A semicircular rods is joined at its end to a straight rod of the same material and the same cross-sectional area. The straight rod forms a diameter of the other rod. The junctions are maintained at different temperatures. Find the ratio of the heat transferred through a cross section of the straight rod in a given time.

Explanation

To solve the problem of finding the ratio of heat transferred through a cross-section of the straight rod in a given time, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**: We have a semicircular rod (Rod 1) joined to a straight rod (Rod 2). Both rods are made of the same material and have the same cross-sectional area. The semicircular rod has a length equal to the arc length of a semicircle, while the straight rod has a length equal to the diameter of the semicircle.

2. **Identify the Lengths**:
- For the semicircular rod (Rod 1), if the radius is \( r \), the length \( L_1 \) is given by the formula for the arc length of a semicircle:
\[ L_1 = \pi r \]
- For the straight rod (Rod 2), the length \( L_2 \) is simply the diameter of the semicircle:
\[ L_2 = 2r \]

3. **Write the Heat Transfer Formula**: The rate of heat transfer \( \frac{Q}{t} \) through a rod is given by Fourier's law of heat conduction:
\[ \frac{Q}{t} = kA \frac{\Delta T}{L} \]
where:
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \Delta T \) is the temperature difference,
- \( L \) is the length of the rod.

4. **Calculate Heat Transfer for Each Rod**:
- For the semicircular rod (Rod 1):
\[ \frac{Q_1}{t} = kA \frac{\Delta T}{L_1} = kA \frac{\Delta T}{\pi r} \]
- For the straight rod (Rod 2):
\[ \frac{Q_2}{t} = kA \frac{\Delta T}{L_2} = kA \frac{\Delta T}{2r} \]

5. **Find the Ratio of Heat Transfer**:
- To find the ratio of heat transfer through the two rods, we take the ratio \( \frac{Q_1/t}{Q_2/t} \):
\[ \frac{Q_1/t}{Q_2/t} = \frac{kA \frac{\Delta T}{\pi r}}{kA \frac{\Delta T}{2r}} = \frac{2r}{\pi r} \]
- Simplifying this gives:
\[ \frac{Q_1/t}{Q_2/t} = \frac{2}{\pi} \]

6. **Conclusion**: The ratio of the heat transferred through a cross-section of the straight rod to that of the semicircular rod is:
\[ \frac{Q_1/t}{Q_2/t} = \frac{2}{\pi} \]