I E Irodov Solutions for Chapter: PHYSICAL FUNDAMENTALS OF MECHANICS, Exercise 4: UNIVERSAL GRAVITATION

A planet of mass $M$ moves along a circle around the Sun with velocity $v=34.9 \mathrm{km} {\mathrm{s}}^{-1}$ (relative to the heliocentric reference frame). Find the period of revolution of this planet around the Sun. $($Take $G=6.67\times {10}^{-11} \mathrm{N} {\mathrm{m}}^2 {\mathrm{kg}}^{-2}$)

There is a uniform sphere of mass $M$, and radius $R$. Find the strength $G$ and the potential $\phi$, of the gravitational field of this sphere, as a function of the distance $r$ from its centre (with $r<R$ and $r>R$). Draw the approximate plots of the functions $G\left(r\right)$ and $\phi \left(r\right)$.

Inside a uniform sphere, of density $\rho$, there is a spherical cavity, whose center is at a distance $l$, from the center of the sphere. Find the strength $G$, of the gravitational field, inside the cavity.

A uniform sphere has a mass $M$, and radius $R$. Find the pressure $p$, inside the sphere, caused by gravitational compression, as a function of the distance $r$, from its centre. Evaluate $p$, at the center of the Earth, assuming it to be a uniform sphere.

$\left(\text{Radius of Earth }R=6.4\times {10}^6 \mathrm{m} \text{and } \text{Mean Density of Earth }\text{ρ}=5.52\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}\right)$

Find the proper potential energy of gravitational interaction of matter forming

(a) a thin uniform spherical layer of mass $m$ and radius $R$;

(b) a uniform sphere of mass $m$ and radius $R$. 

Two Earth's satellites move in a common plane, along circular orbits. The orbital radius of one satellite is $r=7000 \mathrm{km}$, while that of the other satellite is $\mathrm{\Delta }r=70 \mathrm{km}$ less. What time interval separates the periodic approaches of the satellites, to each other, over the minimum distance?

$\left( \text{Mass of Earth}=6\times {10}^{24} \mathrm{kg}\right)$

Calculate the ratios of the following accelerations: The acceleration $w_1$, due to the gravitational force on the Earth's surface, the acceleration $w_2$, due to the centrifugal force of inertia on the Earth's equator, and the acceleration $w_3$, caused by the Sun to the bodies on the Earth.

$\left(G=6.67\times {10}^{-11} \mathrm{N} {\mathrm{kg}}^{-2} {\mathrm{m}}^2, \text{Radius of Earth}=6.37\times {10}^6 \mathrm{m} \text{and} \text{Mass of Earth}=5.96\times {10}^{24} \mathrm{Kg}\right)$

Find approximately the third cosmic velocity $v_3$, i.e., the minimum velocity that has to be imparted to a body, relative to the Earth's surface, to drive it out of the Solar system. The rotation of the Earth about its own axis is to be neglected.

Useful data is given below,

$$\begin{gathered} G=6.67\times {10}^{-11} \mathrm{N} {\mathrm{m}}^2 {\mathrm{kg}}^{-2}, \\ \text{Radius of Earth}=6.37\times {10}^6 \mathrm{m}, \\ \text{Mass of Earth}=5.97\times {10}^{24} \mathrm{kg}, \\ \text{Mass of Sun}=1.99\times {10}^{30} \mathrm{kg} \\ \text{Average distance of Earth from the Sun}=1.5\times {10}^{11} \mathrm{m} \end{gathered}$$