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

Options

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

Explanation

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

### Step-by-Step Solution:

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

2. **Determine the Length and Mass of the Remaining Rod**:
If one-fourth of the rod's length is cut away, the remaining length \( l' \) of the rod is:
\[ l' = l - \frac{l}{4} = \frac{3l}{4} \]
Since the rod is uniform, the mass of the remaining rod \( m' \) is:
\[ m' = m - \frac{m}{4} = \frac{3m}{4} \]

3. **Calculate the Moment of Inertia of the Remaining Rod**:
The moment of inertia \( I' \) of the remaining rod about the same end can be calculated using the formula for the moment of inertia of a rod about one end:
\[ I' = \frac{1}{3} m' (l')^2 \]
Substituting \( m' \) and \( l' \):
\[ I' = \frac{1}{3} \left(\frac{3m}{4}\right) \left(\frac{3l}{4}\right)^2 \]
Simplifying this:
\[ I' = \frac{1}{3} \left(\frac{3m}{4}\right) \left(\frac{9l^2}{16}\right) \]
\[ I' = \frac{3m \cdot 9l^2}{3 \cdot 4 \cdot 16} = \frac{27ml^2}{192} \]
\[ I' = \frac{27}{192} ml^2 = \frac{9}{64} ml^2 \]

4. **Relate to the Original Moment of Inertia**:
Since the original moment of inertia \( I \) was:
\[ I = \frac{1}{3} ml^2 \]
We can express \( I' \) in terms of \( I \):
\[ I' = \frac{9}{64} ml^2 = \frac{9}{64} \cdot \frac{3}{1} I = \frac{27}{64} I \]

### Final Answer:
The moment of inertia of the remaining rod about its one end is:
\[ I' = \frac{27}{64} I \]