UNIT IV STUDY GUIDE
Gravity and Orbital Motion
Course Learning Outcomes for Unit IV
Upon completion of this unit, students should be able to:
3. Explain Newton's laws of motion at work in common phenomena.
3.1 Illustrate the relation of the universal law of gravitation to Newton’s second law.
3.2 Distinguish between gravitational acceleration (g) and gravitational constant (G).
3.3 Evaluate gravitational field strength when mass and radius of an object are given.
4. Explain the concepts and applications of momentum, work, mechanical energy, and general relativity.
4.1 Apply total mechanical energy conservation for orbital motion.
4.2 Calculate escape velocity when gravitational potential energy is balanced with kinetic energy.
4.3 Describe the escape velocity in a black hole, a consequence of Einstein’s general relativity.
Reading Assignment
Chapter 9: Gravity
Chapter 10: Projectile and Satellite Motion
Unit Lesson
Projectile Motion
When an object moves with a curved path near the earth’s surface under the influence of gravity, its motion is
called projectile motion. For example, look at Figures 10.6 and 10.8 on pages 185 to 186 in the textbook.
If we ignore air resistance, the horizontal motion of the projectile does not slow down; its velocity is constant.
In other words, the horizontal component of the acceleration is zero. However, the vertical component of the
velocity is not constant, but changes. In addition, the vertical component of the acceleration is downward
acceleration, gravitational acceleration, (g).
Weightlessness and Free Fall
Suppose you are in an elevator. If the elevator is not accelerating, your weight (W) is just your mass (m) times
the gravitational acceleration (g). In fact, two forces are acting on you; the weight (W) and the normal force
(F). According to Newton’s second law, in the vertical direction, ma=F-W=F-mg. That is, normal force
F=m(g+a). Here, g is positive, but a may be either positive for upward acceleration or negative for downward
acceleration of the elevator. If the elevator is in upward motion, apparent weight (or normal force) is greater
than your true weight. On the other hand, if the elevator is in downward motion, the apparent weight is smaller
than your true weight. In a special case, when the acceleration is equal to g, that is, a=-g, or free fall, the
apparent weight becomes zero: weightless. Please look at Figure 9.9 on p.166 in the textbook for an example
of this. The same phenomena occur when an object is circling around the earth. The orbiting satellite, which
accelerates toward the center of the earth, is also in free fall. See Figure 9.10 on p.167 in the textbook.
Over a long period of time, the weightlessness is harmful for humans, and thus, a rotating space station in a
wheel shape is provided to create artificial gravity. It is balanced with the centripetal force, mv2/r, of the
system. That is mg=mv2/r. Here, m is the mass of an astronaut, r is the distance from axis to the surface of
the station, and v is the rotating speed. For instance, if r is given 1 km, then v=(rg)1/2= 100 m/s.
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Newton’s Law of Universal Gravitation
Newton speculated about the highest reachable point by the force of gravity on the earth. He realized that
there is a limit and concluded that the orbital motion of the moon around the earth is maintained by the
gravitational force (Hewitt, 2015). Suppose you throw a stone horizontally from a high place (See Figure
10.16 on p. 190 in the textbook). The stone falls to the ground because of gravity. However, if you throw the
stone with great speed, it will move further and further away from where you are standing before falling to the
ground. When the speed is great enough, the stone will eventually circle around the earth. This is the
projectile motion, where the projectile falls in the gravitational field but never touches the ground. This logical
consideration can be applied to explain the orbital motion of the moon. Newton concluded that the moon is
falling in its pathway around the Earth because of the gravitational acceleration.
Newton extended the above idea to any two objects in the universe in order to explain the interaction between
them. Newton’s law of universal gravitation postulates that there is an attractive force between the two objects
(Hewitt, 2015). The force between two objects in the universe is proportional to the product of two masses m
and M and is inversely proportional to the square of distance r between two objects; F=GmM/r2 , where G=
(6.6710-11 N m2/kg2) is the universal gravitational constant. This is the case when the gravitational
acceleration (a) is equal to g in the second law of Newton; a=g, and thus, g=GM/r2. The constant, G was
measured by Cavendish 100 years after Newton announced his theory. It was not an easy task because of
the extremely small value of gravitation attraction. The detailed story is in Section 9.2 on pp. 163–164 in the
textbook.
Example: What is the magnitude of the gravitational force between the sun and the earth? The distance
between the sun and the earth is 1AU= 1.501011 m. The mass of the earth is m = 5.9810 24 kg and the
mass of the Sun is M=1.991030 kg.
