Planets move through the sky with More complicated paths than Stars do. We begin with Kepler’s Laws in Order to find out how.
Kepler’s Laws of planetary motion Are stated as… 1)The paths of planets are ellipses, With the sun at one focus. 2) An imaginary line from the sun To a planet sweeps out equal Areas in equal time intervals. Thus planets move faster when They are closer to the sun, and Vise-versa.
3) The square of the ratio of the Periods of any two planets Revolving about the sun is Equal to the cube of the ratio Of their average distances From the sun. TATBTATB = rArBrArB 23
TATBTATB = rArBrArB 23 T = the planets’ periods Of revolution r = the average distance From the sun (or another focus)
The first 2 laws apply to each Individual planet, moon, or Satellite. The 3 rd law relates to the motion Of several satellites about a Single body.
Io has a period of 1.8 days and is 4.2 units away from Jupiter. Callisto has a period of 16.7 days. What is the distance away From Jupiter? r c = 19 units
The moon has a period of 27.3 Days and a mean distance from Earth of 3.9 X 10 5 km. Find the Period of a satellite that is in Orbit 6.7 X 10 3 km. 88.6 mins
Newton also played around With a few formulas for gravitation. Newton said that there was an Attractive force between All objects, this is called the Gravitational force.
Newton was so confidant that He came up with a law to find The quantity of gravity. The Law of Universal Gravitation F = G mAmBd2mAmBd2
mAmBd2mAmBd2 m = mass d = distance between The center of the objects G = 6.67 X 10 -11 Nm 2 / kg 2
Without showing all the math, The force equals something else. F = mp4π2rT2mp4π2rT2 = G msmpd2msmpd2 And rearranged again… T 2 = 4π 2 Gm s r3r3
Jeff (75 kg) and Racheal (60 kg) sit about 4 Meters away from each other. Racheal looks up and feels an Attraction. Find the gravitational Force between them. F = 1.87 X 10 -8 N
An object that is falling at the Same rate at which the earth is Curved is said to be in orbit.
For an object to be in orbit, air resistance Must be taken away, which Means that it must be at least 150 km above the earth. Use Newton’s thought experiment On page 179 to see how.
We can continuously derive New formulas to find out things. v = Gm E r T = 2π r 3 Gm E These formulas only Work for the earth.
A satellite orbits Earth 225 km Above its surface. What is its Speed in orbit and its period? v = 7760 m/sT = 1.5 hr
As you move farther from Earth’s Center, the acceleration due To gravity changes. a = g rEdrEd 2
There are 2 kinds of mass… Inertial mass And Gravitational mass
Inertial mass is a measure of an Object’s resistance to an type Of force. m inertial = F net a
The gravitational mass is the size Of the gravitational force Between the objects. m grav = r 2 F grav Gm
Einstein proposed that gravity Is not a force, but an effect Of space itself. According to Einstein, mass Changes space itself, Making it curved.
Bodies are thus accelerated Because they follow this Curved space toward the center. This is Einstein’s Special Theory of Relativity.