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Question

Two wires of the same material (Young’s modulus Y) and same length Lbut radii R and 2R respectively are joined end to end and a weight w is suspended from the combination as shown in the figure. The elastic potential energy in the system is

  1. fraction numerator 3 w to the power of 2 end exponent L over denominator 4 pi R to the power of 2 end exponent Y end fraction    
  2. fraction numerator 3 w to the power of 2 end exponent L over denominator 8 pi R to the power of 2 end exponent Y end fraction    
  3. fraction numerator 5 w to the power of 2 end exponent L over denominator 8 pi R to the power of 2 end exponent Y end fraction    
  4. fraction numerator w to the power of 2 end exponent L over denominator pi R to the power of 2 end exponent Y end fraction    

The correct answer is: fraction numerator 5 w to the power of 2 end exponent L over denominator 8 pi R to the power of 2 end exponent Y end fraction


    k subscript 1 end subscript equals blank fraction numerator Y pi open parentheses 2 R close parentheses to the power of 2 end exponent over denominator L end fraction comma blank k subscript 2 end subscript equals fraction numerator Y pi open parentheses R close parentheses to the power of 2 end exponent over denominator L end fraction
    Equivalent fraction numerator 1 over denominator k subscript 1 end subscript end fraction plus fraction numerator 1 over denominator k subscript 2 end subscript end fraction equals fraction numerator L over denominator 4 Y pi R to the power of 2 end exponent end fraction plus fraction numerator L over denominator Y pi R to the power of 2 end exponent end fraction
    Since, k subscript 1 end subscript x subscript 1 end subscript equals k subscript 2 end subscript x subscript 2 end subscript equals w
    Elastic potential energy of the system
    U equals fraction numerator 1 over denominator 2 end fraction k subscript 1 end subscript x subscript 1 end subscript superscript 2 end superscript plus fraction numerator 1 over denominator 2 end fraction k subscript 2 end subscript x subscript 2 end subscript superscript 2 end superscript
    U equals fraction numerator 1 over denominator 2 end fraction k subscript 1 end subscript open parentheses fraction numerator w over denominator k subscript 1 end subscript end fraction close parentheses to the power of 2 end exponent plus fraction numerator 1 over denominator 2 end fraction k subscript 2 end subscript open parentheses fraction numerator w over denominator k subscript 2 end subscript end fraction close parentheses to the power of 2 end exponent
    = fraction numerator 1 over denominator 2 end fraction w to the power of 2 end exponent open curly brackets fraction numerator 1 over denominator k subscript 1 end subscript end fraction plus fraction numerator 1 over denominator k subscript 2 end subscript end fraction close curly brackets equals fraction numerator 1 over denominator 2 end fraction w to the power of 2 end exponent open parentheses fraction numerator 5 L over denominator 4 Y pi R to the power of 2 end exponent end fraction close parentheses
    U = fraction numerator 5 w to the power of 2 end exponent L over denominator 8 pi Y R to the power of 2 end exponent end fraction

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