– Universal gravitation law
\[
F=\gamma \frac{m_{1} m_{3}}{r^{3}} \text {. }
\]
– The squares of the perieds of revelution of any twe plasets around the Sun are proportional in the eubes of the major semiases of their erbits (Kepler):
– Strength 6 and potential 9 of the grovitational feld of a mass point:
\[
6=-7 \frac{\mathrm{m}}{r^{3}} \mathrm{r}, \quad 8=-\gamma \frac{\mathrm{m}}{r} \text {. }
\]
– Orbital and eacape velocities:
\[
n_{1}=\sqrt{R R}, \quad n_{1}=\sqrt{2} n_{1} \text {. }
\]
1.200. A planet of mass $M$ moves along a circle around the Sun with velocity $v=34.9 \mathrm{~km} / \mathrm{s}$ (relative to the heliocentric reference frame). Find the period of revolution of this planet around the Sus.
1.201. The Jupiter’s period of revolution around the Sun is 12 times that of the Earth. Assuming the planetary orbits to be circular, find:
(a) how many times the distance between the Jupiter and the Sun exceeds that between the Earth and the Sun:
(b) the velocity and the acceleration of Jupiter in the heliocentric relerence frame.
1.202. A planet of mass $M$ moves around the Sun along an ellipse so that its minimum distance from the Sun is equal to $r$ and the maximum distance to $\boldsymbol{R}$. Making use of Kepler’s laws, find its period of revolution around the Sun.
1.203. A small body starts falling onte the Sun from a distance equal to the radius of the Earth’s orbit. The initial velocity of the body is equal to zero in the heliocentric relerence frame. Making use of Kepler’s laws, find how long the body will be falling.
1.204. Suppose we have made a model of the Solar system scaled down in the ratio $\eta$ but of materials of the same mean density as the actual materials of the planets and the Sun. How will the orbital periods of revolution of planetary models change in this case?
1.206. A double star is a system of twe stars moving around the centre of inertia of the system due to gravitation. Find the distance between the components of the double star, if its total mass equals $M$ and the period of revolution $T$.
1.206. Find the potential energy of the gravitational interaction
(a) of two mass points of masses $m_{1}$ and $m_{\text {, l located at a distance } r}$ from each other;
(b) of a mass point of mass $m$ and a thin uniform rod of mass $M$ and length $b$, if they are located along a straight line at a distance a from each other; also find the force of their interaction.
1.207. A planet of mas $m$ moves along an ellipse around the Sun so that its maximum and misimum distances from the Sun are equal to $r_{1}$ and $r_{1}$ respectively. Find the angular momentum $M$ of this planet relative to the centre of the Sun.
1.208. Using the conservation laws, demonstrate that the total mechanical energy of a planet of mass $m$ moving around the Sun along an ellipse depends only on its semi-major axis a. Find this energy as a function of a.
1.209. A planet $\boldsymbol{A}$ moves along an elliptical arbit around the Sun. At the moment when it was at the distance $r_{\text {g f }}$ from the Sun its velocity was equal to $v_{0}$ and the angle between the radius vector $r_{4}$ and the velocity vector $\mathrm{v}_{\text {, was }}$ equal to $a$. Find the maximum and minimum distances that will separate this planet from the Sun during its orbital motion.
1.210. A cosmic body $A$ moves to the Sun with velocity $v_{0}$ (when far from the Sun) and aiming parameter $l$ the arm of the vector $\mathrm{v}_{0}$
relative to the centre of the Sun (Fig. 1.51). Find the minimum distance by which this body will get to the Sun.
1.211. A particle of mass $m$ is located outside a uniform sphere of mass $M$ at a distance $r$ from its centre. Find:
(a) the potential energy of gravitational interaction of the particle and the sphere:
(b) the gravitational force which the sphere exerts on the particle. 1.212. Demonstrate that the gravitational force acting on a particle $A$ inside a uniform spherical layer of matter is equal to zero. 1.213. A partiele of mass $m$ was transferred from the centre of the base of a uniform hemisphere of mass $M$ and radius $R$ into infinity.
Fig. 1.51.
What work was performed in the process by the gravitational force exerted on the particle by the hemisphere?
1.214. There is a sniform sphere of mass $M$ and radius $R$. Find the strength $G$ and the potential $\varphi$ of the gravitational field of this sphere as a function of the distance $r$ from its centre (with $r<n$ and $r>R$ ). Draw the approximate plots of the functions $G(r)$ and $\varphi(r)$.
