Coordinates of Centers of Gravity and Moments of Inertia of Solids

The coordinates $x_{c}$, $y_{c}$, $z_{c}$ of the center of gravity of a body occupying a volume $V$ are determined by the formulas:

where $\rho$ is the density of the body at the point $(x,y,z)$, and $m$ is the mass of the body. In the following problems the body is assumed to be homogeneous with $\rho =1$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x+y+z=a$, $x=0$, $y=0$, $z=0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $z^{2}=xy$, $x=a$, $y=b$, $z=c$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $abz=c(a-x)(b-y)$, $x=0$, $y=0$, $z=0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $h^{2}(x^{2}+y^{2})=a^{2}z^{2}$; $0<z<h$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $x^{2}+y^{2}+z^{2}=a^{2}$; $x>0$, $y>0$, $z>0$.

Find the coordinates of the center of gravity of a homogeneous body that is a segment of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$; $h<z<a$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}+z^{2}=a^{2}$, $x^{2}+y^{2}=ax$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}=z$, $x+y+z=0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}+z^{2}=a^{2}$, $x^{2}+y^{2}=a(a-2z)$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}=2a^{2}$, $x+y+z=2a$, $x=0$, $y=0$, $z=0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $(x^{2}+y^{2}+z^{2})^{2}=axyz$; $x>0$, $y>0$, $z>0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{3}z$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$, $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z^{2}}{c^{2}}$; $z>0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z}{c}$, $\dfrac {x}{a}+\dfrac {y}{b}=\pm 1$, $\dfrac {x}{a}-\dfrac {y}{b}=\pm 1$, $z=0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}+\sqrt{\dfrac {z}{c}}=1$; $x>0$, $y>0$, $z>0$.

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $\Big(\dfrac {x}{a}\Big)^{2/3}+\Big(\dfrac {y}{b}\Big)^{2/3}+\Big(\dfrac {z}{c}\Big)^{2/3}=1$; $x>0$, $y>0$, $z>0$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, where $l=\sqrt{x^{2}+y^{2}}$ is the distance from the variable point of the body to the axis $Oz$, bounded by the surfaces $x=0$, $x=a$, $y=0$, $y=b$, $z=0$, $z=c$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, bounded by the surfaces $z^{2}=2ax$, $z=0$, $x^{2}+y^{2}=ax$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, bounded by the surface $h^{2}(x^{2}+y^{2})=a^{2}z^{2}$; $0<z<h$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, bounded by the surfaces $x+y+z=a\sqrt{2}$, $x^{2}+y^{2}=a^{2}$, $z=0$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, bounded by the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{3}z$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, bounded by the surface $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, bounded by the surfaces $\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}=1$, $x=0$, $y=0$, $z=0$.

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ($\rho =1$), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, bounded by the surface $\Big(\dfrac {x}{a}\Big)^{2/3}+\Big(\dfrac {y}{b}\Big)^{2/3}+\Big(\dfrac {z}{c}\Big)^{2/3}=1$.

Find the moments of inertia of the homogeneous torus

relative to its axis of rotation.

Find the moment of inertia of the same homogeneous torus

as in the previous problem, relative to its equatorial diameter.

Find the moment of inertia of the homogeneous elliptic cone $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z^{2}}{h^{2}}$, $z=h$, relative to the axis $Ox$.

Find the moment of inertia of a homogeneous circular cylinder of height $H$ and base radius $a$ relative to a diameter of the base.

Find the moment of inertia of a homogeneous circular cone of height $H$ and base radius $R$ relative to a diameter of the base.

The moments of inertia of a homogeneous body ($\rho =1$) relative to the coordinate axes are given by the formulas $I_{x}=\iiint \limits_{V}(y^{2}+z^{2}) \, dx \, dy \, dz$, $I_{y}=\iiint \limits_{V}(z^{2}+x^{2}) \, dx \, dy \, dz$, $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$, and the centrifugal moments of inertia by the integrals $I_{xy}=\iiint \limits_{V}xy \, dx \, dy \, dz$, $I_{yz}=\iiint \limits_{V}yz \, dx \, dy \, dz$, $I_{zx}=\iiint \limits_{V}zx \, dx \, dy \, dz$.

Determine the height $h$ and base radius $a$ of a homogeneous cylinder so that the ellipsoid of inertia for this cylinder becomes a sphere.

The same question as in the previous problem (determine the height $h$ and base radius so that the ellipsoid of inertia becomes a sphere), for a homogeneous cone.

Determine the height $h$ of a rectangular parallelepiped, whose base is a square with side $a$, so that the ellipsoid of inertia for the homogeneous mass filling this parallelepiped becomes a sphere.

Prove that among all homogeneous ellipsoids, only for the sphere is the ellipsoid of inertia itself a sphere.

Prove that for homogeneous bodies symmetric with respect to the plane $yOz$, the quantity $J_{xy}=0$, where $J_{xy}$ is the centrifugal moment of inertia $I_{xy}=\iiint \limits_{V}xy \, dx \, dy \, dz$ (notation from problem 3751).

Prove that for homogeneous regular polyhedra, the ellipsoid of inertia is a sphere.

Prove that a homogeneous ellipsoid can be similar to its own ellipsoid of inertia only in the case where the ellipsoid is a sphere.