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

Answer: (C)

Put the dimensions for each physical quantity in the given relation. Given,
$v=\frac{p}{2 l}\left[\frac{F}{m}\right]^{1 / 2}$
Squaring the equation on either side, we have
$v^{2}=\frac{p^{2}}{4 l^{2}}\left[\frac{F}{m}\right] $
$\Rightarrow m=\frac{p^{2} F}{4 l^{2} v}$
Putting the dimensions of equations on RHS,
we get $F=\left[M L T^{-2}\right], l=[L], v=\left[T^{-1}\right],$
$p$ being a number is dimensionless, we have
${[m]=\frac{\left[M L T^{-2}\right]}{\left[L^{2}\right]\left[T^{-1}\right]^{2}}} $
$=\left[M L^{-1} T^{0}\right]$