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Mechanical Properties Of Fluids

Question
CBSEENPH11020685

A cylindrical metallic rod in thermal contact with two reservoirs of heat at its two ends conducts an amount of heat Q in time t. The metallic rod is melted and the material is formed into a rod of half the radius of the original rod. What is the amount of heat conducted by the new rod when placed in thermal contact with the two reservoirs in time t?

  • Q/4

  • Q/16

  • 2Q

  • Q/2

Solution

B.

Q/16

In steady state, the amount of heat flowing from one face to the other face in time t is given by 

straight Q space equals space fraction numerator KA space left parenthesis straight theta subscript 1 minus straight theta subscript 2 right parenthesis straight t over denominator straight l end fraction space where space straight K space is space coefficient space of space thermal space conductivity space of space material space of space rod
rightwards double arrow space straight Q over straight t space proportional to space straight A over straight l space proportional to space straight r squared over straight I space space.. space left parenthesis straight i right parenthesis

As space the space metallic space rod space is space melted space and space the space material space is space formed space into space straight a space rod space of space half space the space radius
space straight V subscript 1 space equals space straight V subscript 2
πr subscript 1 superscript 2 straight l subscript 1 space equals space πr subscript 2 superscript 2 straight l subscript 2
straight l subscript 1 space equals space straight l subscript 2 over 4 space space space... space left parenthesis iI right parenthesis
Now comma space from space eqs. space left parenthesis straight i right parenthesis thin space and space left parenthesis ii right parenthesis space

fraction numerator straight Q subscript 1 over denominator straight Q subscript 2 space end fraction space equals space fraction numerator straight r subscript 1 superscript 2 over denominator straight l subscript 1 end fraction space straight x space fraction numerator straight l subscript 2 over denominator straight r subscript 2 superscript 2 end fraction space equals space fraction numerator straight r subscript 1 superscript 2 over denominator straight l subscript 1 end fraction space straight x space fraction numerator 4 straight l subscript 1 over denominator left parenthesis straight r subscript 1 divided by 2 right parenthesis squared end fraction
space rightwards double arrow space straight Q subscript 1 space equals space 16 space straight Q subscript 2