Newtonian potentials of bodies and shells

Form differential equations to solve the following problem: with what force does an infinite straight line of constant linear density $\mu_{0}$ attract a particle of mass $m$ at distance $a$ from the line?

Form differential equations to solve the following problem: compute the force with which a circular thin plate of radius $a$ and constant surface density $\delta_{0}$ attracts a particle $P$ of mass $m$, where the particle lies on the perpendicular line through the center $Q$ of the plate and perpendicular to the plane of the plate, with the shortest distance $PQ$ equal to $b$.

Find the gravitational attraction exerted by the uniform cylinder $x^{2}+y^{2}\leqslant a^{2}$, $0\leqslant z\leqslant h$ on a point mass $P(0,0,b)$, given that the mass of the cylinder equals $M$ and the mass of the point equals $m$.

Find the Newtonian potential at the point $(x,y,z)$ of the homogeneous sphere $\xi^{2}+\eta^{2}+\zeta^{2}\leqslant R^{2}$ of density $\rho_{0}$.

Let the density be $\rho =f(R)$, where $f$ is a given function and $R=\sqrt{\xi^{2}+\eta^{2}+\zeta^{2}}$; find the Newtonian potential at the point $P(x,y,z)$ of the spherical shell $R_{1}^{2}\leqslant \xi^{2}+\eta^{2}+\zeta^{2}\leqslant R_{2}^{2}$.

Find the Newtonian potential at the point $P(0,0,z)$ of the cylinder $\xi^{2}+\eta^{2}\leqslant a^{2}$, $0\leqslant \zeta \leqslant h$, of fixed density $\rho_{0}$.

With what force does the homogeneous sphere $\xi^{2}+\eta^{2}+\zeta^{2}\leqslant R^{2}$, of radius $R$ and mass $M$, attract a particle of mass $m$ at the point $P(0,0,a)$?

Find the attractive force exerted by the homogeneous cylinder $\xi^{2}+\eta^{2}\leqslant a^{2}$, $0\leqslant \zeta \leqslant h$, of density $\rho_{0}$, on a particle of unit mass at the point $P(0,0,z)$.

Find the attractive force exerted by a homogeneous solid spherical cone of density $\rho_{0}$ on a particle of unit mass located at its apex, where the radius of the sphere is $R$ and the angle of the sector of the axial cross-section equals $2\alpha$.

Compute the potential of a uniform ball of radius $R$ and density $\rho_{0}$ upon itself, that is, find the integral

where $r_{1,2}=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}}$.

With what force does the uniform truncated conical surface of density $\rho_{0}$

attract a point mass $m$ located at the vertex of the surface?

Find the potential, at the point $M_{0}(x_{0},y_{0},z_{0})$, of the uniform spherical shell of density $\rho_{0}$, $x^{2}+y^{2}+z^{2}=a^{2}$, that is, compute the integral

where

Find at the point $M(x,y,z)$ the Newtonian potential of a homogeneous shell of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ with surface density $\mu$ equal to some constant, i.e., find the integral $\iint_{\sigma }\mu \dfrac {d\sigma }{r}$, where $\sigma$ is the surface of the sphere, and $r$ is the distance from a variable point $(\xi ,\eta ,\zeta )$ on this surface to the given point $M$.

Find at the point $M(x,y,z)$ the Newtonian potential of a homogeneous sphere, i.e., the integral

where $\omega$ is a sphere of radius $a$ centered at the origin, $\mu$ is the constant density, $(\xi ,\eta ,\zeta )$ is a point inside the sphere, $r=\sqrt{(x-\xi )^{2}+(y-\eta )^{2}+(z-\zeta )^{2}}$ is the distance from the point $(\xi ,\eta ,\zeta )$ to the point $(x,y,z)$.

The same question (find at the point $M(x,y,z)$ the Newtonian potential of a sphere, i.e., the integral $P=\iiint_{\omega } \mu \, d\xi \, d\eta \, d\zeta / r$, where $\omega$ is a sphere of radius $a$ centered at the origin, $r$ is the distance from the point $(\xi ,\eta ,\zeta )$ inside the sphere to the point $(x,y,z)$) for a sphere whose density $\mu$ equals a given function $f(\rho )$, where $\rho^{2}=\xi^{2}+\eta^{2}+\zeta^{2}$.

At the point $(0,0,z)$ find the Newtonian potential of the upper half of the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ with constant density $\mu$.

Find the Newtonian potential of a homogeneous spherical shell between two concentric spheres of radii $a$ and $b$, where $a<b$.

Find at the apex of a cone the Newtonian potential of the solid cone with base radius $r$ and height $h$.

Show that among bodies of constant density and given mass, the one that attracts a material point most strongly (by Newton's law) is bounded by the surface $R=a\sqrt{\cos \theta }$ (the material point is taken as the origin, the direction of the attractive force as the axis $Oz$; $R$, $\theta$ are spherical coordinates). Find this force and compare it with the force of attraction by a sphere (of the same mass) on a material point lying on its surface.

