An achromatic convergent lens of focal length `150 cm` is made by combining flint and crown glass lenses. Calculate the focal length of both the lenses and point out which one is divergent if the ratio of the dispersive power of flint and crown glasses is `3:2.`

Explanation

To solve the problem, we need to find the focal lengths of the flint and crown glass lenses that make up the achromatic lens with a total focal length of 150 cm. We also need to determine which lens is divergent.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Focal length of the achromatic lens, \( F = 150 \, \text{cm} \)
- Ratio of dispersive powers of flint glass to crown glass, \( \omega_1 : \omega_2 = 3 : 2 \)

2. **Using the Formula for Dispersive Power**:
- The dispersive power is inversely proportional to the focal length. Thus, we can express this relationship as:
\[ \frac{\omega_1}{\omega_2} = -\frac{f_1}{f_2} \]
- Substituting the given ratio:
\[ \frac{3}{2} = -\frac{f_1}{f_2} \]
- Rearranging gives us:
\[ f_1 = -\frac{3}{2} f_2 \]

3. **Using the Focal Length of the System**:
- For two lenses in combination, the formula for the total focal length \( F \) is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
- Substituting \( F = 150 \, \text{cm} \):
\[ \frac{1}{150} = \frac{1}{f_1} + \frac{1}{f_2} \]

4. **Substituting for \( f_1 \)**:
- Substitute \( f_1 = -\frac{3}{2} f_2 \) into the equation:
\[ \frac{1}{150} = -\frac{2}{3f_2} + \frac{1}{f_2} \]
- Finding a common denominator for the right side:
\[ \frac{1}{150} = \frac{-2 + 3}{3f_2} = \frac{1}{3f_2} \]

5. **Solving for \( f_2 \)**:
- Cross-multiplying gives:
\[ 3f_2 = 150 \implies f_2 = \frac{150}{3} = 50 \, \text{cm} \]

6. **Finding \( f_1 \)**:
- Now substitute \( f_2 \) back to find \( f_1 \):
\[ f_1 = -\frac{3}{2} f_2 = -\frac{3}{2} \times 50 = -75 \, \text{cm} \]

7. **Identifying the Divergent Lens**:
- The lens with a negative focal length is divergent. Thus, \( f_1 = -75 \, \text{cm} \) (flint glass) is the divergent lens, and \( f_2 = 50 \, \text{cm} \) (crown glass) is the convergent lens.

### Final Answer:
- Focal length of flint glass lens, \( f_1 = -75 \, \text{cm} \) (divergent)
- Focal length of crown glass lens, \( f_2 = 50 \, \text{cm} \) (convergent)