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If we introduce the notion of a massless string to denote the fact that net force on a massless string will always be $0$, since it is massless . How can these massless strings ever accelerate when force on them is $0$ .

As an example consider two blocks joined by a massless string , you push a block , the other block accelerates and so does the massless string . But massless string has no net force .

How to resolve this paradox ?

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$F=ma$. If $F=0$, and $m=0$, $a$ can be anything. Most physical laws are not "A causes B". They usually say that "A and B can coexist in these conditions". So, it is not necessarily "Force causes acceleration". It is "an accelerating body can coexist with a force if $F=ma$"

The net force on a massless string is always 0 -- it has to be (otherwise it will have infinite acceleration). Whenever we draw free body diagrams of systems that contain massless strings, we always take a tension force $T$ that represents the string "pulling" the body. Take the reaction force of $T$ on the string and you'll notice that the string is always in equilibrium.

For example, take this system, where someone is pulling a set of two boxes interconnected by a string:

enter image description here

Note that the reaction force of $T$ (in red) on the string balances itself.

For a more complicated system, take the following:

enter image description here

(I've taken a massless smooth pulley here. If the pulley wasn't smooth, then the tensions in the two portions of string would be different. If it wasn't massless, the force from the ceiling would be different)

In this system $T$ balances out on the string as well. This is precisely why we say that a section of string exerts $T$ on both ends — to maintain equilibrium.

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Non-relativistic mechanics can't. Massless objects travel at the speed of light. The only reason to introduce a massless string is so that you can get some effect from the string without having to worry about the string in calculations. As soon as you start worrying about forces on the string causing it to accelerate you've violated the whole reason for making it massless in the first place.

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Yes , so what is a massless object in non-relativistic mechanics ?. I tagged the question with newtonian mechanics to make it clear . –  nonagon May 16 '13 at 7:06
    
Get some effect ? –  nonagon May 16 '13 at 7:08
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A massless object in non-relativistic mechanics is no object at all. It doesn't exist. It's only there to enforce some rule and make calculations simple. –  Brandon Enright May 16 '13 at 7:08
    
But we are always considering constraints due to that and implying force on it is 0 –  nonagon May 16 '13 at 7:09
    
For your massless string connecting two blocks together example, the massless string is just a conceptual short-hand. You don't worry about what happens to the massless string, you just use it as a guide to figure out what force is being applied through it to a second block. Put another way, the "mass" of the massless string connecting the two blocks together is the sum of the masses of the two blocks. You've attached them all together as though they are one system. –  Brandon Enright May 16 '13 at 7:15
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