Physics

Problem

A composite rod consists of a

10\,\text{cm}

segment of copper (

k_1=400\,\text{W}/\text{m}\cdot\text{K}

) joined end-to-end to a

15\,\text{cm}

segment of brass (

k_2=120\,\text{W}/\text{m}\cdot\text{K}

). Both segments have the same cross-sectional area of

1.5\times10^{-4}\,\text{m}^2

. The copper end is kept in contact with a heater at

150^{\circ}\text{C}

, while the brass end is attached to a

0.30\,\text{kg}

iron block initially at

20^{\circ}\text{C}

. Neglect heat losses to the surroundings and assume temperature gradients in each segment are linear. What is the initial rate at which the block’s temperature rises? (Specific heat of iron:

c=450\,\text{J}/\text{kg}\cdot\text{K}

.) a)

1.5\times10^{-1}\,\text{K/s}

9.6\times10^{-3}\,\text{K/s}

9.6\times10^{-2}\,\text{K/s}

1.1\times10^{-1}\,\text{K/s}

Intuitive explanation

Heat conduction through composite materials relies on the principle that in steady state, the same heat current flows through both segments—think of it like a series circuit where the bottleneck determines the flow. Each segment acts as a thermal resistor ( $R_{\text{th}} = L/kA$ ), and you find the total resistance, then use the temperature difference across the rod to calculate the heat flow rate. Once you know how much power is reaching the iron block, you convert that to a temperature-change rate using the block's heat capacity.

Intuitive explanation

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