Question

If the moment of inertia of a circular disc about its central axis is I, then that for the same disc about its diameter is

• A. I
• B. 2I
• C. 4I
• D. I/2
• E. I/4

Hint:

For a disc, the moment of inertia about a diameter is half the moment of inertia about the central perpendicular axis.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires the use of the Perpendicular Axis Theorem to relate the moment of inertia of a planar object about an axis perpendicular to its plane to the moments of inertia about two perpendicular axes lying in its plane.
Step 2: Key Formula or Approach:
The Perpendicular Axis Theorem states that for a planar lamina, the moment of inertia (\(I_z\)) about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia (\(I_x\) and \(I_y\)) about two mutually perpendicular axes lying in the plane of the lamina and intersecting at the point where the perpendicular axis passes through it. \[ I_z = I_x + I_y \] Step 3: Detailed Explanation:
Let the circular disc lie in the x-y plane.
The "central axis" is the axis perpendicular to the plane of the disc and passing through its center. Let's call this the z-axis. The moment of inertia about this axis is given as \(I_z = I\).
A "diameter" of the disc is an axis lying in the plane of the disc and passing through its center. We can choose any two perpendicular diameters as our x-axis and y-axis.
Due to the symmetry of the circular disc, the moment of inertia about any diameter is the same. Therefore, the moment of inertia about the x-axis (\(I_x\)) is equal to the moment of inertia about the y-axis (\(I_y\)). Let's call this value \(I_{diameter}\). \[ I_x = I_y = I_{diameter} \]
Now, we apply the Perpendicular Axis Theorem: \[ I_z = I_x + I_y \] Substituting the known terms: \[ I = I_{diameter} + I_{diameter} \] \[ I = 2 \times I_{diameter} \] Solving for the moment of inertia about the diameter (\(I_{diameter}\)): \[ I_{diameter} = \frac{I}{2} \] Step 4: Final Answer:
The moment of inertia of the disc about its diameter is I/2. This corresponds to option (D).