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Q.

A long uniform steel wire has a diameter of 2.0 mm. what should be the tension in the wire, so that the speed of the transverse wave on it equals the speed of sound at STP (320 m/s) the density of steel is $7800 k g / m^{3}$ .

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a

25.1N

b

120 N

c

130 N

d

2508 N

answer is C.

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Detailed Solution

$V o l u m e o f w i r e V = π r^{2} l = π d^{2} l / 4 M a s s o f w i r e = M = V o l u m e \times d e n s i t y = \frac{π d^{2} l ρ}{4}$

$L i n e a r M a s s D e n s i t y μ = \frac{M}{l} = \frac{π d^{2} ρ}{4}$

$W a v e S p e e d v = \sqrt{\frac{T}{μ}} \begin{array}{l} T e n s i o n i n w i r e T = μ v^{2} = \frac{π d^{2} ρ v^{2}}{4} \\ \Rightarrow T = \frac{3.142 \times {(2 \times 10^{- 3})}^{2} \times 7800 \times {(320)}^{2}}{4} \\ S u b s t i t u t i n g v, ρ, d w e h a v e \\ T = 2508 N \end{array}$

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