To solve the problem step by step, we will analyze the situation involving a glass capillary tube dipped in water, which rises to a certain height. When the tube is broken, we will determine the new height of the water column and the angle of contact.
### Step-by-Step Solution:
1. **Understanding the Initial Situation**:
- The glass capillary tube is dipped in water, and the water rises to a height of \( h_1 = 9 \, \text{cm} \).
- The angle of contact \( \theta_1 \) for water in a glass tube is typically \( 0^\circ \) (since water wets glass).
2. **Using the Capillary Rise Formula**:
- The height of the liquid column in a capillary tube can be described by the formula:
\[
h = \frac{2s \cos \theta}{\rho r g}
\]
- Here, \( s \) is the surface tension, \( \theta \) is the angle of contact, \( \rho \) is the density of the liquid, \( r \) is the radius of the tube, and \( g \) is the acceleration due to gravity.
3. **Setting Up the Ratios**:
- Since the surface tension, density, radius, and gravity are constants for a given liquid and tube, we can set up a ratio for the heights and angles of contact before and after the tube is broken:
\[
\frac{h_1}{\cos \theta_1} = \frac{h_2}{\cos \theta_2}
\]
- Where \( h_1 = 9 \, \text{cm} \) and \( h_2 \) is the new height after breaking the tube at \( 5 \, \text{cm} \).
4. **Substituting Known Values**:
- For the initial situation:
- \( h_1 = 9 \, \text{cm} \)
- \( \theta_1 = 0^\circ \) (thus \( \cos \theta_1 = 1 \))
- For the broken tube:
- \( h_2 = 5 \, \text{cm} \)
- We need to find \( \theta_2 \).
5. **Calculating the New Height and Angle of Contact**:
- Substitute the known values into the ratio:
\[
\frac{9}{1} = \frac{5}{\cos \theta_2}
\]
- Rearranging gives:
\[
\cos \theta_2 = \frac{5}{9}
\]
- To find \( \theta_2 \), take the inverse cosine:
\[
\theta_2 = \cos^{-1} \left( \frac{5}{9} \right)
\]
6. **Final Results**:
- The height of the water column after breaking the tube is \( h_2 = 5 \, \text{cm} \).
- The angle of contact is \( \theta_2 = \cos^{-1} \left( \frac{5}{9} \right) \).
### Summary of Results:
- Height of the water column after breaking the tube: **5 cm**
- Angle of contact after breaking the tube: **\( \cos^{-1} \left( \frac{5}{9} \right) \)**
