A uniform metallic wire of density 8000 kg m ⁻³ is rigidly clamped at both ends when its temperature is 60^ C . The wire is then cooled to 20^ C . If the Young's modulus of the material of the wire is 2.0 10¹¹ N m ⁻² and its coefficient of linear expansion is 10⁻⁵ ^ C ⁻¹ , the speed of transverse waves on the wire at the lower temperature is _____ m s ⁻¹ .

Step-by-step solution

Let the density be , Young's modulus be Y , and coefficient of linear expansion be . The change in temperature is: = 60^ C - 20^ C = 40^ C When the wire is cooled, it tends to contract but is prevented by the rigid clamps, developing a thermal strain: = = 10⁻⁵ 40 = 4 10⁻⁴ The thermal stress developed in the wire is: = Y = (2.0 10¹¹) (4 10⁻⁴) = 8 10^7 N m ⁻² The speed of transverse waves is given by: v = T = Substituting the values: v = 8 10^7 8000 = 10000 = 100 m s ⁻¹ Answer: 100