Solid Cone Moment of Inertia Calculation

The question asks for the moment of inertia of a solid cone about its vertical axis. The vertical axis typically refers to the axis passing through the apex and the center of the circular base.

Understanding Moment of Inertia

Moment of inertia ($I$) is a measure of an object's resistance to changes in its rotational motion. It depends on the object's mass distribution relative to the axis of rotation. For a solid cone, the mass is distributed throughout its volume.

Moment of Inertia Formula for a Solid Cone

The standard formula for the moment of inertia of a solid cone with mass $M$ and base radius $R$, rotating about its central vertical axis (the axis of symmetry passing through the apex and the center of the base) is given by:

$ I = \frac{3}{10} M R^2 $

This formula is derived using calculus by integrating the contributions of infinitesimal mass elements within the cone relative to the axis of rotation.

Comparing with Options

Let's compare this standard formula with the provided options:

• Option 1: $MR^2/10$
• Option 2: $3MR^2/10$
• Option 3: $5MR^2/10$
• Option 4: $7MR^2/10$

The formula derived from physics principles, $I = \frac{3}{10} M R^2$, directly matches Option 2.

Conclusion

Therefore, the moment of inertia of a solid cone about its vertical axis is $ \frac{3}{10} M R^2 $.