Moment of inertia of a rod of mass m and length `l` about its one end is `I`. If one-fourth of its length is cut away, then moment of inertia of the remaining rod about its one end will be

Options

A. `(3)/(4)I`
B. `(9)/(16)I`
C. `(27)/(64)I`
D. `(I)/(16)`

Explanation

To solve the problem of finding the moment of inertia of a remaining rod after cutting one-fourth of its length, we can follow these steps:

### Step 1: Understand the Moment of Inertia of the Original Rod
The moment of inertia \( I \) of a rod of mass \( m \) and length \( l \) about one end is given by the formula:
\[ I = \frac{1}{3} m l^2 \]

### Step 2: Determine the New Length of the Rod
If one-fourth of the rod's length is cut away, the new length \( l' \) of the remaining rod is:
\[ l' = l - \frac{1}{4}l = \frac{3}{4}l \]

### Step 3: Determine the New Mass of the Remaining Rod
Assuming the mass is uniformly distributed along the length of the rod, the new mass \( m' \) of the remaining rod can be calculated as:
\[ m' = \frac{m}{l} \times l' = \frac{m}{l} \times \frac{3}{4}l = \frac{3}{4}m \]

### Step 4: Calculate the Moment of Inertia of the Remaining Rod
Now, we can calculate the moment of inertia \( I' \) of the remaining rod about its one end using the new mass \( m' \) and the new length \( l' \):
\[ I' = \frac{1}{3} m' (l')^2 \]
Substituting the values of \( m' \) and \( l' \):
\[ I' = \frac{1}{3} \left(\frac{3}{4}m\right) \left(\frac{3}{4}l\right)^2 \]
\[ I' = \frac{1}{3} \left(\frac{3}{4}m\right) \left(\frac{9}{16}l^2\right) \]
\[ I' = \frac{27}{64} \cdot \frac{1}{3} m l^2 \]
Since \( \frac{1}{3} m l^2 = I \):
\[ I' = \frac{27}{64} I \]

### Step 5: Conclusion
Thus, the moment of inertia of the remaining rod about its one end is:
\[ I' = \frac{27}{64} I \]

### Final Answer
The correct option is \( \frac{27}{64} I \).

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