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Are below the correct conditions for a string to have variable tension

  1. string has to be massless

  2. String is un-accelerated.

I get the 1st one. I am not able to understand 2nd one. I have seen lot of atwood problems where two un equal masses are hanging over a pulley. when we apply Newtons 2nd law on the both the masses, we use tension to be same in both the strings.?

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  • $\begingroup$ Do you mean variable in space or variable in time? $\endgroup$
    – garyp
    Commented Aug 18, 2016 at 17:52

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I think you have it backwards. To have variable tension along a string you need both conditions

  1. String has linear density (not massless)
  2. String is accelerating

The only mechanism to change the tension internally is via inertial loads. Take a small segment of string and you find that the change in tension across is

$${\rm d}T = \ddot{x} \lambda {\rm d}x$$ where $\lambda$ is the linear density of the string such that ${\rm d}m = \lambda {\rm d}x$. The above state that the difference in tension goes into accelerating the small segment.

So if string is massless $\lambda=0$ and tension is constant ${\rm d}T=0$, or if string is un-accelerating $\ddot{x}=0$ then again ${\rm d}T=0$.

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    $\begingroup$ Condition 2. is not necessary. Take a massive string, attach one end to the ceiling, and hang a mass on the other. At any point on the string the tension force has to support the weight of the hanging mass plus the fraction of the string below the point in question. $\endgroup$
    – garyp
    Commented Aug 18, 2016 at 17:55
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    $\begingroup$ Point taken. Also a catenary will vary the tension along the length when it is static. $\endgroup$ Commented Aug 18, 2016 at 21:13

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