The diameter of an iron wire is 1.20 $m m$ . If the speed of transverse waves in the wire is 50.0 $m / s$ ,what is the tension in the wire? ? The density of iron is $7.7 \times 10^{3} k g / m^{3}$ .

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Solution

Given data, diameter of iron wire $(d) = 1.2 m m = 1.2 \times 10^{- 3} m$ .
$V (s p e e d o f t r a n s v e r s e w a v e s) = 50.0 m / s$ .
$P (d e n s i t y o f i r o n) = 7.7 \times 10^{3} k g / m^{3}$ .
Let Tension in the wire be $T$ , is be mass per unit length of wire.
We know, $V = \sqrt{\frac{T}{μ}}$
$\Rightarrow V = \sqrt{\frac{T}{(m / e)}} = \sqrt{\frac{T l}{m}}$
$\Rightarrow V = \sqrt{\frac{T l}{p . V}} = \sqrt{\frac{T l}{p . A . l}}$
$∴ p = \frac{m a s s (m)}{v o l u m e (V)}$
volume $(V) = A r e a o f c r o s s s e c t i o n (A) \times l e n g t h (l)$
$\Rightarrow V^{2} p . A = T \Rightarrow T = (50)^{2} . (7.7 \times 10^{3}) \times (π) \frac{(1.2 \times 10^{- 3})^{2}}{4} k g m / s^{2}$
$\Rightarrow T = 21.76 m / s^{2} k g = 21.76 N$
$∴$ Tension in the wire $= 21.76 N$

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