In the image below you see a rod with a sphere attached to its right side. The rod-sphere system is rotating around the entire system's center of mass. How do I apply parallel axis theorem to find the moment of inertia of the entire rod-sphere system rotating about the entire system's center of mass?
Here is my thought process and I would appreciate it if the community could give me a hint:
Apply Parallel Axis Theorem to rod and sphere separately and then add their moment of inertia's together to form the entire system's moment of inertia.
Given information:
length of rod = $L$
radius of sphere = $R$
mass of rod = $m_{r}$
mass of sphere= $m_{s}$
$I_{rod} = I_{rod,CM} + m_rd_{rod}^2$ (Parallel Axis Thm. applied to rod)
where $d_{rod}=$ distance from center of rod to center of mass of entire rod-sphere system.
$I_{sphere} = I_{sphere,CM} + m_sd_{sphere}^2 $ (Parallel Axis Thm. applied to sphere)
where $d_{sphere}=$ distance from center of sphere to center of mass of entire rod-sphere system.
So the final answer should be
$I_{tot} = I_{rod} + I_{sphere}$
But this is apparently wrong! So what am I doing wrong here?