Solution: From F=GmM/r2= 6.6710-11 x 5.981024 x 1.991030 / (1.50 x1011)2 = 3.51025 N
Kepler’s Three Empirical Laws for Planetary Motion
Johannes Kepler (1571 - 1630) was a German astronomer and had an endless enthusiasm for researching
the solar system. It took him more than 20 years to realize through his calculations the exact shape of the
planets’ orbitals. He tested many different kinds of models using his teacher Tyco Brache’s enormous data
set. Brache had accumulated very exact planetary data without even the use of telescopes. Kepler
established three important empirical laws of planetary motion: the law of elliptical orbit, the law of areas, and
the law of the relation between period and distance. This is what he used to describe and understand the
motion of the Solar System (Zeilik & Smith, 1987).
Mechanical motion of our solar system obeys gravitational law, and planets are orbiting around the sun, which
is the heaviest mass in the solar system. The orbital shape is not circular, but elliptical. Some comets have
parabolic or hyperbolic orbits. These well-known mechanics were not easily discovered. Since the ancient
times, the sky was considered a realm of gods, so perfectness was assumed. The notion that the orbits of
planets should be a perfect circle was widely accepted, and no scholar would be able to prove otherwise to
the people, even Kepler. For these reasons, it took a long time to accurately describe planetary motion in our
solar system. The famous three empirical laws of planetary motion describe the motion of the solar system as
follows:
First Law—The law of ellipses: The orbit of each planet is an ellipse with the sun at one foci. The
shape of a planet's orbit is an ellipse.
Second Law—The law of areas: The radius vector to a planet sweeps out equal areas in equal
intervals of time. When a planet is closer to the sun, it revolves faster, and, on the other hand, when a
planet is farther away from the sun, it revolves slower.
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Third Law—The law of harmony: The squares of the sidereal periods of the planets are proportional
to the cubes of the semi-major axes (mean radii) of their orbits. Here, the sidereal period is the time it
takes the planet to complete one orbit of the Sun with respect to the stars (Zeilik & Smith, 1987).
Thus, Kepler's laws and Newton's laws taken together imply that a force holds a planet in its orbit by
continuously changing the planet's velocity, so that it follows an elliptical path. The force is directed toward the
sun from the planet and is proportional to the product of the masses of the sun and the planets. Also, the
force is inversely proportional to the square of the planet-sun separation. This is precisely the form of the
gravitational force postulated by Newton. Newton's laws of motion, with a gravitational force used in the
second law, imply Kepler's laws, and the rest of the planets obey the same laws of motion as objects on the
surface of the earth.
Conic Sections and Gravitational Orbits
Hypatia (360 - 415) visualized various shapes of geometric equations using conic sections for the first time in
Alexandria, Egypt (Larson & Edwards, 2010). Conic sections are formed when a cone is cut with a plane at
various angles. For a more detailed description, visit the website about this in the Suggested Reading section
of this unit.
There are various orbits in a gravitational system. The circular orbit is a special case of ellipse. The ellipse
can be formed when the plane intersects opposite “edges” of the cone. In the case of the parabola orbit, the
plane is parallel to one edge of the cone. On the other hand, the hyperbola orbit does not intersect opposite
edges of the cone, and the plane is not parallel to the edge. Planets in our solar system have elliptical orbits
with various eccentricities. The orbital eccentricity (e) determines the shape of orbits. If e=0 (E<0), the orbit is
circular, if 01(E>0), the orbit is
hyperbolic. Here, E is the total energy.
Some comets have elliptical orbits, too. However, other comets have parabolic orbits, and once they pass the
Sun, they will never come back. In the case of two interacting stars, they show a hyperbolic orbit. Like the
parabolic orbit, the hyperbolic one is a one-time encounter.
Geostationary Orbits
Some satellites revolve around the earth at the same speed of earth’s rotation, which is very useful for
communication purposes. They are located at about 36,000 km above the earth’s surface. This position does
not depend on the mass of the satellite.
The centripetal force is equal to gravitational force, mv2/r= GmM/r2, where m is the mass of satellite, M is the
mass of the earth, and r is the distance from the center of the earth’s surface to the satellite. The speed of the
satellite is v=(GM/r)1/2. Also, v=distance/time=the circumference of the circular orbit/the orbital period=2πr/T.
That is, the orbital period is T=2πr3/2/(GM)1/2, or T2=4π2r3/GM. Notice that this is exactly the same with
Kepler’s third law. In other words, the harmonic law is a consequence when the centripetal force is in balance
with the gravitational force.
T is about 24 hours. One hour equals 60 minutes. One minute equals 60 seconds. Therefore, 24 hours are
24x3600=86400 seconds. The earth’s mass is 5.98 x 1024 kg and G = 6.67 x 10-11N m2/kg2. By substituting all
the appropriate values to the above equation, you obtain r=42300 km. The radius of the earth is about 6400
km, so subtract 6400 from 42300: 42300-6400=35900 km, which is about 22300 miles.
