The moment of inertia of a metre stick of mass 300 gm , abut an axis at right angles to the stick and located at 30 cm mark , is

Options

A. `8.3 xx10^(5) g - cm^(2)`
B. `5.8 g - cm^(2)`
C. `3.7 xx10^(5) g - cm^(2)`
D. none of these

Explanation

To find the moment of inertia of a meter stick of mass 300 gm about an axis at right angles to the stick and located at the 30 cm mark, we can follow these steps:

### Step-by-Step Solution

1. **Identify the parameters**:
- Mass of the meter stick, \( m = 300 \, \text{g} = 0.3 \, \text{kg} \) (since we often work in SI units).
- Length of the meter stick, \( L = 100 \, \text{cm} = 1 \, \text{m} \).
- The axis of rotation is at the 30 cm mark.

2. **Find the moment of inertia about the center of mass**:
The moment of inertia \( I_{cm} \) of a uniform rod about an axis through its center of mass is given by the formula:
\[ I_{cm} = \frac{mL^2}{12} \]
For a meter stick, the center of mass is at the 50 cm mark. Therefore, we calculate:
\[ I_{cm} = \frac{0.3 \times (1)^2}{12} = \frac{0.3}{12} = 0.025 \, \text{kg m}^2 \]

3. **Apply the Parallel Axis Theorem**:
The Parallel Axis Theorem states that if you know the moment of inertia about the center of mass, you can find the moment of inertia about any parallel axis by:
\[ I = I_{cm} + md^2 \]
Where \( d \) is the distance between the two axes. The distance \( d \) from the center of mass (50 cm) to the 30 cm mark is:
\[ d = 50 \, \text{cm} - 30 \, \text{cm} = 20 \, \text{cm} = 0.2 \, \text{m} \]

4. **Calculate the moment of inertia about the 30 cm mark**:
Now, substituting the values into the Parallel Axis Theorem:
\[ I = I_{cm} + md^2 = 0.025 + 0.3 \times (0.2)^2 \]
\[ I = 0.025 + 0.3 \times 0.04 = 0.025 + 0.012 = 0.037 \, \text{kg m}^2 \]

5. **Convert to gram centimeter square**:
Since the question asks for the answer in gram centimeter square, we convert:
\[ 0.037 \, \text{kg m}^2 = 0.037 \times 10^4 \, \text{g cm}^2 = 3700 \, \text{g cm}^2 \]

### Final Answer
The moment of inertia of the meter stick about the axis at the 30 cm mark is:
\[ I = 3700 \, \text{g cm}^2 \]