§6.3 Density, mass, and center of mass (continued)

Center of Mass

Summary

For the bar with $\rho(x) = 4+0.1x$ g/cm on $[0,20]$, compute the total mass in grams.

A 20 cm bar has density $\rho(x) = 4+0.1x$ g/cm. Without computing, the center of mass should satisfy

Using $\overline{x} = \frac{\int_0^{20}x(4+0.1x)\,dx}{\int_0^{20}(4+0.1x)\,dx}$ for the bar with $\rho(x) = 4+0.1x$, the exact center of mass is $\frac{32}{3}$. Report the decimal, rounded to three places.

The same 20 cm bar with $\rho(x) = 4+0.1x$ has total mass 100 g. Its average density equals

A 20 cm bar with density $p(x) = 4e^{0.020732x}$ has its center of mass slightly to the right of the center of mass of the bar with $\rho(x) = 4+0.1x$, because $p$ exceeds $\rho$ near the right end.

A 20 cm bar with density $p(x) = 4e^{0.020732x}$ g/cm has center of mass $\frac{\int_0^{20}xp(x)\,dx}{\int_0^{20}p(x)\,dx} \approx 10.689$. Report this value, rounded to three decimals.

A rod with density $\rho(x) = 10e^{-0.1x}$ g/cm has mass $100(1-e^{-0.1a})$ grams when its length is $a$. Find the length $a$ that makes the mass exactly 30 g, given $a = -10\ln(0.7)$. Report the decimal, rounded to three places.

For the same 30 g rod with $\rho(x) = 10e^{-0.1x}$, the equal mass cut satisfies $100(1-e^{-0.1q}) = 15$, so $q = -10\ln(0.85)$. Report the cut location, rounded to three decimals.

For the 30 g rod with $\rho(x) = 10e^{-0.1x}$ g/cm, the center of mass lies to the left of the rod's midpoint.

If a new bar has density $f(x) = \rho(x)+p(x)$ built from two earlier bars, the center of mass of the new bar equals the plain average of the two earlier centers of mass.

The bat generated by revolving $f(x) = 2xe^{-1.25x}+(30-x)e^{-0.25(30-x)}$ on $[0,30]$ has volume about $52.0666$ cubic inches and uniform weight density $0.6$ ounces per cubic inch. Compute its total weight in ounces, rounded to two decimals.

Using the linear weight density $\rho(x) = 0.6\pi f(x)^2$ along the 30-inch bat, the center of mass computes to $\overline{x} \approx 23.214$. This tells us the balance point sits