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.

Homework Statement

A pendulum consists of a rod of length $$L=1 m$$ and mass $$m_1=0.5 kg$$ and a ball (radius $$r=5 cm$$) 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 $$T=2s$$. Find the density $$\rho$$of the ball.

Homework Equations

$$T=2 \pi \sqrt{\frac{I}{mgx}}$$
$$m= \rho V= \rho \frac{4}{3}r^3 \pi$$

The Attempt at a Solution

For this pendulum, the period is:
$$T=2 \pi \sqrt{\frac{I}{mgx}}$$
where $$x$$ is the distance between the axis of oscillation and the center of mass and $$I$$ is the moment of inertia.

So, according to parallel axis theorem, the moment of inertia for the rod $$I_{rod}$$ is equal to:
$$I_{rod}=\frac{1}{3}m_1 L^2 + m_1 (\frac{1}{2}L - x)^2$$
Similarly, the moment of inertia of the ball $$I_{ball}$$ is equal to:
$$I_{ball}=\frac{2}{5}m_2 r^2 + m_2 (L - x + r)^2$$
Considering that $$I=I_{rod} + I_{ball}$$, we are left with two unknowns - $$x$$ and $$\rho$$. 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

 

[anathema]

Dear Bruno,

No need to apologize for your English, it is perfectly understandable. As for your question, you are on the right track with using the parallel axis theorem to calculate the moment of inertia for the rod and ball. However, you are correct in thinking that you need another equation to solve for x and \rho.

One way to approach this problem is to use the equation for the period of oscillation, T=2 \pi \sqrt{\frac{I}{mgx}}, and substitute in the moment of inertia values you have calculated for the rod and ball. This will give you an equation with two unknowns, x and \rho, but you can also use the fact that the period T is given as 2 seconds to solve for one of the unknowns.

Another approach is to use the equation for the total mass of the pendulum, m= \rho V= \rho \frac{4}{3}r^3 \pi, and substitute in the values for the mass of the rod and ball. This will give you an equation with one unknown, \rho, which you can then solve for.

I hope this helps. Keep up the good work in your studies of physics!