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A). directly proportional to the tension in the string.

B). directly proportional to the square root of the tension.

C). inversely proportional to tension.

D). inversely proportional to the square root of the tension.

Answer

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$\sqrt{\dfrac{T}{\mu }}$

The speed of the transverse wave on a stretched string will be dependent on the tension in the string and on the mass per unit length of the string. We will use dimensional analysis to find the exact dependence and will get the answer to the given question. Dimensions of tension will be the same as Force and dimensions of μ i.e. mass per unit length will be of mass divided by length.

$\begin{align}

& v\propto {{T}^{a}}{{\mu }^{b}} \\

& [L{{T}^{-1}}]={{[ML{{T}^{-2}}]}^{a}}{{[M{{L}^{-1}}]}^{b}}=[{{M}^{a+b}}{{L}^{a-b}}{{T}^{-2a}}] \\

\end{align}$

We get the following relations from the expression above

$a+b = 0$

$a-b = 1$

$-2a = -1$

When we solve them we get a = $\dfrac{1}{2}$ and b = \[-\dfrac{1}{2}\]

So, it can be seen that v is proportional to the square root of the tension. Hence, the correct option is B, i.e. directly proportional to the square root of the tension.