5.6 Calculating centers of mass and moments of inertia  (Page 7/8)

$R$ is the trapezoidal region determined by the lines $y=0,y=1,y=x,$ and $y=\text{−}x+3;\rho \left(x,y\right)=2x+y.$

$R$ is the disk of radius $2$ centered at $\left(1,2\right);$ $\rho \left(x,y\right)=x^2+y^2-2x-4y+5.$

$R$ is the unit disk; $\rho \left(x,y\right)=3x^4+6x^2{y}^2+3y^4.$

$R$ is the region enclosed by the ellipse $x^2+4y^2=1;\rho \left(x,y\right)=1.$

$R=\left\{\left(x,y\right)|9x^2+y^2\le 1,x\ge 0,y\ge 0\right\};\rho \left(x,y\right)=\sqrt{9x^2+y^2}.$

$R$ is the region bounded by $y=x,y=\text{−}x,y=x+2,\text{and}y=\text{−}x+2;$ $\rho \left(x,y\right)=1.$

$R$ is the region bounded by $y=\frac{1}{x},y=\frac{2}{x},y=1,\text{and}y=2;\rho \left(x,y\right)=4\left(x+y\right).$

Let $Q$ be the solid unit cube. Find the mass of the solid if its density $\rho$ is equal to the square of the distance of an arbitrary point of $Q$ to the $xy\text{-plane}.$

Let $Q$ be the solid unit hemisphere. Find the mass of the solid if its density $\rho$ is proportional to the distance of an arbitrary point of $Q$ to the origin.

The solid $Q$ of constant density $1$ is situated inside the sphere $x^2+y^2+z^2=16$ and outside the sphere $x^2+y^2+z^2=1.$ Show that the center of mass of the solid is not located within the solid.

Find the mass of the solid $Q=\left\{\left(x,y,z\right)|1\le x^2+z^2\le 25,y\le 1-x^2-z^2\right\}$ whose density is $\rho \left(x,y,z\right)=k,$ where $k>0.$

[T] The solid $Q=\left\{\left(x,y,z\right)|x^2+y^2\le 9,0\le z\le 1,x\ge 0,y\ge 0\right\}$ has density equal to the distance to the $xy\text{-plane}\text{.}$ Use a CAS to answer the following questions.

1. Find the mass of $Q.$
2. Find the moments $M_{xy},M_{xz},\text{and}{M}_{yz}$ about the $xy\text{-plane,}$ $xz\text{-plane,}$ and $yz\text{-plane,}$ respectively.
3. Find the center of mass of $Q.$
4. Graph $Q$ and locate its center of mass.

Consider the solid $Q=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le 2,0\le z\le 3\right\}$ with the density function $\rho \left(x,y,z\right)=x+y+1.$

1. Find the mass of $Q.$
2. Find the moments $M_{xy},M_{xz},\text{and}{M}_{yz}$ about the $xy\text{-plane,}$ $xz\text{-plane,}$ and $yz\text{-plane,}$ respectively.
3. Find the center of mass of $Q.$

[T] The solid $Q$ has the mass given by the triple integral ${\displaystyle \underset{\mathrm{-1}}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{\frac{\pi }{4}}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}{r}^{2}drd\theta dz}}}.$ Use a CAS to answer the following questions.

1. Show that the center of mass of $Q$ is located in the $xy\text{-plane.}$
2. Graph $Q$ and locate its center of mass.

The solid $Q$ is bounded by the planes $x+4y+z=8,x=0,y=0,\text{and}z=0.$ Its density at any point is equal to the distance to the $xz\text{-plane}\text{.}$ Find the moments of inertia $I_y$ of the solid about the $xz\text{-plane}\text{.}$

The solid $Q$ is bounded by the planes $x+y+z=3,$ $x=0,y=0,$ and $z=0.$ Its density is $\rho \left(x,y,z\right)=x+ay,$ where $a>0.$ Show that the center of mass of the solid is located in the plane $z=\frac{3}{5}$ for any value of $a.$

Let $Q$ be the solid situated outside the sphere $x^2+y^2+z^2=z$ and inside the upper hemisphere $x^2+y^2+z^2=R^2,$ where $R>1.$ If the density of the solid is $\rho \left(x,y,z\right)=\frac{1}{\sqrt{x^2+y^2+z^2}},$ find $R$ such that the mass of the solid is $\frac{7\pi }{2}.$

The mass of a solid $Q$ is given by ${\displaystyle \underset{0}{\overset{2}{\int }}{\displaystyle \underset{0}{\overset{\sqrt{4-x^2}}{\int }}{\displaystyle \underset{\sqrt{x^2+y^2}}{\overset{\sqrt{16-x^2-y^2}}{\int }}{\left(x^2+y^2+z^2\right)}^n}}}dzdydx,$ where $n$ is an integer. Determine $n$ such the mass of the solid is $\left(2-\sqrt{2}\right)\pi .$

Let $Q$ be the solid bounded above the cone $x^2+y^2=z^2$ and below the sphere $x^2+y^2+z^2-4z=0.$ Its density is a constant $k>0.$ Find $k$ such that the center of mass of the solid is situated $7$ units from the origin.