1.215. Inside a uniform sphere of density $\rho$ there is a spherical cavity whose centre is at a distance $l$ from the centre of the sphere. Find the strength $\mathbf{G}$ of the gravitational field inside the cavity.
1.216. A uniform sphere has a mass $M$ and radius $\boldsymbol{R}$. Find the pressure $p$ inside the sphere, caused by gravitational compression, is a function of the distance $r$ from its centre. Bvaluate $p$ at the centre of the Earth, assuming it to be a uniform sphere.
1.217. 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 $\boldsymbol{A}$ (make use of the answer to Problem 1.214).
1.218. Two Earth’s satellites move in a common plane along circular orbits. The orbital radius of one satellite $r=7000 \mathrm{~km}$ while that of the ether satellite is $\Delta r=70 \mathrm{~km}$ less. What time interval separates the periodic approaches of the satellites to each other over the minimum distance?
1.219. Calculate the ratios of the following accelerations: the acceleration $w_{1}$ due to the gravitational force on the Earth’s surface,
the acceleration $w$, due to the centrifugal force of inertia on the Earth’s equator, and the acceleration $w_{\text {, }}$ caused by the Sun to the bodies on the Earth.
1.220. At what height over the Earth’s pole the free-fall acceleration decreases by one per cent; by half?
1.221. On the pole of the Earth a body is imparted velocity $v_{4}$ directed vertically op. Knowing the radius of the Earth and the freefall acceleration on its surface, find the beight to which the body will ascend. The air drag is to be neglected.
1.222. An artificial satellite is launched into a circular orbit around the Barth with velocity $v$ relative to the reference frame moving translationally and fixed to the Earth’s rotation axis. Find the distance from the satellite to the Earth’s surface. The radius of the Barth and the free-fall acceleration on its surface are supposed to be known.
1.223. Calculate the radius of the circular orbit of a stationary Earth’s satellite, which remains motionless with respect to its surface. What are its velocity and acceleration is the inertial reference frame fixed at a given moment to the centre of the Earth?
1.224. A satellite revolving in a circular equatorial orbit of $\mathrm{ra}$ dius $\boldsymbol{R}=2.00 \cdot 10^{4} \mathrm{~km}$ from west to east appears over a certain point at the equator every $\mathrm{z}=11.6$ hours. Using these data, calculate the mass of the Earth. The gravitational constant is supposed to be known.
1.225. A satellite revolves from east to west in a circular equatorial orbit of radius $A=1.00-10^{4} \mathrm{~km}$ around the Earth. Find the velocity and the acceleration of the satellite in the reference frame fixed to the Barth.
1.226. A satellite must move in the equatorial plane of the Earth elose to its surface either in the Earth’s rotation direction or against it. Find how many times the kinetic energy of the satellite in the latter case exceeds that in the former case (in the reference frame fixed to the Earth).
1.227. An artificial satellite of the Moon revolves in a circular orbit whose radius exceeds the radius of the Moon $\eta$ times. In the process of motion the satellite experiences a slight resistance due to cosmic dust. Assuming the resistance force to depend on the velocity of the satellite as $F=a v$, where $a$ is a constant, find how long the satellite will stay in orbit until it falls onto the Moon’s surface.
1.228. Calculate the orbital and escape velocities for the Moon. Compare the results obtained with the corresponding velocities for the Earth.
1.229. A spaceship approaches the Moon along a parabolic trajectory which is almost tangent to the Moon’s surface. At the moment of the maximum approach the brake rocket was fired for a short time interval, and the spaceship was transferred into a circular orbit of a Moon satellite. Find how the spaceship velocity modulus increased in the process of braking.
1.230. A spaceship is lausched into a circular orbit close to the
Earth’s surface. What additional velocity has to be imparted to the spaceship to overcome the gravitational pull?
1.231. At what distance from the centre of the Moos is thepoint at which the strength of the resultant of the Earth’s and Moon’s gravitational fields is equal to zero? The Earth’s mass is assumed to be $\eta=81$ times that of the Moon, and the distance between the centres of these planets $n=60$ times greater than the radius of the Earth A.
1.232. What is the minimum work that has to be performed to bring a spaceship of mass $m=2.0-10^{\circ} \mathrm{kg}$ from the surface of the Earth to the Moon?
1.233. Find approximately the third cosmic velocity $v_{2}$, 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.