Check by the method of dimensional analysis whether the following… - Vidyadip

Question

Check by the method of dimensional analysis whether the following relations are correct.
$\text{v}=\sqrt{\frac{\text{P}}{\text{D}}}$ where v = velocity of sound and P = pressure, D = density of medium
$\text{n}=\frac{1}{21}\sqrt{\frac{\text{F}}{\text{m}}},$ where n = frequency of vibration
l = legnth of the string
F = Stretching force
m = mass per unit length of the string.

✓

Answer

$[\text{R.H.S}]=\sqrt{\frac{[\text{P}]}{\text{[D]}}}=\sqrt{\frac{[\text{ML}^{-1}\text{T}^{-2}]}{[\text{ML}^{-3}]}}=[\text{LT}^{-1}]$

$[\text{L.H.S}]=[\text{v}]=[\text{LT}^{-1}]$

$[\text{R.H.S.}]=[\text{L.H.S.}]$

Hence, the relation is correct.

$[\text{R.H.S.}]=\frac{1}{[\text{l}]}\sqrt{\frac{[\text{F}]}{[\text{m}]}}$

$=\frac{1}{\text{L}}\sqrt{\frac{\text{MLT}^{-2}}{\text{ML}^{-1}}}=\frac{1}{\text{L}}[\text{LT}^{-1}]=[\text{T}^{-1}]$

$[\text{L.H.S.}]=\Big[\frac{1}{\text{Time}}\Big]=\frac{1}{\text{T}}=[\text{T}^{-1}]$

Hence, the relation is correct.

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