Kepler's Laws and Gravitation
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Kepler's Laws
Two artificial satellites, S1 and S2 orbit the Earth in a circular
motion at distances, respectively, equal to r1 = R and
r2 = 3R from their center. At a certain instant, the line connecting the centers
of the satellites is tangent to the orbit of S1. Determine at this instant the distance
d between S1 and S2.
Mars is 52% further from the Sun than Earth. Calculate, in Earth years, the period of the revolution of Mars
around the Sun.
Gravitation
Variations in the gravitational field at the Earth's surface can arise from irregularities in the
distribution of its mass. Consider the Earth as a sphere of uniform radius R and density
ρ, with a spherical cavity of radius a, entirely contained within it. The distance
between the centers O, of the Earth, and C, of the cavity, is d, which can vary
from 0 (zero) to R−a, thus causing a variation in the gravitational field at a point
P, on the surface of the Earth, aligned with O and C (see figure). If
G1 is the gravitational force at P without the existence of the cavity on
Earth, and G2, is the gravitational force at the same point, considering the existence
of the cavity. What will be the maximum value of the relative variation:
\( \left(G_{1}-G_{2}\right)/G_{1} \),
obtained by moving the cavity position?
A planet of mass m describes a circular orbit around a star S at a distance R, with a
speed such that the duration of each revolution is T. The movement takes place under the action of a
force F of constant magnitude, directed towards S. Using acp, K and
V to represent the centripetal acceleration, kinetic energy, and velocity of the planet,
respectively.
a) Establish the expression of K as a function of F and R, that is K = f(F, R);
b) Establish as a function of R, T and m the expressions of V, acp and K, that is, V = f(R, T, m), acp = f(R, T, m) and K = f(R, T, m);
c) Show that F is given by the expression \( F=A\dfrac{m}{R^{2}} \) where A is a constant;
d) Apply the expressions found by calculating acp, F e K in the case of the Earth around the Sun. Data are: Earth's speed in its orbit 30 km/s, Earth's orbit radius 15×107 km, and Earth's mass 6×1021 t.
a) Establish the expression of K as a function of F and R, that is K = f(F, R);
b) Establish as a function of R, T and m the expressions of V, acp and K, that is, V = f(R, T, m), acp = f(R, T, m) and K = f(R, T, m);
c) Show that F is given by the expression \( F=A\dfrac{m}{R^{2}} \) where A is a constant;
d) Apply the expressions found by calculating acp, F e K in the case of the Earth around the Sun. Data are: Earth's speed in its orbit 30 km/s, Earth's orbit radius 15×107 km, and Earth's mass 6×1021 t.
In the film 2001: A Space Odyssey (1968 directed by Stanley Kubrick) the spacecraft Discovery One
has a section formed by a centrifuge that rotates in order to produce artificial gravity similar to the
gravity of the Moon.
Assuming that an astronaut has an average height of 1.70 m and that the centrifuge has a diameter of 11.6 m and rotates at a frequency of 5 rpm, check if it is possible to build such a device.
Calculate the acceleration due to gravity on a space station orbiting the Earth at a distance of 360 km
from the surface (e.g. the International Space Station - ISS). Data: radius of Earth
6.37×103 km, mass of Earth 5.97×1024 kg, and Universal Gravitational
Constant 6.67×10−11 N.m2/kg2.
Note: e.g. is the abbreviation of the Latin expression "exempli gratia" which means
"for example".
Determine the angular speed of a satellite around the Earth, assuming a circular orbit, as a function of the
distance from the center of the Earth.
A rocket is launched from the Earth towards the Moon, following a straight trajectory, that joins the
centers of the two bodies. Since the mass of the Earth ME is approximately 81 times the
mass of the Moon MM, determine the point in the trajectory where the magnitude of the
gravitational fields due to the Earth and the Moon canceling each other out. Consider the Earth-Moon system
isolated from the rest of the Universe, the system is stationary and with the total mass of each body
concentrated at its center.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .