Water rises in a capillary upto a height of 8.0 cm. if the capillary is inclined at an angle of `45^(@)` with the vertical then determine the vertical height of water. How much length of the capillary will be occupied by water ? If the length of the capillary is reduced to 4.0 cm and it is held vertically in water then what will be the position of water?

Answer

To solve the problem step by step, we will break it down into three parts as per the questions asked.

### Step 1: Determine the vertical height of water in the inclined capillary.
Given that the water rises in the capillary to a height of 8.0 cm, and this height is measured vertically, the vertical height of water in the capillary is simply:
- **Vertical height of water (h)** = 8.0 cm.

### Step 2: Calculate the length of the capillary occupied by water when inclined.
The capillary is inclined at an angle of 45 degrees with the vertical. The relationship between the vertical height (h) and the length of the capillary (L) can be expressed using trigonometric functions. Specifically, we can use the cosine of the angle:
- \( L = \frac{h}{\cos(\theta)} \)
Where \( \theta = 45^\circ \).

Since \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \), we can substitute this into the equation:
- \( L = \frac{h}{\frac{1}{\sqrt{2}}} = h \cdot \sqrt{2} \).

Now substituting the value of \( h \):
- \( L = 8.0 \, \text{cm} \cdot \sqrt{2} \approx 8.0 \, \text{cm} \cdot 1.414 \approx 11.31 \, \text{cm} \).

### Step 3: Determine the position of water when the capillary length is reduced to 4.0 cm and held vertically.
If the length of the capillary is reduced to 4.0 cm and it is held vertically, we need to check if this length is sufficient to accommodate the height of water that would rise due to capillary action.

Since the original height of water (when the capillary was longer) was 8.0 cm, and now the capillary length is only 4.0 cm, it will not be able to hold the full height of the water. Therefore, the water will rise to the maximum height that the capillary can accommodate, which is equal to the length of the capillary itself.

Thus, the position of water in the capillary will be:
- **Position of water** = 4.0 cm (the entire length of the capillary will be filled with water, but it will not overflow).

### Summary of Answers:
1. Vertical height of water = 8.0 cm.
2. Length of the capillary occupied by water = 11.31 cm.
3. Position of water when the capillary length is 4.0 cm = 4.0 cm.