How Does Temperature Affect the Speed of a Transverse Wave on a Copper Wire?

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SUMMARY

The speed of a transverse wave on a copper wire is influenced by temperature changes due to the material's properties. Given a copper wire with a cross-sectional area of 1.1 x 10^-6 m^2 and a linear density of 7.0 x 10^-3 Kg/m, the initial wave speed at ambient temperature is 46 m/s. The coefficient of linear expansion for copper is 17 x 10^-6, and Young's modulus is 1.1 x 10^11 N/m^2. When the temperature is lowered by 14°C, the wave speed can be recalculated using the derived formula v = √((Y.α.∆T.A)/(m/L)).

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  • Understanding of wave mechanics and transverse waves
  • Familiarity with Young's modulus and its application in material science
  • Knowledge of thermal expansion coefficients and their significance
  • Basic algebra for manipulating equations and solving for variables
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  • Calculate the new wave speed using the formula v = √((Y.α.∆T.A)/(m/L)) with ∆T = -14°C
  • Explore the effects of temperature on wave propagation in different materials
  • Study the relationship between linear density and wave speed in various wire types
  • Investigate the practical applications of wave speed calculations in engineering
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Physics students, materials scientists, and engineers interested in wave mechanics and the thermal properties of materials.

golriz
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A copper wire, whose cross sectional area is 1.1 x 10 ^ -6 m^2, has a linear density of 7.0 x 10^-3 Kg/m and is strung between two walls. At the ambient temperature, a transverse wave travels with a speed of 46 m/s on this wire. The coefficient of linear expansion for copper is 17 x 10^-6 , and Youngs modulus for copper is 1.1 x 10^11 N/m^2. What will be the speed of the wave when the temperature is lowered by 14 C?


v = √(F/(m⁄L)) (1)
F = Y(∆L/L0)A (2)
∆L = α.L0.∆T ⇒ ∆L/L0 = α.∆T (3)



A = 1.1 x 10^-6 m^2
m/L = 7 x 10^-3 Kg/m
α = 17 x 10^-6
Y = 1.1 x 10^11 N/m^2
v = 46 m/s




We can write formula (1) such this:

v = √((Y.α.∆T.A)/(m/L))

and now substitute all the variables in the above formula for finding ∆T.
But now I don't know what do I have to do, ∆T2 to find the speed of the wave.
 
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Please somebody answer my question
 

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