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 A Copper plate is soldered between two steel plates . All the plates have the same cross- section (A) and length l. The coefficient of thermal expansions are alpha subscript c end subscript a n d alpha subscript s end subscript and their young’s moduli are y subscript c end subscript a n d y subscript s end subscript respectively. What force will arise in the plates if the temperature is increased by t to the power of 0 end exponent c ? (Assume that the plates suffer the same net expansion)

  1. fraction numerator 2 A y subscript c end subscript y subscript s end subscript open parentheses alpha subscript c end subscript minus alpha subscript s end subscript close parentheses over denominator 2 y subscript s end subscript plus y subscript c end subscript end fraction    
  2. fraction numerator A y subscript c end subscript y subscript s end subscript open parentheses alpha subscript c end subscript minus alpha subscript s end subscript close parentheses over denominator 2 y subscript s end subscript plus y subscript c end subscript end fraction    
  3. fraction numerator 2 A y subscript c end subscript y subscript s end subscript open parentheses alpha subscript c end subscript minus alpha subscript s end subscript close parentheses over denominator y subscript s end subscript plus y subscript c end subscript end fraction    
  4. fraction numerator A y subscript c end subscript y subscript s end subscript open parentheses alpha subscript c end subscript minus alpha subscript s end subscript close parentheses over denominator y subscript s end subscript plus y subscript c end subscript end fraction    

The correct answer is: fraction numerator 2 A y subscript c end subscript y subscript s end subscript open parentheses alpha subscript c end subscript minus alpha subscript s end subscript close parentheses over denominator 2 y subscript s end subscript plus y subscript c end subscript end fraction


    Thermal expansion of c u equals l alpha subscript c end subscript t Elastic contraction of c u equals fraction numerator F over denominator A end fraction cross times fraction numerator l subscript blank over denominator y subscript c end subscript end fraction Net expansion of c u equals l alpha subscript c end subscript t minus fraction numerator F over denominator A end fraction fraction numerator l over denominator y subscript c end subscript end fraction Either steel plate is subjected to the tensile force fraction numerator F over denominator 2 end fraction from the sides of the cu plate .Net expansion of steel = l alpha subscript s end subscript t plus fraction numerator F over denominator 2 A end fraction cross times fraction numerator l over denominator y subscript s end subscript end fraction The plus sign is on account of the fact that there is elastic expansion, of steel plates. Since the plates suffer same net expansion therefore l alpha subscript c end subscript t minus fraction numerator F over denominator A end fraction. fraction numerator l subscript blank over denominator y subscript c end subscript end fraction equals l alpha subscript s end subscript t plus fraction numerator F over denominator 2 A end fraction. fraction numerator l over denominator y subscript s end subscript end fractionsolving we get F equals fraction numerator 2 A y subscript c end subscript y subscript s end subscript open parentheses alpha subscript c end subscript minus alpha subscript s end subscript close parentheses over denominator 2 y subscript s end subscript plus y subscript c end subscript end fraction

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