A rod is placed along the line,`y=2x` with its centre at origin. The moment of inertia of the rod is maximum about.

Options

A. X-axis
B. Y-axis
C. Z-axis
D. Data insufficient

Explanation

To find the axis about which the moment of inertia of a rod placed along the line \(y = 2x\) with its center at the origin is maximum, we can follow these steps:

### Step 1: Understand the position of the rod
The rod is placed along the line \(y = 2x\). This means that the rod has a slope of 2 and is positioned diagonally in the xy-plane. The center of the rod is at the origin (0,0).

### Step 2: Determine the orientation of the rod
The line \(y = 2x\) can be expressed in vector form. The direction vector of the line can be taken as \((1, 2)\). Therefore, the rod can be represented as extending from \((-L/2, -L)\) to \((L/2, L)\), where \(L\) is the length of the rod.

### Step 3: Moment of Inertia about different axes
The moment of inertia \(I\) of a rod about an axis can be calculated using the following formulas:
- For an axis through the center of mass (perpendicular to the length of the rod):
\[ I_{cm} = \frac{1}{12} ML^2 \]
- For an axis through one end of the rod (perpendicular to the length of the rod):
\[ I_{end} = \frac{1}{3} ML^2 \]

### Step 4: Identify maximum moment of inertia
The moment of inertia is maximum when the axis of rotation is perpendicular to the length of the rod. Since the rod is oriented along the line \(y = 2x\), the axis that is perpendicular to this line will have a slope that is the negative reciprocal of 2, which is \(-\frac{1}{2}\).

### Step 5: Conclusion
The axis that is perpendicular to the rod and passes through the center of the rod will yield the maximum moment of inertia. This axis can be represented as the z-axis in a 3D coordinate system where the rod lies in the xy-plane.

Thus, the moment of inertia of the rod is maximum about the z-axis.