Question

The frequency of vibration of a string is given by $f=\dfrac{n}{2L}\sqrt{\dfrac{T}{m}}$ where $T$ is tension in the string, $L$ is the length, $n$ is number of harmonics. What is the dimensional formula for $m$?
$\text{A}\text{. }[{{M}^{0}}LT]$
$\text{B}\text{. }[{{M}^{0}}{{L}^{-1}}{{T}^{-1}}]$
$\text{C}\text{. }[{{M}^{1}}{{L}^{-1}}{{T}^{0}}]$
$\text{D}\text{. }[{{M}^{0}}L{{T}^{-1}}]$

Answer 1

Frequency is defined as the number of vibrations per unit second by a body in periodic motion. It has SI unit Hertz (Hz) or ${{s}^{-1}}$. Tension is a force and has unit $N$. $n$, the number of harmonics does not have any unit. Dimensional formula can be derived by substituting the dimensional formula of all other quantities in expression of $m$.

Formula used:
$m=\dfrac{{{n}^{2}}T}{4{{L}^{2}}{{f}^{2}}}$

Complete step by step answer:
We are provided with the frequency of vibrations of a string as
$f=\dfrac{n}{2L}\sqrt{\dfrac{T}{m}}$
We first obtain expression for $m$by squaring and rearranging the terms in the given equation.
$m=\dfrac{{{n}^{2}}T}{4{{L}^{2}}{{f}^{2}}}$
In this expression, $n$ is the number of harmonics and is dimensionless. $T$ is tension in the string which is a kind of force, hence, it has dimensions of force. Now as we know, force acting on the moving object is the product of its mass ($m$) and the acceleration ($a$).
Therefore,
$\vec{F}=m\vec{a}$
We know that the dimension of mass is [${{\text{M}}^{1}}$] and dimension of acceleration ($a$) is [${{\text{L}}^{1}}{{\text{T}}^{-2}}$]. Therefore, dimension of force is [${{\text{M}}^{1}}{{\text{L}}^{1}}{{\text{T}}^{-2}}$].
$L$is the length of the string thus has dimension of length i.e. $[{{M}^{0}}{{L}^{1}}{{T}^{0}}]$
$f$ is the frequency and has dimension $[{{T}^{-1}}]$
Now we substitute these dimensions in expression of $m$and get
Dimensional formula of $m=\dfrac{{{[{{M}^{0}}{{L}^{0}}{{T}^{0}}]}^{2}}[ML{{T}^{-2}}]}{{{[{{M}^{0}}{{L}^{1}}{{T}^{0}}]}^{2}}{{[{{M}^{0}}{{L}^{0}}{{T}^{-1}}]}^{2}}}$
On simplifying this, we get
$[m]=[M{{L}^{-1}}{{T}^{0}}]$
Hence, option C is correct.

Note:
Students should read the question carefully, especially the symbols’ meaning given in the questions. Many times symbols used in question are different from their usual meanings.
As an example, students can confuse $T$ with time period. $m$ can also be confused with mass as it is often used to represent mass of a substance, which has dimensional formula $[M{{L}^{0}}{{T}^{0}}]$.