We can now use Newton's Law to derive some results concerning planets in circular orbits. Although we know from Kepler's Laws that the orbits are not circular, in most cases approximating the orbit by a circle gives satisfactory results. When two massive bodies exert a gravitational force on one another, we shall see (in the SparkNote on Orbits) that planets describe circular or elliptical paths around their common center of mass. In the case of a planet orbiting the sun, however, the sun's mass is so much greater than the planets, that the center of mass lies well within the sun, and in fact very close to its center. For this reason it is a good approximation to assume that the sun stays fixed (say at the origin) and the planets move around it. The force is then given by:
= |
v ^{2} = |
v = |
= âá’T ^{2} = |
We can apply the Universal Law of Gravitation to objects near the earth also. For an object at or near the surface of the earth, the force due to gravity acts (for reasons that will become clearer in the section on Newton's Shell Theory) toward the center of the earth. That is, it acts downwards because every particle in the earth is attracting the object. The magnitude of the force on an object of mass m is given by:
F = |
= 9.74 |
9.8 m/sec ^{2}
, but the value varies considerably at different places on the earth's surface). Thus if we rename the constants = g , then we have the familiar equation F = mg which determines all free-fall motion near the earth.We can also calculate the value of g that an astronaut in a space shuttle would feel orbiting at a height of 200 kilometers above the earth:
g _{1} | = | ||
= | (6.67×10^{-11})(5.98×10^{24})(6.4×10^{6} +2×10^{5})^{-2} | ||
= | 9.16 |
Because the gravitational force between everyday-sized objects is very small, the gravitational constant, G , is extremely difficult to measure accurately. Henry Cavendish (1731-1810) devised a clever apparatus for measuring the gravitational constant. A fiber is attached to the center of the beam to which m and m' are attached, as shown in . This is allowed to reach an equilibrium, untwisted state before, the two larger masses M and M' are lowered next to them. The gravitational force between the two pairs of masses causes the string to twist such that the amount of twisting is just balanced by the gravitational force. By appropriate calibration (knowing how much force causes how much twisting), the gravitational force may be measured. Since the masses and the distances between them may also be measured, only G remains unknown in the Universal Law of Gravitation. Thus G can be calculated from the measured quantities. Accurate measurements of G now place the value at 6.673×10^{-11} N.m ^{2} /kg ^{2} .