Question

Theorem of perpendicular axis is used in obtaining the moment of inertia of a

• A. Triangular lamina
• B. Square lamina
• C. Circular lamina
• D. Semi-circular lamina

Hint:

Remember that the Perpendicular Axis Theorem is strictly applicable only to 2D (planar) bodies. Do not attempt to apply it to 3D objects like spheres or cylinders.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The perpendicular axis theorem is a principle in rotational mechanics used to determine the moment of inertia of planar bodies (laminas).
Key Formula or Approach:
The theorem states that the moment of inertia of a planar lamina about an axis (\(z\)-axis) perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes (\(x\) and \(y\) axes) in its own plane, which intersect at the point where the perpendicular axis passes through it: \[ I_{zz} = I_{xx} + I_{yy} \]

Step 2: Detailed Explanation:
Let us see how this applies to a circular lamina:

For a circular lamina of radius \(r\), the polar moment of inertia about the perpendicular axis passing through its center is easily derived using polar coordinates: \[ I_{zz} = \int r^2 \, dA = \int_{0}^{R} r^2 (2\pi r \, dr) = 2\pi \left[ \frac{r^4}{4} \right]_{0}^{R} = \frac{\pi R^4}{2} \]
Due to the perfect rotational symmetry of the circle, the moments of inertia about any two perpendicular diameters in its plane are identical: \[ I_{xx} = I_{yy} \]
Applying the perpendicular axis theorem: \[ I_{zz} = I_{xx} + I_{yy} \implies \frac{\pi R^4}{2} = 2 I_{xx} \] \[ I_{xx} = I_{yy} = \frac{\pi R^4}{4} \] This theorem is classical and directly useful for obtaining the diametral moment of inertia of a circular lamina from its polar moment of inertia.
For triangular or semi-circular laminas, the lack of symmetry about perpendicular axes in the plane makes this theorem less directly useful for finding in-plane moments of inertia.

Step 3: Final Answer:
The theorem is primarily used to obtain the moment of inertia of a Circular lamina, which is Option (C).