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The frequency of vibration of string is given by
$\mathrm{v}=\frac{\mathrm{p}}{2 l}\left[\frac{\mathrm{F}}{\mathrm{m}}\right]^{1 / 2}$
Here $\mathrm{p}$ is number of segments in the string and $l$ is the length. The dimensional formula for $\mathrm{m}$ will be
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$\mathrm{v}=\frac{\mathrm{p}}{2 l}\left[\frac{\mathrm{F}}{\mathrm{m}}\right]^{1 / 2}$
Here $\mathrm{p}$ is number of segments in the string and $l$ is the length. The dimensional formula for $\mathrm{m}$ will be
Solution:
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Verified Answer
The correct answer is:
$\left[\mathrm{ML}^{-1} \mathrm{~T}^{0}\right]$
$\mathrm{v}=\frac{\mathrm{p}}{2 \ell}\left[\frac{\mathrm{F}}{\mathrm{m}}\right]^{1 / 2}$
$v^{2}=\frac{p}{4 \ell^{2}} \frac{F}{m} \Rightarrow m=\frac{p^{2} F}{4 \ell^{2} v^{2}}$
Now, dimensional formula of R.H.S.
$=\frac{\mathrm{MLT}^{-2}}{\mathrm{~L}^{2}\left(\frac{1}{\mathrm{~T}}\right)^{2}}$
[p will have no dimension as it is an integer] $=\mathrm{ML}^{-1} \mathrm{~T}^{0}$
So, dimensions of $\mathrm{m}$ will be $\mathrm{ML}^{-1} \mathrm{~T}^{0}$.
$v^{2}=\frac{p}{4 \ell^{2}} \frac{F}{m} \Rightarrow m=\frac{p^{2} F}{4 \ell^{2} v^{2}}$
Now, dimensional formula of R.H.S.
$=\frac{\mathrm{MLT}^{-2}}{\mathrm{~L}^{2}\left(\frac{1}{\mathrm{~T}}\right)^{2}}$
[p will have no dimension as it is an integer] $=\mathrm{ML}^{-1} \mathrm{~T}^{0}$
So, dimensions of $\mathrm{m}$ will be $\mathrm{ML}^{-1} \mathrm{~T}^{0}$.
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