Physics-
General
Easy

Question

If the ratio of lengths, radii and Young’s modulus of steel and brass wires shown in the figure are a comma blank band c, respectively. The ratio between the increase in lengths of brass and steel wires would be

  1. fraction numerator b to the power of 2 end exponent a over denominator 2 c end fraction    
  2. fraction numerator b c over denominator 2 a to the power of 2 end exponent end fraction    
  3. fraction numerator b a to the power of 2 end exponent over denominator 2 c end fraction    
  4. fraction numerator a over denominator 2 b to the power of 2 end exponent c end fraction    

The correct answer is: fraction numerator a over denominator 2 b to the power of 2 end exponent c end fraction


    Given, fraction numerator l subscript 1 end subscript over denominator l subscript 2 end subscript end fraction equals a comma fraction numerator r subscript 1 end subscript over denominator r subscript 2 end subscript end fraction equals b comma fraction numerator Y subscript 1 end subscript over denominator Y subscript 2 end subscript end fraction equals c

    Let Young’s modulus of steel be Y subscript 1 end subscript, and that of brass be Y subscript 2 end subscript
    therefore Y subscript 1 end subscript equals fraction numerator F subscript 1 end subscript l subscript 1 end subscript over denominator A subscript 1 end subscript increment l subscript 1 end subscript end fraction (i)
    and Y subscript 2 end subscript equals fraction numerator F subscript 2 end subscript l subscript 2 end subscript over denominator A subscript 2 end subscript increment l subscript 2 end subscript end fraction(ii)
    Dividing Equation (i) by Equation (ii), we get
    fraction numerator Y subscript 1 end subscript over denominator Y subscript 2 end subscript end fraction equals fraction numerator F subscript 1 end subscript. A subscript 2 end subscript. l subscript 1 end subscript. increment l subscript 2 end subscript over denominator F subscript 2 end subscript. A subscript 1 end subscript. l subscript 2 end subscript. increment l subscript 1 end subscript end fraction(iii)
    Force on steel wire from free body diagram
    T equals F subscript 1 end subscript equals left parenthesis 2 g right parenthesis Newton
    Force on brass wire from free body diagram
    F subscript 2 end subscript equals T subscript 1 end subscript superscript ´ ´ end superscript equals T plus 2 g equals 4 g Newton
    Now, putting the value of F subscript 1 end subscript comma F subscript 2 end subscript comma in Equation (iii), we get
    fraction numerator Y subscript 1 end subscript over denominator Y subscript 2 end subscript end fraction equals open parentheses fraction numerator 2 g over denominator 4 g end fraction close parentheses. open parentheses fraction numerator pi r subscript 2 end subscript superscript 2 end superscript over denominator pi r subscript 1 end subscript superscript 2 end superscript end fraction close parentheses. open square brackets fraction numerator l subscript 1 end subscript over denominator l subscript 2 end subscript end fraction close square brackets. open parentheses fraction numerator increment l subscript 2 end subscript over denominator increment l subscript 1 end subscript end fraction close parentheses equals fraction numerator 1 over denominator 2 end fraction open parentheses fraction numerator 1 over denominator b subscript 2 end subscript end fraction close parentheses. a open parentheses fraction numerator increment l subscript 2 end subscript over denominator increment l subscript 1 end subscript end fraction close parentheses

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