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