Question

A sonometer wire of length 1.5 m is made of steel. the tension in it produce an elastic strain of 1%.What is the fundamental frequency of steel if density and elasticity of steel are 7.7x10^3 kg/m^3 and 2.2x10^11 N/m^2 respectively
Answer is 178.2Hz.
Plz help to solved out it in simple language

Answer 1

Transverse waves on a Sonometer wire/bridge.

Young's modulus Y = Stress / Strain = 2.2 * 10^11 N/m²
    Stress = Tension / Cross section area = T / A
    Strain = 1 % = 0.01
    2.2 * 10^11 =  (T / A ) / 0.01 =  T / (0.01 A)

       T = 2.2 * 10^9 * A    Newtons
   L = 1.5 meter

     Volume density d =  7.7 * 10³  kg/m³
     Linear density μ = volume density * Area = d * A = 7.7 * 10³ * A  kg /meter

   velocity of transverse wave on the Sonometer wire = √(T/μ)
          v   = √(2.2*10^9 / 7.7*10³)  m/sec
             = 534.52 m/sec

   Fundamental note has a wavelength = λ = 2 * L = 3 meters.  As the given wire vibrates with one loop.  Nodes at both ends and anti-node at the center.  So half of the wavelength is equal to the length of the wire.

       So frequency = v / λ = 178.17 Hz