In a capillary tube C, water rises to a height of 10 cm. How much will the water rise in another tube C' whose radius is one fourth of radius of tube C? If C is inclined at an angle of `60^@` with vertical then what is the position of water in the tube?

Answer

To solve the problem step by step, we will break it down into two parts as described in the question.

### Part 1: Height of Water in Tube C'

1. **Understanding the Relationship**: The height of the liquid column in a capillary tube is inversely proportional to the radius of the tube. This can be expressed mathematically as:
\[ h \propto \frac{1}{r} \]
or
\[ h_1 \cdot r_1 = h_2 \cdot r_2 \]
where \(h_1\) and \(r_1\) are the height and radius of tube C, and \(h_2\) and \(r_2\) are the height and radius of tube C'.

2. **Given Values**:
- For tube C: \(h_1 = 10 \, \text{cm}\)
- Let the radius of tube C be \(r_1\).
- For tube C': \(r_2 = \frac{1}{4} r_1\).

3. **Setting Up the Equation**: Using the relationship established:
\[ h_1 \cdot r_1 = h_2 \cdot r_2 \]
Substituting the known values:
\[ 10 \cdot r_1 = h_2 \cdot \left(\frac{1}{4} r_1\right) \]

4. **Solving for \(h_2\)**: Rearranging the equation gives:
\[ h_2 = \frac{10 \cdot r_1}{\frac{1}{4} r_1} = 10 \cdot 4 = 40 \, \text{cm} \]

### Part 2: Height of Water in Inclined Tube C

1. **Understanding the Inclination**: When tube C is inclined at an angle of \(60^\circ\) with the vertical, we need to find the vertical height of the water column in the inclined position.

2. **Using Trigonometry**: The relationship between the vertical height \(H\) and the length \(L\) of the water column in the inclined tube is given by:
\[ H = L \cdot \cos(\theta) \]
where \(\theta = 60^\circ\).

3. **Finding Length \(L\)**: We know from Part 1 that the height of the water column in tube C is \(h = 10 \, \text{cm}\). Therefore, we can set:
\[ H = h = 10 \, \text{cm} \]
Thus,
\[ 10 = L \cdot \cos(60^\circ) \]
Since \(\cos(60^\circ) = \frac{1}{2}\), we have:
\[ 10 = L \cdot \frac{1}{2} \]

4. **Solving for \(L\)**: Rearranging gives:
\[ L = 10 \cdot 2 = 20 \, \text{cm} \]

### Final Answers:
- The height of water in tube C' is **40 cm**.
- The length of the water column in the inclined tube C is **20 cm**.