An empty barometric tube 1 m long is lowered vertically (mouth downwards) into a tank of water. What will be the depth above the water level in the tube, when the water has risen 20 cm inside the tube ? Take atmospheric pressure as 10.4 m column of water.

Explanation

To solve the problem step by step, we need to analyze the situation involving the barometric tube and the water level.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a barometric tube that is 1 meter long and is lowered into a tank of water with its mouth facing downwards.
- The water rises 20 cm (0.2 m) inside the tube.

2. **Determine Initial Conditions**:
- The atmospheric pressure (P1) is given as equivalent to a 10.4 m column of water.
- Therefore, P1 = 10.4 m of water.

3. **Calculate Initial Volume (V1)**:
- The volume of the tube (V1) when it is empty can be considered as the entire length of the tube, which is 1 meter long.
- If we denote the cross-sectional area of the tube as A, then V1 = 1 * A = A m³.

4. **Calculate Final Volume (V2)**:
- When the water rises by 20 cm (0.2 m), the volume of the water inside the tube will be the remaining volume of the tube.
- Thus, V2 = V1 - Volume of water that has risen = A - (0.2 * A) = 0.8 * A m³.

5. **Apply Boyle's Law**:
- Since the temperature is constant, we can use Boyle's Law, which states that P1 * V1 = P2 * V2.
- Substituting the known values:
- P1 = 10.4 m of water,
- V1 = A,
- V2 = 0.8 * A.
- This gives us the equation:
\[ 10.4 \cdot A = P2 \cdot (0.8 \cdot A) \]

6. **Solve for P2**:
- Cancel A from both sides (assuming A ≠ 0):
\[ P2 = \frac{10.4}{0.8} = 13 \text{ m of water} \]

7. **Calculate the Depth Above Water Level**:
- The depth above the water level in the tube can be found by calculating the difference between P2 and P1:
\[ \text{Depth} = P2 - P1 = 13 \text{ m} - 10.4 \text{ m} = 2.6 \text{ m} \]

### Final Answer:
The depth above the water level in the tube is **2.6 meters**.

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