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PART - 1 CH - 1 Units and Measurements — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsPART - 1 CH - 1 Units and Measurements2 Marks
Question
The frequency of a tunning fork is given by $n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$. Here resonance length is $l$, tension generated due to weight is $T$ and mass of unit length of wire is $m$. Check the correctness of this formula by dimensional method.
✓

Answer

Formula,
$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$
Dimension of frequency in left side $=\left[ M ^0 L^0 T^{-1}\right]$ on R.H.S. :
$\text { Dimension of } l=\left[M^0 L^1 T^0\right]$
Dimension of Tension $T=\left[ M ^1 L^1 T^{-2}\right]$
Dimension of mass per unit length $(m)=\left[ M ^1 L^{-1} T^0\right]$
Hence, dimensional formula of $\frac{1}{2 l} \sqrt{\frac{T}{ m }}$
$\begin{array}{l}
=\frac{1}{\left[M^0 L^1 T^0\right]}\left[\frac{M^1 L^1 T^{-2}}{M^1 L^{-1} T^0}\right]^{\frac{1}{2}} \\
=\frac{\left[M^0 L^2 T^{-2}\right]^{\frac{1}{2}}}{\left[M^0 L^0 T^0\right]}=\frac{\left[M^0 L^1 T^{-1}\right]}{\left[M^0 L^1 T^0\right]} \\
=\left[M^0 L^0 T^{-1}\right]
\end{array}$
Hence, Dimensions of LHS = Dimensions of RHS Hence, the formula is dimensionally correct.

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