Moment of inertia of a ball filled with water

Question

Let's say, I have a ball filled with water fixed to a massles rod undergoing oscillations

I've been told that the moment of inertia can be taken as $m d^2$, $d$ being the distance from the pivot

I don't understand why it's not: $\frac{2}{5} mr^2 + md^2 $

Answer 1

If the ball is a sphere of radius $r$ with uniform density (i.e. its shell is very thin or has the same density as its contents) then, yes, the correct expression for the moment of inertia about an axis at a distance $d$ from the centre of the sphere is

$I = \frac 2 5 mr^2 + md^2$

But if $d$ is large compared to $r$ (say $d > 10r$) then the second term is much larger than the first and we can approximate the moment of inertia by

$ I \approx md^2$