Exercise 6

Consider two thin bars of constant cross-sectional area, each of length 10 cm, with respective mass density functions $\rho(x) = \frac{1}{1+x^2}$ and $p(x) = e^{-0.1x}$.

1. Find the mass of each bar.

2. Find the center of mass of each bar.

3. Now consider a new 10 cm bar whose mass density function is $f(x) = \rho(x) + p(x)$.

   1. Explain how you can easily find the mass of this new bar with little to no additional work.

   2. Similarly, compute $\int_0^{10} xf(x) \, dx$ as simply as possible, in light of earlier computations.

   3. True or false: the center of mass of this new bar is the average of the centers of mass of the two earlier bars. Write at least one sentence to say why your conclusion makes sense.