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Physics-

General

Easy

Question

One end of a copper rod of uniform cross-section and of length 3.1 m is kept in contact with ice and the other end with water at 100°C. At what point along it's length should a temperature of 200°C be maintained so that in steady state, the mass of ice melting be equal to that of the steam produced in the same interval of time. Assume that the whole system is insulated from the surroundings. Latent heat of fusion of ice and vaporisation of water are 80 cal/gm and 540 cal/gm respectively

40 cm from 100°C end
40 cm from 0°C end
125 cm from 100°C end
125 cm from 0°C end

The correct answer is: 40 cm from 100°C end

Rate of flow of heat is given by $fraction numerator d Q over denominator d t end fraction equals fraction numerator capital delta theta over denominator l divided by K A end fraction$ also $fraction numerator d Q over denominator d t end fraction equals L fraction numerator d m over denominator d t end fraction$ (where L = Latent heat)
Þ $fraction numerator d m over denominator d t end fraction equals fraction numerator K A over denominator l end fraction open parentheses fraction numerator capital delta theta over denominator L end fraction close parentheses$ . Let the desire point is at a distance x from water at 100°C.

Q Rate of ice melting = Rate at which steam is being produced
Þ $open parentheses fraction numerator d m over denominator d t end fraction close parentheses subscript S t e a m end subscript equals open parentheses fraction numerator d m over denominator d t end fraction close parentheses subscript I c e end subscript$ Þ $open parentheses fraction numerator capital delta theta over denominator L l end fraction close parentheses subscript S t e a m end subscript equals open parentheses fraction numerator capital delta theta over denominator L l end fraction close parentheses subscript I c e end subscript$
Þ $fraction numerator left parenthesis 200 minus 100 right parenthesis over denominator 540 cross times x end fraction equals fraction numerator left parenthesis 200 minus 0 right parenthesis over denominator 80 left parenthesis 3.1 minus x right parenthesis end fraction$ Þ x = 0.4 m = 40 cm

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