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In string theory, it is assumed that a string is about the size of a Planck length, $$\ell_{string} \sim \ell_{Pl} \simeq 10^{-35}\,\text m.$$ Why that length? Why not for example a hundred times larger?

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In string theory, the string length is not taken to be the Planck length,

$$\ell_P = \sqrt{\frac{\hbar G}{c^3}}.$$

Rather, we expect strings, should they exist, to be approximately around this scale but from my knowledge of the literature and many introductory texts, there is no claim it is precisely $\ell_P$.

The energy scale of the quantum theory itself is believed to also be around the Planck mass, $M_P$. The Einstein-Hilbert action can be expanded in powers of $M_P^{-1}$ and this is believed to be the relevant coupling in the quantum theory.

The reason quantum gravity does not appear to affect every day life is that the scale the LHC probes, that is, the electroweak scale $M_{EW}$ has $M_{EW}/M_P \sim 10^{-15}$.

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@JamalS 's point that the association is a purely dimensional argument cannot get you more precisely than a factor of 100 to $\ell_{Pl}$. The "freshman physics" standard Nambu 1970 string picture, repurposed to gravity by Schwarz-Scherk, is that the string's Regge slope is $\alpha'\sim \ell_{Pl}^2 \sim \frac{1}{2\pi T}$.

The $10^{40}$ton string tension T is related to the Regge slope of the gravity-applied Regge trajectory $$ \frac {dJ}{dE^2}=\alpha' $$ if you consider a freely spinning string of length L, since $E\sim TL$, and $J\sim TL^2\sim \alpha' E^2$. While L can grow indefinitely with E, for particle spins, such as J ~ 2, ..., we need $L\sim \ell_{Pl}$.

(You might flesh the argument out in more details in a paper of mine & collaborators, 1986; or String Theory and M-Theory, A Modern Introduction, by Becker, Becker & Schwarz, Cambridge UP, 2007, Ch. 2)

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