Q 11-01-046NEETJEE MainMedium
The frequency $f$ of a vibrating string depends on its length $l$, the tension $T$ and its mass per unit length $\mu$ as $f \propto l^a T^b \mu^c$. The values of $a$, $b$ and $c$ are
Answer: (C) $-1,\ \tfrac12,\ -\tfrac12$
$$[T^{-1}] = [L]^a\,[MLT^{-2}]^b\,[ML^{-1}]^c$$
M: $b + c = 0$
T: $-2b = -1 \Rightarrow b = \tfrac12$, so $c = -\tfrac12$
L: $a + b - c = 0 \Rightarrow a = -1$
So $f \propto \dfrac{1}{l}\sqrt{\dfrac{T}{\mu}}$, which matches $f = \dfrac{1}{2l}\sqrt{\dfrac{T}{\mu}}$ for the fundamental mode.
Solution by Sreeraj P, M.Sc Physics