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(a) Three rod of lengths 20 cm each and area of cross-section 1 cm^(2) are joined to form a triangle ABC. The conductivities of the rods are K_(AB) = 50 J//m-s-.^(@)C, K_(BC) = 200 J//m-s-.^(@)C and K_(AC) = 400 J//m-s-.^(@)C. The junctions A, B and C are maintained at 40^(@)C, 80^(@)C and 80^(@)C respectively. Find the rate of heat flowing through the rods AB, AC and BC. (b) A semicircular rod is joineted as its end to a straight rod of the same material and the same cross-sectional area. The straight rod forms a diameter of the other rod. The junctions are maintained at different temperatures. Find the ratio of the heat transferred through a cross-section of the simicircular rod to the heat transferred through a cross-section of the straight rod in a given time. |
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Answer» SOLUTION :`R_(AB) = (l)/(K_(AB) A) = (20 xx 10^(-2))/(50 xx 1 xx 10^(-4)) = 40^(@) C//W` `R_(BC) = (l)/(K_(BC) A) = 10^(@)C//W` `R_(CA) = (l)/(K_(CA) A) = 5^(@)C//W` Road `AB : i_(1) = (80 - 40)/(R_(AB)) = (40)/(40) = 1 W` Rod `AC :i_(1) = (80 - 40)/(R_(AC)) = (40)/(5) = 8 W` Rod `BC : i_(3) = (80 - 80)/(R_(BC)) = 0` (ii) `R_("semicircular"), R_(1) prop pi r` `R_("straight"), R_(2) prop 2 pi r` `(i_(1))/(i_(2)) = ((theta_(1) - theta_(2)) // R_(1))/((theta_(1) - theta_(2)) // R_(2)) = (R_(2))/(R_(1)) = (2r)/(pi r)`  
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