Final answer:
To find the mass and moments of inertia of a solid in various coordinate systems, one must perform integration over the object's volume in the respective coordinates. The moment of inertia involves the distance squared from each mass element to the axis of rotation and is calculated differently for each coordinate system, often using the parallel-axis theorem when needed.
Step-by-step explanation:
Finding Mass and Moments of Inertia in Various Coordinates
The mass (m) of a solid can be found using different coordinate systems, which include spherical, cylindrical, rectangular, and polar coordinates. Each system will have its method of integration to account for the mass distribution throughout the solid. For moments of inertia (Iₓ, I_y, and I_z), the integrals become more complex as they involve the distance squared from every differential mass element to the axis of rotation.
The moment of inertia is the rotational equivalent of mass in translational motion and depends on how mass is distributed relative to the axis of rotation. For a solid body, one typically finds the moment of inertia by integrating r²dm over the volume of the object, where r is the distance from the axis and dm is the mass element of the object.
For instance, the moment of inertia of a rod about an axis through one end is ML²/3 and for a solid sphere about its center is (2/5)mR². Additionally, the parallel-axis theorem is often used when the axis of rotation does not pass through the center of mass, and it states that I = I_cm + Md², where I_cm is the moment of inertia about the center of mass, M is the mass of the object, and d is the distance from the center of mass to the new axis.