Final answer:
To find the center of mass position of the rod, we calculate the weighted average of the centers of mass for two segments with different densities, resulting in a final position that reflects the non-uniform density distribution.
Step-by-step explanation:
To find the center of mass position of a rod with variable density, we must consider the density distribution along the length of the rod and use the definition of center of mass. The rod is divided into two segments: the first half from 0 to L/2 with density 2ρ0 and the second half from L/2 to L with density ρ0.
We know that the center of mass x_cm can be calculated using the formula:
x_cm = (∫x dm) / M
We calculate the integral for each section separately, since the density is different for each. The mass M is the total mass of the rod, which is the sum of the masses of both segments. Here's how each part of the rod contributes to the center of mass:
- For the first segment (0 to L/2): dm = 2ρ0 dx, the center of mass is at L/4.
- For the second segment (L/2 to L): dm = ρ0 dx, the center of mass is at 3L/4.
After integrating and including the mass of each segment, we find the weighted average of the two centers of mass to get the overall center of mass position of the entire rod.