The solid $Q=\left\{(x,y,z)|0\le x^2+y^2\le 16,x\ge 0,y\ge 0,0\le z\le x\right\}$ has the density $\rho \left(x,y,z\right)=k.$ Show that the moment $M_{xy}$ about the $xy\text{-plane}$ is half of the moment $M_{yz}$ about the $yz\text{-plane}\text{.}$

The solid $Q$ is bounded by the cylinder $x^2+y^2=a^2,$ the paraboloid $b^2-z=x^2+y^2,$ and the $xy\text{-plane,}$ where $0<a<b.$ Find the mass of the solid if its density is given by $\rho \left(x,y,z\right)=\sqrt{x^2+y^2}.$

Let $Q$ be a solid of constant density $k,$ where $k>0,$ that is located in the first octant, inside the circular cone $x^2+y^2=9{\left(z-1\right)}^2,$ and above the plane $z=0.$ Show that the moment $M_{xy}$ about the $xy\text{-plane}$ is the same as the moment $M_{yz}$ about the $xz\text{-plane}\text{.}$

The solid $Q$ has the mass given by the triple integral ${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{\pi \text{/}2}{\int }}{\displaystyle \underset{0}{\overset{r^2}{\int }}\left(r^4+r\right)}}}dzd\theta dr.$

1. Find the density of the solid in rectangular coordinates.
2. Find the moment $M_{xy}$ about the $xy\text{-plane}\text{.}$

The solid $Q$ has the moment of inertia $I_x$ about the $yz\text{-plane}$ given by the triple integral ${\displaystyle \underset{0}{\overset{2}{\int }}{\displaystyle \underset{\text{−}\sqrt{4-y^2}}{\overset{\sqrt{4-y^2}}{\int }}{\displaystyle \underset{\frac{1}{2}\left(x^2+y^2\right)}{\overset{\sqrt{x^2+y^2}}{\int }}\left(y^2+z^2\right)}}}\left(x^2+y^2\right)dzdxdy.$

1. Find the density of $Q.$
2. Find the moment of inertia $I_z$ about the $xy\text{-plane.}$

The solid $Q$ has the mass given by the triple integral ${\displaystyle \underset{0}{\overset{\pi \text{/}4}{\int }}{\displaystyle \underset{0}{\overset{2\text{sec}\theta }{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}\left(r^3\text{cos}\theta \text{sin}\theta +2r\right)}}}dzdrd\theta .$

1. Find the density of the solid in rectangular coordinates.
2. Find the moment $M_{xz}$ about the $xz\text{-plane.}$

Let $Q$ be the solid bounded by the $xy\text{-plane},$ the cylinder $x^2+y^2=a^2,$ and the plane $z=1,$ where $a>1$ is a real number. Find the moment $M_{xy}$ of the solid about the $xy\text{-plane}$ if its density given in cylindrical coordinates is $\rho \left(r,\theta ,z\right)=\frac{d^{2}f}{d{r}^2}\left(r\right),$ where $f$ is a differentiable function with the first and second derivatives continuous and differentiable on $\left(0,a\right).$

A solid $Q$ has a volume given by ${\displaystyle \underset{D}{\iint }{\displaystyle \underset{a}{\overset{b}{\int }}dAdz,}}$ where $D$ is the projection of the solid onto the $xy\text{-plane}$ and $a<b$ are real numbers, and its density does not depend on the variable $z.$ Show that its center of mass lies in the plane $z=\frac{a+b}{2}.$

Consider the solid enclosed by the cylinder $x^2+z^2=a^2$ and the planes $y=b$ and $y=c,$ where $a>0$ and $b<c$ are real numbers. The density of $Q$ is given by $\rho \left(x,y,z\right)=f\prime \left(y\right),$ where $f$ is a differential function whose derivative is continuous on $\left(b,c\right).$ Show that if $f\left(b\right)=f\left(c\right),$ then the moment of inertia about the $xz\text{-plane}$ of $Q$ is null.

[T] The average density of a solid $Q$ is defined as ${\rho }_{ave}=\frac{1}{V\left(Q\right)}{\displaystyle \underset{Q}{\iiint }\rho \left(x,y,z\right)}dV=\frac{m}{V\left(Q\right)},$ where $V\left(Q\right)$ and $m$ are the volume and the mass of $Q,$ respectively. If the density of the unit ball centered at the origin is $\rho \left(x,y,z\right)=e^{\text{−}{x}^2-y^2-z^2},$ use a CAS to find its average density. Round your answer to three decimal places.

Show that the moments of inertia $I_x,I_y,\text{and}{I}_z$ about the $yz\text{-plane,}$ $xz\text{-plane,}$ and $xy\text{-plane,}$ respectively, of the unit ball centered at the origin whose density is $\rho \left(x,y,z\right)=e^{\text{−}{x}^2-y^2-z^2}$ are the same. Round your answer to two decimal places.