# Due to cold weather a 1m water pipe of cross-sectional area

Question:

Due to cold weather a $1 \mathrm{~m}$ water pipe of cross-sectional area $1 \mathrm{~cm}^{2}$ is filled with ice at $-10^{\circ} \mathrm{C}$. Resistive heating is used to melt the ice. Current of $0.5 \mathrm{~A}$ is passed through $4 \mathrm{k} \Omega$ resistance. Assuming that all the heat produced is used for melting, what is the minimum time required?

(Given latent heat of fusion for water/ice $=3.33 \times 10^{5} \mathrm{~J} \mathrm{~kg}^{-1}$, specific heat of ice $=2 \times 10^{3} \mathrm{~J}$

$\mathrm{kg}^{-1}$ and density of ice $=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$

1. $0.353 \mathrm{~s}$

2. $35.3 \mathrm{~s}$

3. $3.53 \mathrm{~s}$

4. $70.6 \mathrm{~s}$

Correct Option: , 2

Solution:

mass of ice $\mathrm{m}=\rho \mathrm{A} \ell=10^{3} \times 10^{-4} \times 1=10^{-1} \mathrm{~kg}$

Energy required to melt the ice

$\mathrm{Q}=\mathrm{ms} \Delta \mathrm{T}+\mathrm{mL}$

$=10^{-1}\left(2 \times 10^{3} \times 10+3.33 \times 10^{5}\right)=3.53 \times 10^{4} \mathrm{~J}$

$\mathrm{Q}=\mathrm{i}^{2} \mathrm{RT} \Rightarrow 3.53 \times 10^{4}=\left(\frac{1}{2}\right)^{2}\left(4 \times 10^{3}\right)(\mathrm{t})$

Time $=35.3 \mathrm{sec}$

Option $(2)$