The momentum of inertia of a body rotating about a given axis is `12.0 kg m^(2)` in the SI system . What is the value of the moment of inertia in a system of units in which the unit of lengths is `5 cm` and the unit of mass is `10 g`?

Options

A. a. ` 2.4 xx 10^(3)`
B. b. `6.0 xx 10^(3)`
C. c. `5.4 xx 10^(5)`
D. d. `4.8 xx 10^(5)`

Explanation

To find the moment of inertia in a new system of units where the unit of length is 5 cm and the unit of mass is 10 g, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Dimensions of Moment of Inertia**:
The moment of inertia (I) has the dimensions of mass times length squared, which can be represented as:
\[ [I] = M L^2 \]

2. **Identify the Given Values**:
- Moment of inertia in SI units: \( I_1 = 12.0 \, \text{kg m}^2 \)
- Unit of mass in the new system: \( M_2 = 10 \, \text{g} \)
- Unit of length in the new system: \( L_2 = 5 \, \text{cm} \)

3. **Convert Units to Consistent SI Units**:
- Convert \( M_2 \) to kilograms:
\[ M_2 = 10 \, \text{g} = 0.01 \, \text{kg} \]
- Convert \( L_2 \) to meters:
\[ L_2 = 5 \, \text{cm} = 0.05 \, \text{m} \]

4. **Set up the Conversion Formula**:
The relationship between the moment of inertia in the two systems can be expressed as:
\[ I_2 = I_1 \times \left( \frac{M_1}{M_2} \right) \times \left( \frac{L_1}{L_2} \right)^2 \]
where \( M_1 = 1 \, \text{kg} \) and \( L_1 = 1 \, \text{m} \) in the SI system.

5. **Substitute the Values**:
Substitute the known values into the equation:
\[ I_2 = 12.0 \, \text{kg m}^2 \times \left( \frac{1 \, \text{kg}}{0.01 \, \text{kg}} \right) \times \left( \frac{1 \, \text{m}}{0.05 \, \text{m}} \right)^2 \]

6. **Calculate the Ratios**:
- Calculate \( \frac{1 \, \text{kg}}{0.01 \, \text{kg}} = 100 \)
- Calculate \( \frac{1 \, \text{m}}{0.05 \, \text{m}} = 20 \)
- Therefore, \( \left( \frac{1 \, \text{m}}{0.05 \, \text{m}} \right)^2 = 20^2 = 400 \)

7. **Combine the Values**:
Now, substitute these values back into the equation:
\[ I_2 = 12.0 \times 100 \times 400 \]

8. **Final Calculation**:
\[ I_2 = 12.0 \times 40000 = 480000 \, \text{kg cm}^2 \]

9. **Convert to the Desired Units**:
Since we want the moment of inertia in terms of the new units, we can express it as:
\[ I_2 = 4.8 \times 10^5 \, \text{g cm}^2 \]

### Final Answer:
The moment of inertia in the new system of units is:
\[ I_2 = 4.8 \times 10^5 \, \text{g cm}^2 \]