By the theorem of parallel axes:

Options

A. `I=I_(G)-md^(2)`
B. `I_(G)=I+md^(2)`
C. `I+I_(G)=md^(2)`
D. `I_(G)=I-md^(2)`

Answer

To solve the problem using the theorem of parallel axes, we will follow these steps:

### Step 1: Understand the Parallel Axis Theorem
The parallel axis theorem states that the moment of inertia \( I \) about any axis parallel to an axis through the center of mass can be calculated using the formula:
\[ I = I_g + md^2 \]
where:
- \( I \) = moment of inertia about the new axis
- \( I_g \) = moment of inertia about the center of mass axis
- \( m \) = mass of the body
- \( d \) = distance between the two parallel axes

### Step 2: Identify the Components
In this scenario, we need to identify:
- The moment of inertia about the center of mass \( I_g \)
- The mass \( m \) of the body
- The distance \( d \) between the center of mass axis and the new axis

### Step 3: Apply the Formula
Using the identified components, we can apply the formula:
\[ I = I_g + md^2 \]
This equation allows us to calculate the moment of inertia about the new axis if we know the moment of inertia about the center of mass and the distance between the axes.

### Step 4: Select the Correct Option
According to the options provided, we need to find which option correctly represents the parallel axis theorem. The correct representation is:
\[ I = I_g + md^2 \]
This corresponds to option 4 in the question.

### Step 5: Conclusion
Thus, the correct answer is option 4, which states:
\[ I = I_g + md^2 \]

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