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Question
A semicircular rod is joined at 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 semicircular rod to the heat transferred through a cross section of the straight rod in a given time.
Solution
Let A be the area of cross section and K be the thermal conductivity of the material of the rod.
Let q1 be the rate of flow of heat through a semicircular rod.
Rate of flow of heat is given by
`q1 = (dQ) / dt = (K.A (T_1 - T_2 ))/(pi r)`
Let q2 be the rate of flow of heat through a straight rod.
`q_2 = (dQ)/(dt) = (KA (T_1 - T_2))/ (2r)`
Ratio of the rate of flow of heat through the 2 rods
= `(q1)/(q2) =(2r)/(pir) = 2/pi`
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