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

Center of Mass

Summary

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

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

For a rod on $[a,b]$ with density $\rho(x)$, the center of mass equals

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

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.

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