If units in two systems of measurments are in the ratio 2 : 3, the ratio of the units of angular momentum in those two systems in

Options

A. `2:3`
B. `9 : 4`
C. `1:1`
D. `4:9`

Answer

To solve the problem, we need to find the ratio of the units of angular momentum in two systems of measurement where the units are in the ratio of 2:3.

### Step-by-Step Solution:

1. **Understanding Angular Momentum**:
The unit of angular momentum (L) is given by the formula:
\[ L = \text{mass} \times \text{distance}^2 \div \text{time} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1} \]

2. **Identifying the Ratios**:
We are given that the units in two systems of measurement are in the ratio of 2:3. This means:
- For mass (kg), the ratio is \( \frac{2}{3} \)
- For distance (m), the ratio is \( \frac{2}{3} \)
- For time (s), the ratio is \( \frac{2}{3} \)

3. **Calculating the Ratio of Angular Momentum**:
We can express the ratio of angular momentum in the two systems as follows:
\[ \text{Ratio of Angular Momentum} = \left(\frac{\text{mass in system 1}}{\text{mass in system 2}}\right) \times \left(\frac{\text{distance in system 1}^2}{\text{distance in system 2}^2}\right) \div \left(\frac{\text{time in system 1}}{\text{time in system 2}}\right) \]

Substituting the ratios:
\[ = \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right)^2 \div \left(\frac{2}{3}\right) \]

4. **Simplifying the Expression**:
Now, we simplify the expression:
\[ = \left(\frac{2}{3}\right) \times \left(\frac{4}{9}\right) \div \left(\frac{2}{3}\right) \]

The division by \( \frac{2}{3} \) can be rewritten as multiplication by its reciprocal:
\[ = \left(\frac{2}{3}\right) \times \left(\frac{4}{9}\right) \times \left(\frac{3}{2}\right) \]

5. **Cancelling Terms**:
The \( \frac{2}{3} \) in the numerator and denominator cancels out:
\[ = \frac{4}{9} \]

6. **Final Result**:
Therefore, the ratio of the units of angular momentum in the two systems is:
\[ \text{Ratio} = \frac{4}{9} \quad \text{or} \quad 4:9 \]

### Final Answer:
The ratio of the units of angular momentum in the two systems is \( 4:9 \).

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