So,
"a uniform steel rod of length 1.20 meters and mass 6.40 kg has attached to each end a small ball of mass 1.06 kg. The rod is constrained to rotate in a horizontal plane about a vertical axis through its midpoint."
There's more to the question, but the only part I'm really stuck on is Inertia. My friend tried to explain to me that the inertia of the system here is
I(tot) = I(ball) + I(rod)
= MR^2 + (1/12)MR^2
= M(L/2)^2 + (1/12)M(L)^2
...My question is why you can assume that the ball apparently has a radius of L/2. That seems like such an illogical, random assumption to make.
The ball doesn't have a radius of L/2 but it sits at a radius of L/2 from the axis of rotation. Inertia looks at the position of a certain mass from the axis of rotation.
Determining the inertia of a system can be a complex task, but in this case, we can make an assumption about the radius of the balls based on the given information in the problem.
The problem states that the rod is attached to each end of the small balls. We can interpret this situation as if the balls are point masses connected to the ends of the rod. In reality, the balls have a certain radius, but for the purpose of this analysis, we are assuming that the mass of each ball is concentrated at a single point.
Now, if we assume that the balls are point masses, it becomes easier to calculate the inertia of the system. The formula for inertia of a point mass rotating about an axis is I = MR^2, where M is the mass of the object and R is the radius from the axis of rotation to the mass.
For each ball, the radius from the axis of rotation (which is the midpoint of the rod) to the mass is defined as L/2. This assumption is based on the fact that the rod's length is given as 1.20 meters, and the balls are attached to each end of the rod.
Therefore, to find the inertia of the system, we sum the individual inertias of the ball and the rod using the equation I(tot) = I(ball) + I(rod). The inertia of the ball is calculated as MR^2, where R = L/2, and the inertia of the rod is calculated using the formula (1/12)ML^2.
It's important to note that this assumption simplifies the analysis and allows us to calculate the inertia of the system without knowing the exact dimensions of the balls. However, if the problem provided additional information about the size or radius of the balls, we would adjust our calculations accordingly.