What is the formula for moment of inertia of a semi hollow sphere?
The concept of a "semi-hollow sphere" is not standard in physics, as the typical definitions include either a solid sphere or a hollow sphere (spherical shell). However, if we interpret a "semi-hollow sphere" as a sphere with a uniform thickness but not completely hollow, we can approach this by considering it as a composite of a solid outer sphere and a solid inner sphere.
For a hollow sphere (or spherical shell) with a mass ( M ) and radius ( R ), the moment of inertia about an axis through its center and perpendicular to a diameter is given by:
To calculate the moment of inertia of a semi-hollow sphere, you would need to consider the moments of inertia of the outer and inner solid spheres and then subtract the moment of inertia of the inner sphere from that of the outer sphere.
For a solid sphere, the moment of inertia about its central axis is:
[ I = \frac{2}{5} MR^2 ]
If you have a semi-hollow sphere with an inner radius ( R_i ) and an outer radius ( R_o ), you would calculate the moments of inertia for the solid spheres with these radii and then subtract the inner from the outer:
[ I{\text{semi-hollow}} = I{\text{outer}} - I{\text{inner}} ] [ I{\text{semi-hollow}} = \frac{2}{5} M_o R_o^2 - \frac{2}{5} M_i R_i^2 ]
Here, ( M_o ) and ( M_i ) are the masses of the outer and inner solid spheres, respectively. However, since the mass distribution is uniform, you can express ( M_o ) and ( M_i ) in terms of the density and volumes of the spheres.
For a uniform density ( \rho ):
[ M_o = \rho V_o = \rho \frac{4}{3} \pi R_o^3 ] [ M_i = \rho V_i = \rho \frac{4}{3} \pi R_i^3 ]
Substituting these into the equation for ( I_{\text{semi-hollow}} ):
[ I_{\text{semi-hollow}} = \frac{2}{5} \rho \frac{4}{3} \pi R_o^5 - \frac{2}{5} \rho \frac{4}{3} \pi R_i^5 ]
This approach allows you to calculate the moment of inertia for a semi-hollow sphere based on the moments of inertia of solid spheres and the principles of uniform mass distribution45.