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Easy

Question

Three rods of equal length l are joined to form an equilateral triangle P Q R. O is the mid point of P Q. Distance O R remains same for small change in temperature. Coefficient of linear expansion for P R and R Q is same, i. e. comma blank alpha subscript 2 end subscript but that for P Q is alpha subscript 1 end subscript. Then

  1. alpha subscript 2 end subscript equals 3 alpha subscript 1 end subscript    
  2. alpha subscript 2 end subscript equals 4 alpha subscript 1 end subscript    
  3. alpha subscript 1 end subscript equals 3 alpha subscript 2 end subscript    
  4. alpha subscript 1 end subscript equals 4 alpha subscript 2 end subscript    

The correct answer is: alpha subscript 1 end subscript equals 4 alpha subscript 2 end subscript


    open parentheses O R close parentheses to the power of 2 end exponent equals open parentheses P R close parentheses to the power of 2 end exponent minus open parentheses P O close parentheses to the power of 2 end exponent equals l to the power of 2 end exponent minus open parentheses fraction numerator l over denominator 2 end fraction close parentheses to the power of 2 end exponent
    equals open square brackets l open parentheses 1 plus alpha subscript 2 end subscript t close parentheses close square brackets to the power of 2 end exponent minus open square brackets fraction numerator l over denominator 2 end fraction open parentheses 1 plus alpha subscript 1 end subscript t close parentheses close square brackets to the power of 2 end exponentl to the power of 2 end exponent minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction equals l to the power of 2 end exponent open parentheses 1 plus alpha subscript 2 end subscript superscript 2 end superscript t to the power of 2 end exponent plus 2 alpha subscript 2 end subscript t close parentheses minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction left parenthesis 1 plus alpha subscript 1 end subscript superscript 2 end superscript t to the power of 2 end exponent plus 2 alpha subscript 1 end subscript t right parenthesis
    Neglecting alpha subscript 2 end subscript superscript 2 end superscript t to the power of 2 end exponent and alpha subscript 1 end subscript superscript 2 end superscript t to the power of 2 end exponent
    0 equals l to the power of 2 end exponent open parentheses 2 alpha subscript 2 end subscript t close parentheses minus fraction numerator l to the power of 2 end exponent over denominator 4 end fraction open parentheses 2 alpha subscript 1 end subscript t close parentheses rightwards double arrow 2 alpha subscript 2 end subscript equals fraction numerator 2 alpha subscript 1 end subscript over denominator 4 end fraction rightwards double arrow alpha subscript 1 end subscript equals 4 alpha subscript 2 end subscript

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