PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics
Problem
Two homogeneous bars, X and Y, are made of the same material and are used as thermal connectors between identical hot and cold reservoirs. Bar X has length L and cross-sectional area A. Bar Y has length 21L and cross-sectional area 31A. Both bars reach steady-state conduction.
(a) Without performing numerical calculations, identify which bar conducts thermal energy at a greater rate. Provide a qualitative explanation based on geometric considerations.
(b) Derive an algebraic expression for the ratio of the conduction rates (ΔtQ)Y/(ΔtQ)X in terms of the given geometric factors.
(c) Evaluate the ratio obtained in part (b) and interpret the physical meaning of your result.
(d) Suppose both bars are replaced with a single composite connector that is required to match the combined conduction rate of the original two bars operating in parallel. If the composite connector has length L and is made of the same material, determine the cross-sectional area it must have.