. Slower than c.

Is it true that the length of the spring must be limited by the fact that its right-hand end cannot travel faster than the speed of light?

No. Instead, the properties of springs are limited. It can be of any length, but it will not behave as you described.

 

[johne1618]

No. For example, the left hand side of the spring wouldn't know that you released the other side until the signal of that disturbation has traveled to it. Slower than c

What happens if instead of springs I have a line of hydraulic pistons all fed from perpendicular hydraulic lines in such a way that each expands simultaneously?

 

[yuiop]

What happens if instead of springs I have a line of hydraulic pistons all fed from perpendicular hydraulic lines in such a way that each expands simultaneously?

If we have a mass at the leading end of each piston then the energy required to accelerate the leading mass from rest would be 1/√(1-v²/c²) which is infinite if we try to accelerate the leading mass to v=c. I just won't happen. If the first mass is accelerated to v1=0.2 then the second mass will accelerate to a velocity v2<v1+0.1 and the third mass will accelerate to v3<v2+0.1 and so on. If the proper force or energy used to accelerate each successive mass is equal in their respective instantaneous co-moving reference frames, then the velocity of each mass can be calculated using the relativistic velocity equation and no matter how many successive masses you have the velocity of the fastest mass in the original rest frame is always less than c.

 

[johne1618]

If we have a mass at the leading end of each piston then the energy required to accelerate the leading mass from rest would be 1/√(1-v²/c²) which is infinite if we try to accelerate the leading mass to v=c. I just won't happen. If the first mass is accelerated to v1=0.2 then the second mass will accelerate to a velocity v2<v1+0.1 and the third mass will accelerate to v3<v2+0.1 and so on. If the proper force or energy used to accelerate each successive mass is equal in their respective instantaneous co-moving reference frames, then the velocity of each mass can be calculated using the relativistic velocity equation and no matter how many successive masses you have the velocity of the fastest mass in the original rest frame is always less than c.

Ok - but what happens with massless pistons (if that makes sense)?

I think the speed of the end of the chain will be limited to c (and hence its length will be limited) only because the speed limit is c and not for reasons of relativistic increase of mass.

 

[yuiop]

Ok - but what happens with massless pistons (if that makes sense)?

I think the speed of the end of the chain will be limited to c (and hence its length will be limited) only because the speed limit is c and not for reasons of relativistic increase of mass.

Massless pistons can only move at c relative to any inertial reference frame and can never be initially at rest, so in the context of your question it does not make sense.

 

[johne1618]

Massless pistons can only move at c relative to any inertial reference frame and can never be initially at rest, so in the context of your question it does not make sense.

Ok - how about pistons with a mass that tends to zero?

 

[yuiop]

Ok - but what happens with massless pistons (if that makes sense)?

I think the speed of the end of the chain will be limited to c (and hence its length will be limited) only because the speed limit is c and not for reasons of relativistic increase of mass.

Ok - how about pistons with a mass that tends to zero?

It is better to think of the relativistic mass as an indication of the minimum amount of energy to boost a given mass to a given relative velocity, rather than as actual increase in mass. For any non zero rest mass the energy required to accelerate that mass to c is infinite, no matter how tiny that mass is, as long as it is not actually zero. The relativistic energy requirement is how the the speed limit of c is enforced, for objects with non zero rest mass.