An insulated container is filled with ice at 0,^oC , and another container is filled with water that

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10-2.Transmission of Heat

hard

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

In the experiment $I$ : a copper rod is used and all ice melts in $20$ 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 and all ice melts in $t_{20}$ minutes.

The value of $t_{10}$ is $100$ minutes

The value of $t_{10}$ is $50$ minutes

The value of $t_{20}$ is $32$ minutes

The value of $t_{20}$ is $8$ minutes

Solution

$\mathrm{P}_{1}=\frac{\mathrm{K}_{1} \mathrm{A}(100-0)}{\mathrm{L}} \Rightarrow Q=\mathrm{P}_{1} \mathrm{t}_{1}$

$\mathrm{P}_{2}=\frac{\mathrm{K}_{2} \mathrm{A}(100-0)}{\mathrm{L}} \Rightarrow \mathrm{Q}=\mathrm{P}_{2} \mathrm{t}_{2}$

$\mathrm{P}_{3}=\frac{\mathrm{P}_{1} \mathrm{P}_{2}}{\mathrm{P}_{1}+\mathrm{P}_{2}}=\frac{\frac{\mathrm{Q}}{\mathrm{t}_{1}} \times \frac{\mathrm{Q}}{\mathrm{t}_{2}}}{\frac{\mathrm{Q}}{\mathrm{t}_{1}}+\frac{\mathrm{Q}}{\mathrm{t}_{2}}}=\frac{\mathrm{Q}}{\mathrm{t}_{1}+\mathrm{t}_{2}}$

$\Rightarrow Q=P_{3}\left(t_{1}+t_{2}\right)$

$\Rightarrow \mathrm{t}_{10}=\mathrm{t}_{1}+\mathrm{t}_{2}=100 \mathrm{minutes}$

$\mathrm{P}_{2}=\mathrm{P}_{1}+\mathrm{P}_{2}=\frac{\mathrm{Q}}{\mathrm{t}_{1}}+\frac{\mathrm{Q}}{\mathrm{t}_{2}}=\frac{\mathrm{Q}\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right)}{\mathrm{t}_{1} \mathrm{t}_{2}}$

$\Rightarrow Q=P_{4}\left(\frac{t_{1} t_{2}}{t_{1}+t_{2}}\right)$

$\Rightarrow \mathrm{t}_{20}=\frac{\mathrm{t}_{1} \mathrm{t}_{2}}{\mathrm{t}_{1}+\mathrm{t}_{2}}=\frac{20 \times 80}{100}=16 \mathrm{min}$

Standard 11

Physics

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medium

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medium

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Solution

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