Download Fizix Chapter 7 Notes.cwk

Notes: _____
Chapter 7 Notes
Date: _______
Gravitation
7.1 Planetary Motion
Since humans first walked this planet we have wondered about the motion of the many
objects in the nighttime sky. The earliest models used a series of concentric spheres that
rotated about the earth in the center. In the Ptolemaic model (around 100 A.D.) the
planets and sun orbit around the Earth (also called the _____________________ model) and
require small circles in their orbits called ___________________ to describe the backwards
(____________________) motion of planets across the sky. It wasn’t until nearly 1550 A.D.
that Nicolas Copernicus revolutionized modern astronomy with his
______________________ model that correctly placed the sun in the center of the solar
system and the planets, asteroids, comets and meteoroids orbit the sun. A few years after
Copernicus published his work another astronomer, Tycho Brahe, started the
painstaking task of measuring the locations of planets and stars over a period of more
than 3 decades. Brahe expanded on Copernicus’ model but he still required the use of
epicycles. His assistant for many of those years was Johannes Kepler who, upon Brahe’s
death, acquired all of Brahe’s data. Kepler was very good in mathematics and with all of
Brahe’s data he was able to finally answer the riddle, “how does the solar system work?”
Kepler’s Laws
Kepler’s 3 laws of planetary motion describe mathematically how objects travel in the
solar system. Kepler’s First Law states that all orbiting objects travel in _______________
around the sun (or larger body), located at one _________ (note: an ellipse always has
two foci; in the case of the solar system the second foci is just an empty point):
Kepler’s Second Law states that planets move faster when they are closer to the sun and
slow as they are more distant. The second law goes further: If one draws a line from the
sun to a planet at two different points in its orbit and repeat this at two other points in
the orbit, then if the the time between the two points are equal the area “swept out” are
equal.
Kepler’s Third Law states that the period of revolution of a planet (called its _______) is
related to its average distance from the sun:
where
T is the orbital period, and
r is the average orbital radius
(see page 173 for data)
The above formula is used to compare one planet with another! For example, the star
Formalhaut B has two planets that orbit it. One of them orbits in 4.5 days at a distance
of 110 times the diameter of Formalhaut. The other orbits in 9.6 days. What is the
orbital distance of the second planet?
18
3
2
Kepler’s Third law alternately uses the constant, k, 3.35 x 10 m /s . to solve problems
involving planets orbiting the sun in the form:
where
T is the orbital period in seconds, and
r is the average orbital radius in meters
This version of Kepler’s Third Law is useful if one only has data of one object.
For example, what is the orbital period of a satellite orbiting the sun at a distance of
2.00 x1011 m?
After Kepler published his laws others began to test them, including Galileo, who
discovered among other things, that the sun rotated (he discovered sunspots), the moon
wasn’t smooth (he discovered craters), Saturn had a ring, Jupiter had at least 4 moons
that orbited it (now called the Galilean moons), and most importantly, that Venus
underwent phases (like our moon) thus finally proving planets orbited the sun.
Finally, nearly a half century after Kepler’s laws were established, another scientist,
Isaac Newton formulated his 3 laws of motion (Inertia, Acceleration and Interaction) and
turned his attention to the solar system. He realized that if an object accelerates in its
orbit around the sun that the force between the two objects must be inversely
proportional to the square of the distance between them (hence
F ∝ 12 ) but directly
r
proportional to the product of their masses. Newton’s Law of Universal Gravitation, like
the alternative to Kepler’s Third Law, requires a constant. called the Universal
Gravitation Constant, represented as G:
€
where
m1 and m2 are the masses in kg;
r is the distance between centers of mass in m;
G = 6.67 x 10
-11
N·m2 /kg2
Newton’s Law of Gravitation can be applied to ANY two objects!! For example, determine
the force of gravitation between two students, one 75.0 kg the other 85.0 kg seated 1.0 m
apart:
7.2 Satellites
With a little bit of algebra it can be shown that if we use Newton’s 2nd Law (F = m·a)
and realize the acceleration is actually centripetal acceleration we can substitute into
the Law of Gravitation the centripetal force equation and solve for T to get the period of
time it takes a planet to orbit the sun:
where T is the period in seconds;
r is the orbital radius in meters;
and ms is the mass of the sun in kg
30
(= 1.99 x 10 kg)
Note: this is the same equation as Kepler’s Third Law that uses k but instead uses G and
the mass of the sun instead.
Recently we discussed the critical velocity needed for an object to complete a vertical
circle. ( vmin =
r • g ). If the “vertical circle” is the orbit of an object about the Earth at a
distance of r meters we get the speed of an orbiting satellite as:
€
where v is the speed in m/s;
r is the orbital radius in meters;
and me is the mass of the earth in kg
(= 5.97 x 1024 kg)
Lastly, we can determine the orbital period of a satellite around Earth using the same
equation as the period of a planet around the sun, substituting the mass of the earth for
the sun in the equation at the bottom of the previous page:
where T is the period in seconds;
r is the orbital radius in meters;
and me is the mass of the earth in kg
(= 5.97 x 1024 kg)
Note, the orbital radius of an object is always to the CENTER of mass so you must add
the radius of the Earth (r = 6.38 x 106 m) to the distance the satellite orbits above the
Earth!
The Hubble Space Telescope orbits the Earth at an average distance of 569 km.
(a) What is the HST’s orbital speed?
(b) What is Hubble’s orbital period?