1.5E: Exercises

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This page is a draft and is under active development.

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Moments and Centers of Mass

For the following exercises, calculate the center of mass for the collection of masses given.

Exercise $\PageIndex{1}$

$\displaystyle m_1=2$ at $\displaystyle x_1=1$ and $\displaystyle m_2=4$ at $\displaystyle x_2=2$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{2}$

$\displaystyle m_1=1$ at $\displaystyle x_1=−1$ and $\displaystyle m_2=3$ at $\displaystyle x_2=2$

Answer: $\displaystyle \frac{5}{4}$

Exercise $\PageIndex{3}$

$\displaystyle m=3$ at $\displaystyle x=0,1,2,6$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{4}$

Unit masses at $\displaystyle (x,y)=(1,0),(0,1),(1,1)$

Answer: $\displaystyle (\frac{2}{3},\frac{2}{3})$

Exercise $\PageIndex{5}$

$\displaystyle m_1=1$ at $\displaystyle (1,0)$ and $\displaystyle m_2=4$ at $\displaystyle (0,1)$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{6}$

$\displaystyle m_1=1$ at $\displaystyle (1,0)$ and $\displaystyle m_2=3$ at $\displaystyle (2,2)$

Answer: $\displaystyle (\frac{7}{4},\frac{3}{2})$

For the following exercises, compute the center of mass x–.

Exercise $\PageIndex{7}$

$\displaystyle ρ=1$ for $\displaystyle x∈(−1,3)$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{8}$

$\displaystyle ρ=x^2$ for $\displaystyle x∈(0,L)$

Answer: $\displaystyle \frac{3L}{4}$

Exercise $\PageIndex{9}$

$\displaystyle ρ=1$ for $\displaystyle x∈(0,1)$ and $\displaystyle ρ=2$ for $\displaystyle x∈(1,2)$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{10}$

$\displaystyle ρ=sinx$ for $\displaystyle x∈(0,π)$

Answer: $\displaystyle \frac{π}{2}$

Exercise $\PageIndex{11}$

$\displaystyle ρ=cosx$ for $\displaystyle x∈(0,\frac{π}{2})$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{12}$

$\displaystyle ρ=e^x$ for $\displaystyle x∈(0,2)$

Answer: $\displaystyle \frac{e^2+1}{e^2−1}$

Exercise $\PageIndex{13}$

$\displaystyle ρ=x^3+xe^{−x}$ for $\displaystyle x∈(0,1)$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{14}$

$\displaystyle ρ=xsinx$ for $\displaystyle x∈(0,π)$

Answer: $\displaystyle \frac{π^2−4}{π}$

Exercise $\PageIndex{15}$

$\displaystyle ρ=\sqrt{x}$ for $\displaystyle x∈(1,4)$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{16}$

$\displaystyle ρ=lnx$ for $\displaystyle x∈(1,e)$

Answer: $\displaystyle \frac{1}{4}(1+e^2)$

For the following exercises, compute the center of mass $\displaystyle (\bar{x},\bar{y})$. Use symmetry to help locate the center of mass whenever possible.

Exercise $\PageIndex{17}$

$\displaystyle ρ=7$ in the square $\displaystyle 0≤x≤1, 0≤y≤1$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{18}$

$\displaystyle ρ=3$ in the triangle with vertices $\displaystyle (0,0), (a,0)$, and $\displaystyle (0,b)$

Answer: $\displaystyle (\frac{a}{3},\frac{b}{3})$

Exercise $\PageIndex{19}$

$\displaystyle ρ=2$ for the region bounded by $\displaystyle y=cos(x), y=−cos(x), x=−\frac{π}{2}$, and $\displaystyle x=\frac{π}{2}$

Answer: Add texts here. Do not delete this text first.

For the following exercises, use a calculator to draw the region, then compute the center of mass $\displaystyle (\bar{x},\bar{y})$. Use symmetry to help locate the center of mass whenever possible.

