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 Average earth-moon

= 3,955 miles distance:

384,000,000 m

Or

230,600 miles

On Earth:On Moon:

In one second an object will fall… 6O X the radius of the Earth…

d = ½ gt2

So. The moon should fall ______

A satellite at 30 X radius of the Earth should fall

Universal Gravitation Practice Problems

Directions: Solve the following problems and questions. Remember to usethe 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 ofG 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:

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)

Complete the following Chart:

( g = Gm/r2)

Distance Strength of the field (N/Kg) 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.)

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

Gravitational Fields Within The Planet

IMAGINE: A hole dug through the earth. Forget about the impracticalities such as lava and very high temperatures…

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!!

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…

g = 4π2 r / Ƭ2eqn 2

Look familiar???...ac = 4π2 r / Ƭ2eqn 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 ______%

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%

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 ofHarmonies) Share this onDiggFacebookdel.icio.usreddit

StumbleUpon

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

Can't remember the 11 planets? 4th-grader offers help

GREAT FALLS, Montana (AP) -- Those having trouble remembering the newly assigned 11 planets, including three dwarfs, are getting help from a fourth-grader.

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.

Did you know that all but two planets (Mercury and Pluto)

have orbits in the same plane?

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. Determinethe 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).

PLANETARY DATA
NAME / MASS (kg) / PERIHELION DIST. (megamiles) / APHELION DIST. (megamiles) / PERIHELION Date / APHELIONDate
Sun / 1.991 x 1030
Mercury / 3.2 x 1023 / 28.6 / 43.4 / 10/16/95 / 11/29/95
Venus / 4.88 x 1024 / 66.8 / 67.7 / 8/11/95 / 12/1/95
Earth / 5.979 x 1024 / 91.4 / 94.5 / 12/21/95 / 6/21/96
Mars / 6.42 x 1023 / 128.4 / 154.9 / 2/19/96 / 1/28/97
Jupiter / 1.901 x 1027 / 460.3 / 507.2 / 5/5/99 / 3/29/2005
Saturn / 5.68 x 1028 / 837.6 / 936.2 / 5/26/03 / 2/8/2018
Uranus / 8.68 x 1026 / 1699.0 / 1868.0 / 3/1/2050 / 4/17/2008
Neptune / 1.03 x 1026 / 2771.0 / 2819.0 / 3/2030 / 2/2112
Pluto / 1.2 x 1022 / 2756.0 / 4555.0 / 8/1989 / 8/2113

Concepts of PhysicsNAME:______

Mr. KufferPeriod: ______

Orbit Lab Work

*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:

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.
  7. What is the force of gravitational attraction between Kristi V. and the Earth?
  8. What is Kristi’s weight?
  9. 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.
  10. 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?
  11. 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,
  12. Calculate the mass of Earth.
  13. Calculate the average density of the Earth.
  14. 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 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.
  15. 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?
  16. 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?
  17. 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?
  18. 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.
  19. 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.

Universal Gravitation Review

Universal G / Effects of Gravity / Problem Solving / Catch-All
Why did Newton think the moon was falling? / Name 3 things that would happen to the human body in a 0g environment. / A person weighs 600 N on earth. What would they weigh on Mars? / What are Kepler’s three laws?
Calculation for the falling moon? / Are you taller in the morning or in the afternoon?
Why? / What is Mercury’s gravitational field strength? / Which planet has a shorter year?
Neptune?
Saturn?
What does Universal Gravitation mean? / The earth and the moon are gravitationally attracted to each other. Which pulls with a greater force? / What is the attractive force between
M1 300 kg
M2 30 kg
d = 3m ????? / What is a perturbation?
Who discovered Universal Grav.? / What would g be at twice the earth’s radius? / What is the period of 1.36 m pendulum on the surface of the moon? / Age of the universe?
Describe the Cavendish Experiment! / Why is the earth round? / On Planet Y, g = 19 m/s2. What is the period of an 85 cm long pendulum? / Age of the earth?
What was the result of the Cav. Exp.? / Jupiter is 300 times more massive as the earth. But an object on Jupiter only weighs about 2.5 times more… why? / If two objects are attracted to each other with a force of 1.9 x 10-9 N, and the masses are 45 and 60 kg, what distance separates them? / What is a field?
What is the Inverse-Square-Law? / If the earth were to shrink in volume, but not mass, what would happen to your weight? / What is the gravitation force between the earth and a 150 g apple at sea level? / Can a gravitational field exist within a planet?
What is the Fg between…
M1 = 100 kg
M2 = 85 kg
d = 5 m ???? / Why don’t you feel the gravitational effects of large masses like buildings? / Calculate the gravitational field strength on planet X:
M = 3.97 x 1022 kg
r = 2.48 x 105 m / Jump out of a airplane! Are you truly weightless?
Why?
Why not?
What is meant by the quote “Pick a flower, move the farthest planet.” / Why doesn’t the moon crash into the earth? / Determine g on Pluto! / Who had a several metallic noses? What is he best known for?

Review Solutions

Universal G / Effects of Gravity / Problem Solving / Catch-All
It was not moving in a straight line at a constant speed. / Heart shrinks, bones become brittle, become bloated, muscles weaken / gmars = 3.72 m/s2
Fg = 223.2 N /
  1. Elliptical Orbit
  2. Equal Area in Equal Time
  3. r3 / Ƭ2 = Constant

1/602 x 4.9 =
1.4 mm / Morning, gravity compresses your spinal disks during the day. / Gmercury = 3.7 m/s2 / Saturn is shorter because it is closer to the sun.
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
Ƭ = 5.68 s / 13.7 billion years
See chapter 12 / G-R-A-V-I-T-Y!!! / Ƭ = 2π√0.85 m / 19 m/s2
Ƭ = 1.32 s / 4 – 5 billion years
G = 6.67 x10-11 Nm2/kg2 / It has a very large radius! As r↑, Fg↓↓ / d = 9.37 m / See definition in text
Fg α 1/d2
d↑, Fg↓↓ / Your weight would increase. Same mass, but closer to the center of mass. / Fg = mg
F = 1.47 N / Yes. See 13.3
Fg = 2.27 x 10-8 N / They are there… but they are negligible (too small to feel) / g = GM/r2
g = 43.1 m/s2 / No. The earth would still be exerting a force on you.
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 / Tycho Brahe
(1546-1601)

1