The moment of inertia of a thin uniform rod of mass `M` and length `L` about an axis passing through its mid-point and perpendicular to its length is`I_0`. Its moment of inertia about an axis passing through one of its ends perpendicular to its length is.

Options

A. `I_(0)+ML^(2)//4`
B. `I_(0)+2ML^(2)`
C. `I_(0)+ML^(2)`
D. `I_(0)+ML^(2)//2`

Answer

To solve the problem of finding the moment of inertia of a thin uniform rod about an axis passing through one of its ends and perpendicular to its length, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have a thin uniform rod of mass `M` and length `L`.
- The moment of inertia about an axis passing through its midpoint and perpendicular to its length is given as `I_0`.

2. **Identify the Required Moment of Inertia**:
- We need to find the moment of inertia `I_1` about an axis passing through one of its ends and perpendicular to its length.

3. **Use the Parallel Axis Theorem**:
- The Parallel Axis Theorem states that:
\[ I = I_{cm} + Mh^2 \]
where:
- \(I\) is the moment of inertia about the new axis,
- \(I_{cm}\) is the moment of inertia about the center of mass axis,
- \(M\) is the mass of the object,
- \(h\) is the distance between the two axes.

4. **Determine the Values**:
- Here, the moment of inertia about the center of mass (midpoint) is `I_0`.
- The distance \(h\) from the center of mass to the end of the rod is \(L/2\).

5. **Substitute the Values into the Parallel Axis Theorem**:
- Now we can substitute the known values into the equation:
\[ I_1 = I_0 + M \left(\frac{L}{2}\right)^2 \]

6. **Simplify the Expression**:
- Calculate \(M \left(\frac{L}{2}\right)^2\):
\[ M \left(\frac{L}{2}\right)^2 = M \frac{L^2}{4} \]
- Therefore, the moment of inertia about the end of the rod becomes:
\[ I_1 = I_0 + \frac{ML^2}{4} \]

7. **Final Result**:
- Thus, the moment of inertia of the rod about the end is:
\[ I_1 = I_0 + \frac{ML^2}{4} \]