Water rises in a capillary tube to a height 4 cm. If the tube is inclined at an angle of `45^(@)` with the vertical , find the position of the water in the tube.

Explanation

To solve the problem of finding the position of water in a capillary tube inclined at an angle of 45 degrees with the vertical, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Height of water rise in the vertical capillary tube, \( h = 4 \) cm.
- Angle of inclination with the vertical, \( \beta = 45^\circ \).

2. **Identify the Formula:**
- When a capillary tube is inclined at an angle \( \beta \) with the vertical, the effective length \( L \) of the water column in the tube can be calculated using the formula:
\[ L = \frac{h}{\cos \beta} \]

3. **Substitute the Known Values:**
- Substitute \( h = 4 \) cm and \( \beta = 45^\circ \) into the formula:
\[ L = \frac{4 \, \text{cm}}{\cos 45^\circ} \]

4. **Calculate \( \cos 45^\circ \):**
- We know that \( \cos 45^\circ = \frac{1}{\sqrt{2}} \).

5. **Perform the Calculation:**
- Substitute \( \cos 45^\circ \) into the equation:
\[ L = \frac{4 \, \text{cm}}{\frac{1}{\sqrt{2}}} = 4 \, \text{cm} \times \sqrt{2} \]
- This simplifies to:
\[ L = 4\sqrt{2} \, \text{cm} \]

6. **Calculate the Numerical Value:**
- Using the approximate value of \( \sqrt{2} \approx 1.414 \):
\[ L \approx 4 \times 1.414 \approx 5.656 \, \text{cm} \]

7. **Final Result:**
- Therefore, the position of the water in the inclined tube is approximately:
\[ L \approx 5.65 \, \text{cm} \]