The frequency of vibration of a string depends on length $l$, tension $T$, and mass per unit length $m$. Using dimensional analysis, the correct relation is?

Solution

1. The dimensions of the variables are: frequency $[f]=[T^{-1}]$, length $[l]=[L]$, tension $[T]=[ML{T}^{-2}]$, and mass per unit length $[m]=[M{L}^{-1}]$.
2. Calculate the dimensions of the ratio $\frac{T}{m}$: $\frac{[ML{T}^{-2}]}{[M{L}^{-1}]}=[L^2{T}^{-2}]$, so $\sqrt{\frac{T}{m}}$ has dimensions $[L{T}^{-1}]$.
3. To obtain the dimensions of frequency $[T^{-1}]$, we must divide the result by length $[L]$, yielding $\frac{1}{l}\sqrt{\frac{T}{m}}$.
4. Hence the answer is (A).