Find density of a ball on a pendulum

brahle

First of all, sorry for my bad English as it isn't my native language. Also, sorry for poor formating.

1. The problem statement, all variables and given/known data
A pendulum consists of a rod of length [tex]L=1 m[/tex] and mass [tex]m_1=0.5 kg[/tex] and a ball (radius [tex]r=5 cm[/tex]) attached to one end of the rod. The axis of oscillation is perpendicular to the rod, and goes through the other end of the rod. The period of oscillation is [tex]T=2s[/tex]. Find the density [tex]\rho[/tex]of the ball.


2. Relevant equations
[tex]T=2 \pi \sqrt{\frac{I}{mgx}}[/tex]
[tex]m= \rho V= \rho \frac{4}{3}r^3 \pi[/tex]


3. The attempt at a solution
For this pendulum, the period is:
[tex]T=2 \pi \sqrt{\frac{I}{mgx}}[/tex]
where [tex]x[/tex] is the distance between the axis of oscillation and the center of mass and [tex]I[/tex] is the moment of inertia.

So, according to parallel axis theorem, the moment of inertia for the rod [tex]I_{rod}[/tex] is equal to:
[tex]I_{rod}=\frac{1}{3}m_1 L^2 + m_1 (\frac{1}{2}L - x)^2[/tex]
Similarly, the moment of inertia of the ball [tex]I_{ball}[/tex] is equal to:
[tex]I_{ball}=\frac{2}{5}m_2 r^2 + m_2 (L - x + r)^2[/tex]
Considering that [tex]I=I_{rod} + I_{ball}[/tex], we are left with two unknowns - [tex]x[/tex] and [tex]\rho[/tex]. That means we need another equation, and that's the part that isn't clear to me.

I suspect it has to do something with the fact that this is pretty much a one-dimensional problem and certain properties of center of mass. Unfortunately, I do not know what to do next.

Thanks a lot,
Bruno