Question

A copper wire is held at the two ends by rigid supports. At 50°C the wire is just taut, with negligible tension.
If Y = 1.2 x 10^11 N/m2, a = 1.6 × 10^-5/°C and p = 9.2 x 10^3kg/m^3 then the speed of transverse waves in
this wire at 30°C is
(1) 64.6 m/s
(2) 16.2 m/s
(3) 23.2 m/s
(4) 32.2 m/s
​

Answer 1

We know that the velocity of a transverse wave is given by $v=\sqrt{\frac{T}{m} }$

where, T is the tension and

             m is the linear mass density

By thermal expansion,

ΔL = LaΔθ

ΔL/L = aΔθ

where, ΔL is the change in length

            L is the initial length

            a is the coefficient of linear expansion\

            Δθ is the change in temp

We know that $Y=\frac{FL}{Al}$

where, F is the force

            L is the initial length

            A is the area of cross section

            l or ΔL is the change in length

ΔL/L = F/AY

aΔθ  = F/AY

F = AYaΔθ

Here, F = T = AYaΔθ

m = mass/length = mass*area/length*area     (Multiply and divide by area)

m = mass*area/volume    (area*length = volume)

m = p*area                 (mass/volume = density)

m = pA

Putting all these in the equation of wave velocity $v=\sqrt{\frac{T}{m} }$

we get

v = √(AYaΔθ/pA) = √(YaΔθ/p)

Put in the given values and the answer will be approximately equal to the option (3) 23.3 m/s

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