Multiple choice

A metallic rod of cross-sectional area $20 cm^2$, with the lateral surface insulated to prevent heat loss, has one end immersed in boiling water and the other in ice water mixture. The heat conducted through the rod melts the ice at the rate of $1$ gm for every $84$ sec. The thermal conductivity of the rod is $160 Wm^{1}K^{1}$. Latent heat of $ice = 80 cal/gm, 1 cal = 4.2 joule$. What is the length (in m) of the rod?

  1. 4 m

  2. 5 m

  3. 8 m

  4. 10 m
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Melting 1 g of ice requires 80 × 4.2 = 336 J, so the heat current is 336/84 = 4 W. Using Q/t = kAΔT/L with A = 0.002 m^2 and ΔT = 100 K gives L = 8 m.

AI explanation

The heat required to melt the ice is found using the latent heat formula, where the mass is 1 gm and the latent heat is 80 cal/gm, giving 80 calories. Converting this to joules using the provided 4.2 joules per calorie, the total heat conducted through the rod in 84 seconds is 336 joules, making the rate of heat transfer 336 / 84, which is 4 watts. Using the thermal conduction formula H = K*A*dT/L, where H is 4 W, K is 160 W/mK, A is 20*10^-4 m^2 and the temperature difference dT is 100 K, we get 4 = 160 * (20*10^-4) * 100 / L. Solving this equation gives L = 8 meters. The result is 8 m.

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