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1.Units, Dimensions and Measurement
hard
Wave pulse can travel along a tense string like a violin spring. A series of experiments showed that the wave velocity $V$ of a pulse depends on the following quantities, the tension $T$ of the string, the cross-section area A of the string and then as per unit volume $\rho$ of the string. Obtain an expression for $V$ in terms of the T, A and $\rho$ using dimensional analysis.
A$v=k \sqrt{\frac{T}{A \rho}}$
B$v=k \sqrt{\frac{A \rho}{T}}$
C$v=k \sqrt{\frac{A \rho}{T}}$
DNone of these
Solution
Let $V = kT ^{ a } A ^{ b } \rho^{ c }$, $k =$ dimensional constant
Writing dimension on both we side
$\begin{aligned}{\left[ LT ^{-1}\right] } &=\left[ MLT ^{-2}\right]^{ a }\left[ L ^2\right]^{ b }\left[ ML ^{-3}\right]^{ c } \\&=\left[ M ^{ a + c } L ^{ a +2 b -3 c } T ^{-2 a }\right]\end{aligned}$
Comparing power on both sides we have
$\begin{array}{l}a+c=0, a+2 b-3 c=1, \quad-2 a=-1 \\\therefore a=\frac{1}{2}, c=-\frac{1}{2} \Rightarrow b=-\frac{1}{2} \therefore V=k \sqrt{\frac{T}{A \rho}}\end{array}$
Writing dimension on both we side
$\begin{aligned}{\left[ LT ^{-1}\right] } &=\left[ MLT ^{-2}\right]^{ a }\left[ L ^2\right]^{ b }\left[ ML ^{-3}\right]^{ c } \\&=\left[ M ^{ a + c } L ^{ a +2 b -3 c } T ^{-2 a }\right]\end{aligned}$
Comparing power on both sides we have
$\begin{array}{l}a+c=0, a+2 b-3 c=1, \quad-2 a=-1 \\\therefore a=\frac{1}{2}, c=-\frac{1}{2} \Rightarrow b=-\frac{1}{2} \therefore V=k \sqrt{\frac{T}{A \rho}}\end{array}$
Standard 11
Physics
