Question

An insulated container is filled with ice at $0^{\circ} C$, and another container is filled with water that is continuously boiling at $100^{\circ} C$. In series of experiments, the container connected by various, thick metal rods that pass through the walls of container as shown in the figure-4.46.

In the experiment-I: A copper rod is used and all ice melts is 20 minutes.
In the experiment-II: A steel rod of identical dimensions is used and all ice melts in $80$ minutes.
In the experiment-III: Both the rods are used in series and all ice melts in $t_{10}$ minutes.
In the experiment-IV: Both rods are used in parallel all ice melts in $t_{20}$ minutes.

• A. The value of $t_{10}$ is $100$ minutes
• B. The value of $t_{10}$ is $50$ minutes
• C. The value of $t_{20}$ is $16$ minutes
• D. The value of $t_{20}$ is $8$ minutes

Answer 1

Answer: (A), (C)

$P_{1}=\frac{K_{1} A(100-0)}{L}$
As $Q=P_{1} t_{1}$
$\Rightarrow P_{2}=\frac{K_{2} A(100-0)}{L}$
As $Q=P_{2} t_{2}$
$\Rightarrow P_{3}=\frac{P_{1} P_{2}}{P_{1}+P_{2}}=\frac{\frac{Q}{t_{1}} \times \frac{Q}{t_{2}}}{\frac{Q}{t_{1}}+\frac{Q}{t_{2}^{\prime}}}=\frac{Q}{t_{1}+t_{2}}$
and $ Q=P_{3}\left(t_{1}+t_{2}\right)$
$\Rightarrow t_{10}=t_{1}+t_{2}=100\, \min$
$\Rightarrow P_{2}=P_{1}+P_{2}=\frac{Q}{t_{1}}+\frac{Q}{t_{2}}=\frac{Q\left(t_{1}+t_{2}\right)}{t_{1} t_{2}}$
$\Rightarrow Q=P_{4}\left(\frac{t_{1} t_{2}}{t_{1}+t_{2}}\right)$
$\Rightarrow t_{20}=\frac{t_{1} t_{2}}{t_{1}+t_{2}}$
$=\frac{20 \times 80}{100}=16\, \min$