Question

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A rod of length L and square cross-section with side length a conducts heat between two machine parts. In order to modify the machine but maintain the same rate of heat conduction, you replace the rod with a new one of the same material that has a square cross-section with side length 3a/2. What is the new rod's length? A. 3L/2 B. 9L/4 C. 2L/3 D. 4L/9

Answer 1

The rate of heat transfer is given by the equation:

$\frac{Q}{t} = \frac{k A (T_2-T_1)}{L}$

Where Q/t is the rate of heat transfer, L is the length of rod, T is the temperature and A is the area of cross section,

Since the material is same hence k will be same for both cases.

The side of square is now 3/2 times of the earlier hence the area will be of (3/2)^2 = 9/4 times of the earlier.

The temperature difference also same in both cases.

Hence the ratio of length will be equal to the ratio of the area of cross section.

L_2 = L1*A_2/A_1

L_2 = 9L/4