Find at the point $M(x,y)$ the Newtonian potential of a non-homogeneous segment $(-a,a)$, i.e., the integral $\displaystyle \int_{-a}^{a}\frac{\mu (\xi ) \, d\xi }{\sqrt{(\xi -x)^{2}+y^{2}}}$, if $\mu (\xi )=\dfrac {\xi }{a}$.

Find the Newtonian potential of a homogeneous circular plate $x^{2}+y^{2}=a^{2}$, $z=0$, at the point $(0,0,z)$.

Prove the equality

Here $r_{1}^{2}=x^{2}+y^{2}+z^{2}$, $r$ is the distance from the point $(x,y,z)$ to an arbitrary point $(\xi ,\eta ,\zeta )$ inside the volume $\omega$.

The projections on the axes $Ox$, $Oy$, $Oz$ of the force with which a point of mass $m$ is attracted, by Newton's law, by a body of density $\mu$, are expressed by the formulas:

where $P=\iiint_{\omega } \mu \dfrac {d\omega }{r}$, $r=\sqrt{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}}$, $f$ is the gravitational constant. Using this, compute the force of attraction by a sphere of radius $a$ with density $\mu$ on a point of mass $m$ located on its surface.

The points $M(x,y,z)$ and $M_{1}(\xi ,\eta ,\zeta )$ independently of each other range over the entire volume of a body of density $p=p(x,y,z)$. The self-potential of the body is called the integral

Find the value of the self-potential of a homogeneous sphere $x^{2}+y^{2}+z^{2}=a^{2}$. Its value measures the work done in bringing from infinity into a single body particles that attract each other by Newton's law.

Find the magnitude of the force with which a homogeneous sphere of radius $R$ and density $\rho = \rho_{0}$ attracts a point mass $m$.

Find the potential at the point $M_{0}\left(x_{0} ; y_{0} ; z_{0}\right)$ of a simple layer (item 1), distributed:

Find, at the point $(0 ; 0 ; z)$, the potential of a simple layer distributed with density $\mu$ :

Let the function $\mu$ be continuous in the closure $\bar{G}$ of a bounded measurable domain $G$, $N(\xi ; \eta ; \zeta ) \in G, M(x ; y ; z) \in R^{3}$,

Prove that

Find the Newtonian potential at the point $M_{0}$ created by a ball of density $\rho_{0}$¹ and radius $R$.

Find the Newtonian potential at the point $M_{0}$ created by a hollow ball of density $\rho_{0}$, if its inner and outer radii are $R_{1}$ and $R_{2}, R_{1} < R_{2}$.

Find the Newtonian potential created by the ball $x^{2}+y^{2}+z^{2} = R^{2}$ with density $\rho (x ; y ; z) = \sqrt{x^{2}+y^{2}+z^{2}}$.

Find the Newtonian potential at the center of the base of a cylinder with radius $R$, height $H$, and density $\rho_{0}$.

A material cone with slant height $l$ and height $h$ has density $\rho_{0}$. Find the potential of the gravitational field of the cone:

Find, at the point $(0 ; 0, h)$, the Newtonian potential of the hemisphere $x^{2} + y^{2}+z^{2} \leqslant R^{2}, z \geqslant 0$ with density $\rho_{0}$.

Find, at the point $(0 ; 0 ; h)$, the Newtonian potential of the cylinder $x^{2} + y^{2} \leqslant R^{2}, 0 \leqslant z \leqslant H$ with density $\rho_{0}$.

Find the Newtonian potential of the ellipsoid $\left(x^{2}+y^{2}\right) / a^{2}+z^{2} / b^{2} = 1$ with density $\rho_{0}$ at its center.

Find the force of attraction of a material point $M_{0}$ of mass $m$ by a ball of radius $R$ and density $\rho_{0}$.

A spherical sector with angle $2 \alpha$ in the axial cross-section is cut out of a material ball of radius $R$ and density $\rho_{0}$. Find the force with which this sector attracts a point of mass $m$ located at its vertex.

For a material ball with density $\rho = f(r)$, prove that:

Find the force with which a cylinder of density $\rho_{0}$, height $H$, and radius $R$ attracts a point of mass $m$ located at the center of the base of the cylinder.

Find the force with which a cone of density $\rho_{0}$, height $H$, and base radius $R$ attracts a point of mass $m$ located at the vertex of the cone.

The distance between the centers of two balls equals $a$; the mass of one ball equals $M_{1}$, that of the other $M_{2}$; the density of each ball is constant.

Starting from formulas (21), (20), prove that the force of attraction of one ball by the other equals the force of attraction between two material points placed at the centers of the balls and having masses $M_{1}$ and $M_{2}$ respectively. Find this force.

Considering the Earth to be a liquid ball with average density $\rho_{0}$ and radius $R$, find the pressure in it as a function of the distance $r$ to the center.

Evaluate the integral $\int_{\| y\| \leqslant 1} \frac{d y}{\| x-y\| } \quad \left(x, y \in \mathbb {R}^{3}\right)$.