Moment of inertia of a uniform rod of length `L` and mass `M`, about an axis passing through `L//4` from one end and perpendicular to its length is

Options

A. `(7)/(36)ML^(2)`
B. `(7)/(48)ML^(2)`
C. `(11)/(48)ML^(2)`
D. `(ML^(2))/(12)`

Explanation

To find the moment of inertia of a uniform rod of length \( L \) and mass \( M \) about an axis passing through \( \frac{L}{4} \) from one end and perpendicular to its length, we can follow these steps:

### Step 1: Identify the Moment of Inertia about the Center of Mass
The moment of inertia \( I_{CM} \) of a uniform rod about an axis passing through its center of mass and perpendicular to its length is given by the formula:
\[ I_{CM} = \frac{1}{12} ML^2 \]

### Step 2: Determine the Distance from the Center of Mass to the New Axis
The center of mass of the rod is located at \( \frac{L}{2} \) from one end. The new axis is located at \( \frac{L}{4} \) from the same end. The distance \( x \) between the center of mass and the new axis is:
\[ x = \frac{L}{2} - \frac{L}{4} = \frac{L}{4} \]

### Step 3: Apply the Parallel Axis Theorem
According to the parallel axis theorem, the moment of inertia \( I \) about the new axis can be calculated using:
\[ I = I_{CM} + Md^2 \]
where \( d \) is the distance from the center of mass to the new axis. Substituting the values we have:
\[ I = \frac{1}{12} ML^2 + M\left(\frac{L}{4}\right)^2 \]

### Step 4: Calculate \( Md^2 \)
Now, calculate \( Md^2 \):
\[ Md^2 = M\left(\frac{L}{4}\right)^2 = M \cdot \frac{L^2}{16} = \frac{ML^2}{16} \]

### Step 5: Combine the Results
Now combine the results from Step 3 and Step 4:
\[ I = \frac{1}{12} ML^2 + \frac{ML^2}{16} \]

### Step 6: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple of 12 and 16 is 48. Therefore:
\[ I = \frac{4ML^2}{48} + \frac{3ML^2}{48} = \frac{7ML^2}{48} \]

### Final Result
Thus, the moment of inertia of the uniform rod about the given axis is:
\[ I = \frac{7ML^2}{48} \]