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A steel wire of mass 400 gm, length 1 m and area of cross-section $2 mm 2$ is rigidly fixed at both ends. The temperature of wire is lowered by $10 ∘ C$ . If transverse waves are set up by plucking the string in the middle, determine the frequency of the fundamental mode of vibration in Hz.
Young’s modulus of steel $Y = 2 × 1011 N / m 2$ and coefficient of linear expansion of steel $α = 10 − 5 / ∘ C$

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answer is 5.

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Detailed Solution

Tension in the wire

T = A Y α Δ θ = 2 × 10 − 6 × 2 × 1011 × 10 − 5 × 10 = 40 N

Linear mass density

μ = M L = 400 × 10 − 3 1 = 0.4 kg / m

Fundamental frequency

f = 1 2 L T μ = 1 2 × 1 40 0.4

102 = 5 H z

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