Answer: To find the moment of inertia of multiple bodies, we need to use the parallel axis theorem. The parallel axis theorem states that the moment of inertia of a body about any axis parallel to its centroidal axis is equal to the moment of inertia about the centroidal axis plus the product of the body's mass and the square of the distance between the two axes.
Here are the steps to find the moment of inertia of multiple bodies:
Find the moment of inertia of each body about its centroidal axis using the appropriate formula for that body. For example, the moment of inertia of a solid cylinder about its central axis is 1/2MR^2, where M is the mass of the cylinder and R is the radius of the cylinder.
Determine the centroid of the system of bodies. This is the point through which the axis of rotation passes and is the point about which the moment of inertia is calculated.
Calculate the distance between the centroid and each body's centroidal axis.
Use the parallel axis theorem to find the moment of inertia of each body about the axis passing through the centroid of the system. Add up all the moments of inertia of each body about the centroid of the system to get the total moment of inertia.
The formula for the parallel axis theorem is:
I = I_cm + md^2
where I is the moment of inertia about the axis passing through the centroid of the system, I_cm is the moment of inertia about the centroidal axis of the body, m is the mass of the body, and d is the distance between the two axes.
By using this formula, we can find the moment of inertia of any system of bodies with respect to any axis parallel to the centroidal axis.
Explanation: