Mr. Kuffer
Universal Gravitation
Chapter 12
The Falling Moon
Concepts of Physics
According to Newton, there were two things needed for an object to fall
around or orbit the earth. Label the diagram below with these two things:
____________________
_______________
Newton had the foresight to compare the falling apple to the moon. He knew
that objects tended to move in a straight line unless they were acted upon
by __________________________. His idea was that gravity was
universal. That is- it existed… _______________________________.
Mass doesn’t Count for much! Newton knew that in the absence of air
resistance, a heavy and a light object dropped at the same time on the
surface of the earth would fall at the same rate. In other words, they were
affected by gravity the same way.
Distance was the important thing!
Re = 6,370,000 m
= 3,955 miles
Average earth-moon
distance:
384,000,000 m
Or
230,600 miles
2
On Earth:
In one second an object will fall…
d = ½ gt2
On Moon:
6O X the radius of the Earth…
So. The moon should fall _____________________
A satellite at 30 X radius of the Earth should fall
Main Idea:
________________________
________________________
________________________
________________________
________________________
3
Universal Gravitation Practice Problems
Directions: Solve the following problems and questions. Remember to use the
correct equation and also watch sig. figs. and units.
1. Earth is attracted to the sun by the force of gravity. Why doesn’t the Earth fall
into the sun? Explain.
2. If the Earth begins to shrink but its mass remains the same, what would happen
to the value of g on Earths surface?
3. Cavendish did his experiment using lead balls. Suppose he had used equal masses
of copper instead. Would his value of G been the same or different?
4. Why did Newton think that a force must act on the moon?
5. What provides the force that causes the centripetal acceleration of a satellite in
orbit?
6. How do you answer the question, “What keeps a satellite up?”
7. What is the force of gravity between a student with a mass of 75 kg and another
student with a mass of 95 kg, if they are standing 0.50 m apart?
8. What is the gravitational force between a 15 g squirrel and the earth if the
squirrel is in a tree 5.0 m above the earth?
9. A 150 kg person experiences a gravitational force of 7.80 x 109 N. Where is the
person standing?
10. Solve for the gravitational force between each planet and the sun:
4
Gravity and Distance Problems
Given the equation:
F = G (m1 m2) / d2
Consider a 1 kg apple at various places with respect to the earth’s center of
gravity. (1r = 6,380,000 m)
1r
2r
3r
4r
5r…
Complete the following Chart:
( g = Gm/r2 )
Strength of the field (N/Kg)
Distance
Weight of Apple (N)
1r
2r
3r
4r
5r
6r
*Graph the above data on a separate sheet of paper!! (Weight vertical and
Distance horizontal)*
1. Given the fact that 2.2 lbs = 1 kg, find your mass in kg.
Your mass = ___________ kg
2. What would your weight be in Ocean City, MD? (It is at sea level, so r =
6.38 x 106 m)
3. What would your weight be in Denver, CO? (“Mile high city” – 1 mi.)
5
Universal Gravitation
Gravity is the other common force. Newton was the first person to study it
seriously, and he came up with the law of universal gravitation:
Each particle of matter attracts every other particle with a force
which is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them.
The standard formula for gravity is:
Gravitational force = (G * m1 * m2) / (d2)
where G is the gravitational constant, m1 and m2 are the masses of the two
objects for which you are calculating the force, and d is the distance
between the centers of gravity of the two masses.
G has the value of 6.67 x 10-11 N·m2/kg2
The Inverse Square Law
6
Gravitational Fields Within The Planet
IMAGINE: A hole dug through the earth. Forget about the
impracticalities such as lava and very high temperatures…
a = _____________
Notes:
a = _____________
How long would a one-way trip
take? _________
(If you stepped into the hole
bored completely through the
earth and made no attempt to
grab the edge at either end)
a = _____________
a = _____________
a = _____________
7
Gravitational Interactions
CHAPTER 13
Complete the table below using your understanding of gravitational fields
(refer to page 183 of Hewitt text).
g = GM/R2
G = 6.67 x 10-11 N·m2/kg2
Q: What is Universal Gravitation?
A: Simply put, Everything is attracted
to everything else!!
8
Using a Pendulum to Determine “g”
Objective:
Recall that a Period (Ƭ) is the time an object takes to make one complete cycle,
whether that be a complete circle or a swing back and forth. (spinning student demo).
The equation to calculate the period of a pendulum is…
Ƭ = 2 π √(r/g)
eqn 1
Rearrange this equation to solve for “g” and get…
Look familiar???...
g = 4π2 r / Ƭ2
eqn 2
ac = 4π2 r / Ƭ2
eqn 3
Remember that “Ƭ” is the period, which is the time for ONE COMPLETE CYCLE of the
pendulum… BACK AND FORTH!!
Procedure and Analysis:
1. Set up the equipment as shown.
2. Varying the length of the pendulum decreasing them by 5cm (0.05m) each trial.
Swing the .pendulum and time how long it takes for 20 complete cycles. Do half
of the cycles with one pendulum bob, and change the bob to another one for the
second half of the trials
3. Complete the Data Table.
4. Make a graph of radial distance (length of string) vs. Ƭ2
5. Calculate the slope of the graph (r / Ƭ2)
6. Multiply the slope by 4π2 (refer to eqn 2 above)
7. Calculate the percent error between the experimental “g” obtained in step 6 and
the accepted value of 9.8m/s2.
Report the percent error here ___________%
o o o
9
Mass
(kg)
Length
(m)
Time for
20 Swings
(sec)
Period = T
(sec)
Period2 =
T2
(sec2)
Experimental
Gravitational
Acceleration
(m/s2)
%
Error
% Error = Experimental -Actual / Actual X 100%
10
Johannes Kepler
(Creepy looking guy to the right)
Danish astronomer Tycho Brahe
(1546-1601) spent years
cataloguing the stars and planets
with great accuracy. His assistant
Johannes Kepler (1571-1630) put
his observations to good use. He
developed three important laws of
astronomy. His first law describes
the shapes of planetary orbits. His
second law describes the speed at
which the planets travel along
their orbits. His third law relates
the different planetary orbits to
one another. FYI: Newton, born in
1642, came after Kepler.
Kepler’s Laws
Kepler's three laws of planetary motion can be described as follows:
1. The path of the planets about
the sun are elliptical in shape,
with the center of the sun being
located at one focus. (The Law
of Ellipses)
2. An imaginary line drawn from
the center of the sun to the
center of the planet will sweep
out equal areas in equal intervals
of time. (The Law of Equal
Areas)
3. The ratio of the squares of the
periods of any two planets is
equal to the ratio of the cubes
of their average distances from
the sun. (The Law of Harmonies)
11
Determining Planetary Gravitational Forces
Instructions: Using the data in your packet, complete the following chart.
Make sure you change miles to meters and do not forget to square the
distance in the denominator.
Fg = Gm1m2/ d2
Planet
Fg @ Perihelion (N)
Fg @ Aphelion
(N)
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
12
Can't remember the 11
planets? 4th-grader offers help
STORY HIGHLIGHTS
Montana student comes up
GREAT FALLS, Montana (AP) -- Those having trouble remembering
with phrase to help people
the newly assigned 11 planets, including three dwarfs, are getting help
remember the planets.
from a fourth-grader.
Scientists now recognize
11 planets in our solar
system. List includes three
that are considered dwarf
planets: Ceres, Pluto and
Eris. Award-winning
mnemonic will be published
in National Geographic
book
Astronomy buffs have a new way to remember the 11 planets. Updated 7:36 a.m.
EST, Wed February 27, 2008
Maryn Smith, the winner of the National Geographic planetary mnemonic contest, has created a handy way
to remember the planets and their order in distance from the sun.
Her award-winning phrase is: My Very Exciting Magic Carpet Just Sailed Under Nine Palace Elephants.
The 11 recognized planets are Mercury, Venus, Earth, Mars, Ceres, Jupiter, Saturn, Uranus, Neptune, Pluto
and Eris. Ceres, Pluto and Eris are considered dwarf planets.
13
Did you know that all but two planets (Mercury and Pluto)
have orbits in the same plane?
14
The Orbit Lab
Objective: The student will draw an ellipse to simulate the orbit of a planet and then
analyze how the gravitational force varies with position in the orbit.
Important terms:
Perihelion
Aphelion
Materials:
2 thumbtacks, 21 cm x 28 cm piece of cardboard,
Sheet of unlined paper, 30 cm of string or thread
Procedure:
1. Push the thumbtacks into the paper and cardboard so that they are between 6
and 10 cm apart.
2. Make a loop with the string. Place the loop over the two thumbtacks. Keep the
loop tight as you draw the ellipse.
3. Remove the tacks and string. Draw a small star centered as one of the tack
holes.
Observation and Data:
1. Draw the position of the planet in the orbit where it is farthest from the star.
2. Draw the position of the planet when it is nearest the star.
3. Determine the distance from these positions to the star’s center (below).
Analysis:
1. Choose one of the planets in the solar system.
2. Calculate the gravitational force when the planet is at perihelion and aphelion.
You will need to use the enclosed charts to find the distances and masses
required. Draw your planet at the perihelion and aphelion distances and label
the force vectors accordingly.
3. Draw your planet at two additional phases. Draw the tangential velocity vector
at each phase (all four phases).
NAME
MASS (kg)
Sun
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
1.991 x 1030
3.2 x 1023
4.88 x 1024
5.979 x 1024
6.42 x 1023
1.901 x 1027
5.68 x 1028
8.68 x 1026
1.03 x 1026
1.2 x 1022
PLANETARY DATA
PERIHELION
APHELION DIST.
DIST. (megamiles) (megamiles)
PERIHELION
Date
APHELION
Date
28.6
66.8
91.4
128.4
460.3
837.6
1699.0
2771.0
2756.0
10/16/95
8/11/95
12/21/95
2/19/96
5/5/99
5/26/03
3/1/2050
3/2030
8/1989
11/29/95
12/1/95
6/21/96
1/28/97
3/29/2005
2/8/2018
4/17/2008
2/2112
8/2113
43.4
67.7
94.5
154.9
507.2
936.2
1868.0
2819.0
4555.0
15
Concepts of Physics
Mr. Kuffer
Orbit Lab Work
NAME:_________________
Period: ________
16
*Show all work below. This should include several conversion for aphelion and
perihelion from Megamiles to meters and the gravitational force of attraction at those two
points. Every number should have a unit attached to it. If it does not… IT IS WRONG!
Recall:
1 megamile = _______ x 106 miles
1 mile = 1609 m
Planet Chosen: _______________________
Distance at Aphelion:
Distance at Perihelion:
Difference in Distance:
Fg at Aphelion:
Fg at Perihelion:
Difference in Fg:
17
Universal Gravitation
***** Refer to pages 4 and 8 of packet for planetary data *****
1. An apparatus like the one Cavendish used to find G has a large lead
ball that is 5.9 kg and a small one that is 0.047 kg. Their centers are
separated by 0.055 m. Find the force of attraction between them.
2. Use the date on pages 4 and 7 of the packet to compute the
gravitational force the sun exerts on Jupiter.
3. Counahan has a mass of 70.0 kg and Libby has a mass of 50.0 kg.
Counahan and Libby are standing 20.0 m apart on the dance floor.
Counahan looks up and sees her and feels an attraction. If the
attraction is gravitational, find its size.
4. Two spheres have their centers 2.0 m apart. One has a mass of 8.0 kg.
The other has a mass of 6.0 kg. What is the gravitational force
between them?
5. Two bowling balls each have a mass of 6.8 kg. They are located next to
one another with their centers 21.8 cm apart. What gravitational
force do they exert on each other?
6. Kristi V. has a mass of 50.0 kg and Earth has a mass of 5.98 x 1024 kg.
The radius of the Earth is 6.38 x 106 m.
a. What is the force of gravitational attraction between Kristi V.
and the Earth?
b. What is Kristi’s weight?
7. The gravitational force between two electrons 1.0 m apart is 5.42 x
10-71 N. Find the mass of one of the electrons.
8. Two spherical balls are placed so their centers are 2.6 m apart. The
force between them is 2.75 x 10-12 N. What is the mass of each ball if
one ball is twice the mass of the other?
9. Using the fact that a 1.0 kg mass weighs 9.8 N on the surface of
Earth and the radius of Earth is roughly 6.4 x 106 m,
a. Calculate the mass of Earth.
b. Calculate the average density of the Earth.
10. The moon is 3.9 x 105 km from Earth’s center and 1.5 x 108 km from
the sun’s center. If the masses of the moon, Earth, and sun are 7.3 x
18
1022 kg, 6.0 x 1024 kg, and 2.0 x 1030 kg, respectively, find the ratio of
the gravitational forces exerted by Earth and the sun on the moon.
11. What is the force of attraction between two metal spheres, each of
which has a mass of 2.0 x 104 kg, if the distance between their
centers is 4.0 m?
12. Two students in a physics class sit 80 cm apart. Their masses are 42
kg and 58 kg. By how much are they attracted to each other?
13. The force of gravitational attraction between two lead spheres 2.00
m apart is 4.832 x 10-3 N. The mass of one sphere is 4500 kg. What is
the mass of the other?
14. Calculate the gravitational force of attraction between a proton and a
neutron separated by a distance of 1.2 x 10-11 cm if the masses of the
two particles are 1.673 x 10-24 g and 1.675 x 10-24 g respectively.
15. The gravitational force between the moon and the Earth is 1.9 x 1020
N. The masses of these two bodies are 7.36 x 1022 kg and 5.98 x
1024 kg respectively. The distance between them is 3.80 x 105 km.
From this information, calculate the value of G, the gravitational
constant.
19
Universal G
Why did Newton
think the moon was
falling?
Calculation for the
falling moon?
What does
Universal
Gravitation mean?
Who discovered
Universal Grav.?
Describe the
Cavendish
Experiment!
What was the result
of the Cav. Exp.?
What is the InverseSquare-Law?
What is the Fg
between…
M1 = 100 kg
M2 = 85 kg
d=5m
????
What is meant by
the quote “Pick a
flower, move the
farthest planet.”
Universal Gravitation Review
Effects of Gravity Problem Solving
Name 3 things that
A person weighs
would happen to the 600 N on earth.
human body in a 0g What would they
environment.
weigh on Mars?
Are you taller in the What is Mercury’s
morning or in the
gravitational field
afternoon?
strength?
Why?
The earth and the
What is the
moon are
attractive force
gravitationally
between
attracted to each
M1 300 kg
other. Which pulls
M2 30 kg
with a greater force? d = 3m
?????
What would g be at What is the period
twice the earth’s
of 1.36 m pendulum
radius?
on the surface of the
moon?
Why is the earth
On Planet Y, g = 19
round?
m/s2. What is the
period of an 85 cm
long pendulum?
Jupiter is 300 times If two objects are
more massive as the attracted to each
earth. But an object other with a force of
on Jupiter only
1.9 x 10-9 N, and the
weighs about 2.5
masses are 45 and
times more… why? 60 kg, what distance
separates them?
If the earth were to
What is the
shrink in volume,
gravitation force
but not mass, what
between the earth
would happen to
and a 150 g apple at
your weight?
sea level?
Why don’t you feel Calculate the
the gravitational
gravitational field
effects of large
strength on planet
masses like
X:
buildings?
M = 3.97 x 1022 kg
r = 2.48 x 105 m
Why doesn’t the
Determine g on
moon crash into the Pluto!
earth?
Catch-All
What are Kepler’s
three laws?
Which planet has a
shorter year?
Neptune?
Saturn?
What is a
perturbation?
Age of the universe?
Age of the earth?
What is a field?
Can a gravitational
field exist within a
planet?
Jump out of a
airplane! Are you
truly weightless?
Why?
Why not?
Who had a several
metallic noses?
What is he best
known for?
20
Review Solutions
Universal G
Effects of Gravity Problem Solving
It was not moving in Heart shrinks, bones gmars = 3.72 m/s2
a straight line at a
become brittle,
Fg = 223.2 N
constant speed.
become bloated,
muscles weaken
Catch-All
1. Elliptical
Orbit
2. Equal Area
in Equal
Time
3. r3 / Ƭ2 =
Constant
Saturn is shorter
because it is closer
to the sun.
1/602 x 4.9 =
1.4 mm
Morning, gravity
compresses your
spinal disks during
the day.
Gmercury = 3.7 m/s2
Everything is
attracted to
everything else!!!
SAME… SAME!!
Action-reaction
pairs
Fg = 6.67 x 10-8 N
A wobble in the
orbit of a planet due
to gravitational
interaction with a
nearby passing
planet.
Newton
¼ g!! (2.45 m/s2)
Ƭ = 2π√1.36 m /
1.66 m/s2
13.7 billion years
Ƭ = 5.68 s
See chapter 12
G-R-A-V-I-T-Y!!!
Ƭ = 2π√0.85 m / 19
m/s2
4 – 5 billion years
Ƭ = 1.32 s
G = 6.67 x10-11
Nm2/kg2
Fg α 1/d2
d↑, Fg↓↓
It has a very large
radius! As r↑, Fg↓↓
Your weight would
increase. Same
mass, but closer to
the center of mass.
d = 9.37 m
Fg = mg
See definition in
text
Yes. See 13.3
F = 1.47 N
Fg = 2.27 x 10-8 N
They are there… but g = GM/r2
they are negligible
g = 43.1 m/s2
(too small to feel)
As d↓, Fg ↑↑, as in
Fg = G m1m2 / d2
It has inertia (a.k.a.
tangential velocity)
g = GM/r2
g = 0.757 m/s2
No. The earth would
still be exerting a
force on you.
Tycho Brahe
(1546-1601)
21