Question

The frequency of vibration of string is given by
$v=\frac{p}{2l}{\left[\frac{F}{m}\right]}^{1/2}$
Here $p$ is number of segments in the string and $l$ is the length. The dimensional formula for $m$ will be

• A. $\left[M^{0}L{T}^{-1}\right]$
• B. $\left[M{L}^0{T}^{-1}\right]$
• C. $\left[M{L}^{-1}{T}^0\right]$
• D. $\left[M^0{L}^0{T}^0\right]$

Answer 1

The correct option is C $\left[M{L}^{-1}{T}^0\right]$

$v=\frac{p}{2l}{\left[\frac{F}{m}\right]}^{1/2}$
$v^2=\frac{p}{4l^2}\frac{F}{m}\Rightarrow m=\frac{p^{2}F}{4l^2{v}^2}$
Now, dimensional formula of R.H.S.
$=\frac{ML{T}^{-2}}{L^2{\left(\frac{1}{R}\right)}^2}$
$[p$ will have no dimension as it is an integer]
$=M{L}^{-1}{T}^0$.
So, dimensions of $m$ will be $M{L}^{-1}{T}^0$.