1. Axis of rotation passing through the center of the hemisphere and perpendicular to the base:
In this case, the moment of inertia (I) is:
I = (2/5)MR²
where:
* M is the mass of the hemisphere
* R is the radius of the hemisphere
2. Axis of rotation passing through the center of the base of the hemisphere:
In this case, the moment of inertia (I) is:
I = (83/320)MR²
Derivation:
These formulas are derived using integration and the definition of moment of inertia:
I = ∫ r² dm
where:
* r is the distance of a small mass element (dm) from the axis of rotation
The derivation involves dividing the hemisphere into infinitesimally small mass elements and integrating their contributions to the total moment of inertia.
Note:
The moment of inertia of a solid hemisphere is always greater than the moment of inertia of a solid sphere with the same mass and radius. This is because the mass is distributed further from the axis of rotation in the hemisphere.
