The moment of inertia of a ring about its geometrical axis is I, then its moment of inertia about its diameter will be

Options

A. `I`
B. `I/2`
C. `(3)/(4)I`
D. `(5)/(4)I`

Answer

To find the moment of inertia of a ring about its diameter given that its moment of inertia about its geometrical axis is \( I \), we can use the perpendicular axis theorem. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Understand the Problem**: We are given the moment of inertia of a ring about its geometrical axis (which is perpendicular to the plane of the ring) as \( I \). We need to find the moment of inertia about a diameter of the ring.

2. **Draw the Ring and Axes**: Visualize the ring in a coordinate system. Let’s denote the axis perpendicular to the plane of the ring as the \( z \)-axis, and the two diameters in the plane of the ring as the \( x \)-axis and \( y \)-axis.

3. **Apply the Perpendicular Axis Theorem**: The perpendicular axis theorem states that for a planar body, the moment of inertia about an axis perpendicular to the plane (in this case, the \( z \)-axis) is equal to the sum of the moments of inertia about two perpendicular axes lying in the plane (the \( x \)-axis and \( y \)-axis). Mathematically, this can be expressed as:
\[ I_z = I_x + I_y \]
where \( I_z \) is the moment of inertia about the \( z \)-axis, \( I_x \) is the moment of inertia about the \( x \)-axis, and \( I_y \) is the moment of inertia about the \( y \)-axis.

4. **Recognize Symmetry**: Since the ring is symmetric, the moments of inertia about the \( x \)-axis and \( y \)-axis are equal:
\[ I_x = I_y \]

5. **Substitute into the Equation**: Substitute \( I_y \) with \( I_x \) in the perpendicular axis theorem:
\[ I = I_x + I_x = 2I_x \]

6. **Solve for \( I_x \)**: Rearranging gives us:
\[ I_x = \frac{I}{2} \]

7. **Conclusion**: The moment of inertia of the ring about any diameter (either \( x \) or \( y \)) is:
\[ I_x = \frac{I}{2} \]

### Final Answer:
The moment of inertia of the ring about its diameter is \( \frac{I}{2} \).

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