Question

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. $V = k\sqrt{\frac{T}{A}}$
• C. $V =k \sqrt{\frac{A\rho}{T}}$
• D. None of these

Answer 1

Answer: (A)

Let $V=k T^{a} A^{b} \rho^{c}$,
$k=$ dimensional constant
Writing dimension on both we side
$\left[L T^{-1}\right]=\left[M L T^{-2}\right]^{a}\left[L^{2}\right]^{b}\left[M L^{-3}\right]^{c}$
$=\left[M^{a+ c} L^{a+2 b-3 c T^{-2 a}}\right]$
Comparing power on both sides we have
$a+c=0, a+2 b-3 c=1,-2 a=-1$
$\therefore a=\frac{1}{2}, c=-\frac{1}{2}$
$\Rightarrow b=-\frac{1}{2}$
$\therefore V=k \sqrt{\frac{T}{A \rho}}$