A wire is made out of a special steel alloy with Y= 2.00X10^11 Pa, density ρ=7800 kg/m^3, breaking stress 1.00X10^9 Pa and coefficient of linear thermal expansion ω=2.50X10^-5 K^-1. The initial temperature of the wire is 20oC and the wire is measured to be exactly 5.000m long. The wire is then secured at one end to the ceiling of a room and hangs vertically. When a 100 kg mass is then attached to the bottom end of the wire, it elongates by 3.00 mm.
a.) What is the radius of the wire, assuming it has a circular cross-section?
b.) By how much would the temperature in the room have to change so that the wire returns to it original length?
c.) If you were to pluck the wire at the center, what will be the dominant frequency of the sound produced (fundamental frequency)? (assume tension is provided by the 100 kg mass) Where should you pluck the wire to achieve third harmonic(second overtone) without also exciting the fundamental mode, and what is its frequency.