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A sonometer wire of length 1.5 m is made of steel. The tension in it produces an elastic strain of 1 %. What is the fundamental frequency of steel if density and elasticity of steel are 7.7 × 10^ 3 kg/m^ 3 and 2.2 × 10^ 11 N/m^ 2 respectively?

A sonometer wire of length 1.5 m is made of steel. The tension in it produces an elastic strain of 1 %. What is the fundamental frequency of steel if density and elasticity of steel are 7.7 × 10^ 3 kg/m^ 3 and 2.2 × 10^ 11 N/m^ 2 respectively?

Grade:11

2 Answers

Saurabh Koranglekar
askIITians Faculty 10341 Points
11 months ago
576-542_2.PNG
Vikas TU
14149 Points
11 months ago
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
 

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