Energy Conservation and Escape Velocity
For a circular orbit at distance (r) from the center of the Earth (r = RE + h, if h is the altitude of the orbit), the
circular speed (vc) can be found by equating the centripetal force (mv2/r) and gravitational (GmM/r2) force, or v
= (GM/r)1/2 . Also, from the conservation of total energy, TE = (KE + PE)ground= (KE+PE)h =constant.
By evaluating TE at h = 0 and at maximum height (h), we find that mv2/2=mgh, or h=v2/2g. Notice that when
h= RE, v = 11.2 km/s, the projectile escapes to h = . This critical speed is called the escape speed.
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Escape velocity is defined as the minimum velocity an object must have in order to escape the gravitational
field of the earth, to escape the earth without ever falling back. The object must have greater energy than its
gravitational binding energy to escape the earth's gravitational field.
Black Holes and Einstein’s Relativity
A black hole is the final stage of massive stellar evolution. A direct observation of a black hole is impossible
because no light can escape from it. If you get too close to a black hole, the speed you would need to escape
from it would exceed the speed of light. Because nothing can travel faster than light, nothing—not even light—
can escape from a black hole. The existence of black holes was predicted by Einstein’s general theory of
relativity, which is the best theory to describe what gravity is and how it behaves. In 1915, Einstein published
the general theory of relativity. Click here for more information on the general theory of relativity. The
curvature of space and time is influenced by gravity. The more massive, the more distortions of space and
time.
References
Hewitt, P. G. (2015). Conceptual physics (12th ed.). Upper Saddle River, NJ: Pearson.
Larson, R., & Edwards, B. H. (2010). Calculus (9th ed.). Belmont, CA: Brooks/Cole.
Zeilik, M., & Smith, E. I. (1987). Introductory astronomy & astrophysics (2nd ed.). Philadelphia, PA: Saunders
College Publishing.
Suggested Reading
This link will take you to the website mentioned in the unit lesson. It provides more information about conic
sections.
Nave, R. (n.d.). Conic sections. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/math/consec.html
The following document further explains black holes and Einstein’s theory of relativity.
Click here to access a pdf version of the document.
Learning Activities (Nongraded)
Nongraded Learning Activities are provided to aid students in their course of study. You do not have to submit
them. If you have questions, contact your instructor for further guidance and information.
To practice what you have learned in this unit, complete the following problems and questions from the
textbook. The answers to each problem can be found in the “Odd-numbered Answers” section in the back of
the textbook. The question number from the textbook is indicated in parentheses after each question.
1. Calculate the force of Earth’s gravity on a 1-kg mass at Earth’s surface. The mass of Earth is
6.0x1024kg and its radius is 6.4 x 106m. Does the result surprise you? (Textbook #33 on p. 178)
2. Suppose you stood atop a ladder so tall that you were three times as far from Earth’s center as you
presently are. Show that your weight would be one ninth of its present value. (Textbook #39 on p.
178)
3. An astronaut lands on a planet that has the same mass as Earth, but twice the diameter. How does
the astronaut’s weight differ from that on Earth? (Textbook #59 on p. 179)
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4. The force due to gravity on you is mg. Under what condition is mg also your weight? (Textbook #71
on p. 159)
5. If our Sun shrank in size to become a black hole, discuss and show from the gravitational force
equation that Earth’s orbit would not be affected. (Textbook #107 on p. 181)
6. With your friends, whirl a bucket of water in a vertical circle fast enough so that water doesn’t spill out.
As it happens, the water in the bucket is falling, but with less speed that you give to the bucket. Tell
how your bucket swing relates to satellite motion — that satellites in orbit continuously fall toward
Earth, but not with enough vertical speed to get closer to the curved Earth below. Remind your friends
that physics is about finding the connections in nature! (Textbook #25 on page 201)
7. At a particular point in its orbit, a satellite in an elliptical orbit has a gravitational potential energy of
5000 MJ with respect to Earth’s surface and a kinetic energy of 4500 MJ. Later in its orbit, the
satellite’s potential energy is 6000 MJ. What is its kinetic energy at that point? (Textbook #31 on page
202)
8. Fragments of fireworks beautifully illuminate the night sky. (a) What specific path is ideally traced by
each fragment? (b) What paths would be fragments trace in a gravity-free region? (Textbook #45 on
page 203)
9. Which planets have a more-than-one-Earth-year period: planets nearer than Earth to the Sun, or
planets farther from the Sun than Earth? (Textbook #55 on page 204)
10. Can a satellite maintain an orbit in the plane of the Arctic Circle? Why or why not? (Textbook #95 on
page 205)
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