Exercise $\PageIndex{20}$

The region bounded by $\displaystyle y=cos(2x), x=−\frac{π}{4}$, and $\displaystyle x=\frac{π}{4}$

Answer: $\displaystyle (0,\frac{π}{8})$

Exercise $\PageIndex{21}$

The region between $\displaystyle y=2x^2, y=0, x=0,$ and $\displaystyle x=1$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{22}$

The region between $\displaystyle y=\frac{5}{4}x^2$ and $\displaystyle y=5$

Answer: $\displaystyle (0,3)$

Exercise $\PageIndex{23}$

Region between $\displaystyle y=\sqrt{x}, y=ln(x), x=1,$ and $\displaystyle x=4$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{24}$

The region bounded by $\displaystyle y=0, \frac{x^2}{4}+\frac{y^2}{9}=1$

Answer: $\displaystyle (0,\frac{4}{π})$

Exercise $\PageIndex{25}$

The region bounded by $\displaystyle y=0, x=0,$ and $\displaystyle \frac{x^2}{4}+\frac{y^2}{9}=1$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{26}$

The region bounded by $\displaystyle y=x^2$ and $\displaystyle y=x^4$ in the first quadrant

Answer: $\displaystyle (\frac{5}{8},\frac{1}{3})$

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Exercise $\PageIndex{27}$

Rotating $\displaystyle y=mx$ around the $\displaystyle x$-axis between $\displaystyle x=0$ and $\displaystyle x=1$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{28}$

Rotating $\displaystyle y=mx$ around the $\displaystyle y$-axis between $\displaystyle x=0$ and $\displaystyle x=1$

Answer: $\displaystyle \frac{mπ}{3}$

Exercise $\PageIndex{29}$

A general cone created by rotating a triangle with vertices $\displaystyle (0,0), (a,0),$ and $\displaystyle (0,b)$ around the $\displaystyle y$-axis. Does your answer agree with the volume of a cone?

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{30}$

A general cylinder created by rotating a rectangle with vertices $\displaystyle (0,0), (a,0),(0,b),$ and $\displaystyle (a,b)$ around the $\displaystyle y$ -axis. Does your answer agree with the volume of a cylinder?

Answer: $\displaystyle πa^2b$

Exercise $\PageIndex{31}$

A sphere created by rotating a semicircle with radius $\displaystyle a$ around the $\displaystyle y$-axis. Does your answer agree with the volume of a sphere?

Answer: Add texts here. Do not delete this text first.

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $\displaystyle M$ and the centroid $\displaystyle (\bar{x},\bar{y})$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

Exercise $\PageIndex{32}$

Quarter-circle: $\displaystyle y=\sqrt{1−x^2}, y=0$, and $\displaystyle x=0$

Answer: $\displaystyle (\frac{4}{3π},\frac{4}{3π})$

Exercise $\PageIndex{33}$

Triangle: $\displaystyle y=x, y=2−x$, and $\displaystyle y=0$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{34}$

Lens: $\displaystyle y=x^2$ and $\displaystyle y=x$

Answer: $\displaystyle (\frac{1}{2},\frac{2}{5})$

Exercise $\PageIndex{35}$

Ring: $\displaystyle y^2+x^2=1$ and $\displaystyle y^2+x^2=4$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{36}$

Half-ring: $\displaystyle y^2+x^2=1, y^2+x^2=4,$ and $\displaystyle y=0$

Answer: $\displaystyle (0,\frac{28}{9π})$

Exercise $\PageIndex{37}$

Find the generalized center of mass in the sliver between $\displaystyle y=x^a$ and $\displaystyle y=x^b$ with $\displaystyle a>b$. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{38}$

Find the generalized center of mass between $\displaystyle y=a^2−x^2, x=0$, and $\displaystyle y=0$. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Answer: Center of mass: $\displaystyle (\frac{a}{6},\frac{4a^2}{5}),$ volume: $\displaystyle \frac{2πa^4}{9}$

Exercise $\PageIndex{39}$

Find the generalized center of mass between $\displaystyle y=bsin(ax), x=0,$ and $\displaystyle x=\frac{π}{a}.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{40}$

Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius $\displaystyle a$ is positioned with the left end of the circle at $\displaystyle x=b, b>0,$ and is rotated around the y-axis.

$alt$

Answer: Find the center of mass $\displaystyle (\bar{x},\bar{y})$ for a thin wire along the semicircle $\displaystyle y=\sqrt{1−x^2}$ with unit mass. (Hint: Use the theorem of Pappus.)

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