Moment of inertia of the rod about an axis passing through the end and perpendicular to the length of the rod is `I_(0)`. If axis of rotation is inclined at an angle `30^(@)` with the length then moment of inertia is found to be `I`. Calculate value of `I_(0)//I`

Explanation

To solve the problem, we need to calculate the ratio of the moment of inertia of the rod about an axis passing through one end and perpendicular to its length (denoted as \( I_0 \)) to the moment of inertia of the rod about an axis inclined at an angle of \( 30^\circ \) with the length of the rod (denoted as \( I \)).

### Step-by-Step Solution:

1. **Identify the Moment of Inertia \( I_0 \)**:
The moment of inertia of a rod about an axis passing through one end and perpendicular to its length is given by the formula:
\[ I_0 = \frac{1}{3} m l^2 \]
where \( m \) is the mass of the rod and \( l \) is its length.

2. **Determine the Moment of Inertia \( I \)**:
When the axis is inclined at an angle \( \theta = 30^\circ \) with the length of the rod, we need to find the moment of inertia \( I \). The formula for the moment of inertia when the axis is inclined is:
\[ I = I_0 \sin^2 \theta \]
Substituting \( \theta = 30^\circ \):
\[ I = I_0 \sin^2(30^\circ) \]
We know that \( \sin(30^\circ) = \frac{1}{2} \), thus:
\[ I = I_0 \left(\frac{1}{2}\right)^2 = I_0 \cdot \frac{1}{4} \]

3. **Calculate the Ratio \( \frac{I_0}{I} \)**:
To find the ratio \( \frac{I_0}{I} \):
\[ \frac{I_0}{I} = \frac{I_0}{I_0 \cdot \frac{1}{4}} = 4 \]

### Final Result:
The value of \( \frac{I_0}{I} \) is \( 4 \).