The correct answer is 8:9. Key Points • Heat flow in a rod is proportional to its cross-sectional area, the temperature difference between the ends, and inversely proportional to its length. • The formula for heat flow is Q = kAΔT/L, where: • Q is the heat flow. • k is the thermal conductivity of the material. • A is the cross-sectional area of the rod. • ΔT is the temperature difference between the ends. • L is the length of the rod. • The problem provides the ratios of the lengths to radii of two rods as 1:2 for the first rod and 2:3 for the second rod. • Cross-sectional area (A) of a rod is proportional to the square of its radius (r²). Hence, the cross-sectional areas will be proportional to the square of the radii ratios. • For the first rod, the radius ratio is 2, so the cross-sectional area ratio becomes 2² = 4. • For the second rod, the radius ratio is 3, so the cross-sectional area ratio becomes 3² = 9. • Lengths are given directly as ratios. The first rod has a length ratio of 1, and the second rod has a length ratio of 2. • Heat flow ratio: Using the formula Q = kAΔT/L, heat flow is proportional to A/L. The ratio of heat flows between the two rods will be: • First rod: A₁/L₁ = 4/1 = 4 • Second rod: A₂/L₂ = 9/2 = 4.5 • Thus, the ratio of heat flows is 4:4.5, which simplifies to 8:9. Hence, the correct answer is 8:9. Additional Information • Thermal conductivity: • Thermal conductivity (k) is a measure of a material's ability to conduct heat. It is constant for a given material. • Metal rods are usually good conductors of heat due to their high thermal conductivity, making them ideal for experiments involving heat flow. • Relationship between area, length, and heat flow: • Heat flow is directly proportional to the cross-sectional area of a rod and inversely proportional to its length. • Increasing the radius of a rod increases its cross-sectional area exponentially (proportional to r²). • Hence, rods with larger radii can conduct more heat, assuming all other factors remain constant. • Practical applications of heat flow: • Understanding heat flow principles is critical for designing thermal systems such as heat exchangers, radiators, and insulation materials. • These principles are widely used in industries like automotive engineering, HVAC systems, and electronic cooling systems. Important Points • Key concepts in heat transfer: • Heat transfer occurs through conduction, convection, and radiation. The problem specifically addresses conduction. • Conduction depends on material properties, dimensions of the object, and the temperature gradient. • Experimental verification: • Theoretical ratios like 8:9 can be verified experimentally using controlled setups that measure heat flow under constant temperature gradients. • In laboratories, metal rods of varying lengths and cross-sectional areas are commonly used to study conduction.