(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.

Explanation

`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)`