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201The 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 reference frame.
202A 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 R. Making use of Kepler s laws, find its period of revolution around the Sun.
203A small body starts falling onto the Sun from a distance equal to the radius of the Earth orbit. The initial velocity of the body is equal to zero in the heliocentric reference frame. Making use of Kepler s laws, find how long the body will be falling.
204Suppose we have made a model of the Solar system scaled down in the ratio 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?
205A double star is a system of two 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.
206Find the potential energy of the gravitational interaction (a) of two mass points of masses m1 and m2 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 L, if they are located along a straight line at a distance a from each other; also find the force of their interaction.
207A planet of mass m moves along an ellipse around the Sun so that its maximum and minimum distances from the Sun are equal to r1 and r2 respectively. Find the angular momentum M of this planet relative to the centre of the Sun.
208Using 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.
209A planet A moves along an elliptical orbit around the Sun. At the moment when it was at the distance r0 from the Sun its velocity was equal to v0 and the angle between the radius vector r0 and the velocity vector vo was equal to a. Find the maximum and minimum distances that will separate this planet from the Sun during its orbital motion.
210A cosmic body A moves to the Sun with velocity vo (when far from the Sun) and aiming parameter L the arm of the vector v0 relative to the centre of the Sun (Fig. 1.51). Find the minimum distance by which this body will get to the Sun.
211A particle of mass in 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.
212Demonstrate that the gravitational force acting on a particle A inside a uniform spherical layer of matter is equal to zero.
213A particle of mass m was transferred from the centre of the base of a uniform hemisphere of mass M and radius R into infinity. What work was performed in the process by the gravitational force exerted on the particle by the hemisphere?
214There is a uniform sphere of mass M and radius R. Find the strength G and the potential qo 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 (r) and q (r).
215Inside a uniform sphere of density p there is a spherical cavity whose centre is at a distance 1 from the centre of the sphere. Find the strength G of the gravitational field inside the cavity.
216A 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 centre of the Earth, assuming it to be a uniform sphere.
217Find 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 (make use of the answer to Problem 1.214).
218Two Earth's satellites move in a common plane along circular orbits. The orbital radius of one satellite r = 7000 km while that of the other satellite is dr = 70 km less. What time interval separates the periodic approaches of the satellites to each other over the minimum distance?
219Calculate the ratios of the following accelerations: the acceleration zv i due to the gravitational force on the Earth's surface, the acceleration w2 due to the centrifugal force of inertia on the Earth's equator, and the acceleration w3 caused by the Sun to the bodies on the Earth.
220At what height over the Earth's pole the free-fall acceleration decreases by one per cent; by half?
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