Center of a mass distribution Practice Problems Online

A bar with length $60\text{ cm}$ has linear density $\lambda=6x+12,$ where $x$ is in meters and $\lambda$ is in $\text{kg/m}.$ What is the center of mass of this bar?

A rod of length $5\text{ m}$ has density $\lambda(x)=3+x^2\text{ kg/m},$ and is positioned along the $x$ -axis with left endpoint at $x=0.$ What is the center of mass of the rod?

\displaystyle{ \frac{1119}{136}\text{ m}}

\displaystyle{ \frac{465}{136}\text{ m}}

\displaystyle{ \frac{369}{68}\text{ m}}

\displaystyle{ \frac{813}{68}\text{ m}}

Find the center of mass of a semicircular plate of radius $4$ with uniform density.

\displaystyle{ \left(0,\frac{16}{3\pi}\right)}

\displaystyle{ \left(0,\frac{8}{3\pi}\right)}

\displaystyle{ \left(0,\frac{16}{9\pi}\right)}

\displaystyle{ \left(0,\frac{8}{9\pi}\right)}

Find the center of mass of the region bounded by the line $y=4x$ and the parabola $y=4x^2.$

\displaystyle{ \left(\frac{1}{2},\frac{8}{5}\right)}

\displaystyle{ \left(\frac{1}{3},\frac{12}{5}\right)}

\displaystyle{ \left(\frac{1}{3},\frac{8}{5}\right)}

\displaystyle{ \left(\frac{1}{2},\frac{12}{5}\right)}

What is the center of mass of the region bounded by the curves $\begin{array}{c}&y=\cos 7x, &y=0, &x=0, &x=\pi/14\end{array},$ given that the mass has uniform density?

\displaystyle{ \left(\frac{\pi}{7}-\frac{1}{14},\frac{\pi}{14}\right)}

\displaystyle{ \left(\frac{\pi}{7}-\frac{1}{14},\frac{\pi}{8}\right)}

\displaystyle{ \left(\frac{\pi}{14}-\frac{1}{7},\frac{\pi}{14}\right)}

\displaystyle{ \left(\frac{\pi}{14}-\frac{1}{7},\frac{\pi}{8}\right)}

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