Density, mass, and center of mass

Consider the following situations in which mass is distributed in a non-constant manner.

1. Suppose that a thin rod with constant cross-sectional area of 1 cm$^2$ has its mass distributed according to the density function $\rho(x) = 2e^{-0.2x}$, where $x$ is the distance in cm from the left end of the rod, and the units on $\rho(x)$ are g/cm. If the rod is 10 cm long, determine the exact mass of the rod.

2. Consider the cone that has a ba…

For quantities of equal weight, such as two children on a teeter-totter, the balancing point is found by taking the average of their locations. When the weights of the quantities differ, we use a weighted average of their respective locations to find the balancing point.

1. Suppose that a shelf is 6 feet long, with its left end situated at $x = 0$. If one book of weight 1 lb is placed at $x_1 = 0$, and another book…

Consider a thin bar of length 20 cm whose density is distributed according to the function $\rho(x) = 4 + 0.1x$, where $x = 0$ represents the left end of the bar. Assume that $\rho$ is measured in g/cm and $x$ is measured in cm.

1. Find the total mass, $M$, of the bar.

2. Without doing any calculations, do you expect the center of mass of the bar to be equal to 10, less than 10, or greater than 10? Why?

3. Compute $\overline{x}$, the exact center of mass…

Let a thin rod of length $a$ have density distribution function $\rho(x) = 10e^{-0.1x}$, where $x$ is measured in cm and $\rho$ in grams per centimeter.

1. If the mass of the rod is 30 g, what is the value of $a$?

2. For the 30g rod, will the center of mass lie at its midpoint, to the left of the midpoint, or to the right of the midpoint? Why?

3. For the 30g rod, find the center of mass, and compare your prediction in (b).

4. At what value of $x$…

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 ear…