Final answer:
The mass and center of mass of a rod with a linear density given by μ(x) = 50.0 + 21.0x can be found by integration of the linear density over the rod's length for mass, and by using the definition of center of mass for a continuous distribution, divided by the total mass, for the position of the center of mass.
Step-by-step explanation:
To find the mass of a rod with variable linear density, we integrate the linear density function over the length of the rod. In this case, the linear density (μ) is given as a function of x: μ(x) = 50.0 + 21.0x grams/meter. The length of the rod (L) is 30.50 cm or 0.305 meters. Therefore, the mass (m) is found by integrating the linear density function from 0 to L:
m = ∫0L μ(x) dx = ∫00.305 (50.0 + 21.0x) dx
Thus, the mass is given by the antiderivative evaluated from 0 to 0.305 meters.
To find the center of mass (xcm) of the rod, we use the definition of center of mass for a continuous distribution:
xcm = (1/m) ∫0L x μ(x) dx
We plug in the given linear density function and integrate from x=0 to x=L. After finding the integral, we divide by the total mass of the rod to find the position of the center of mass.