1. Gravitation Applications Lecturer: Professor Stephen T. Thornton

2. Reading Quiz Astronauts float in the space shuttle because:
A) They are so far from Earth that Earth’s gravity doesn’t act any more.
B) Gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward.
C) While gravity is trying to pull them inward, they are trying to continue on a straight-line path.
D) Their weight is reduced in space so the force of gravity is much weaker.
Astronauts float in the space shuttle because:

3. Reading Quiz Astronauts float in the space shuttle because:
A) They are so far from Earth that Earth’s gravity doesn’t act any more.
B) Gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward.
C) While gravity is trying to pull them inward, they are trying to continue on a straight-line path.
D) Their weight is reduced in space so the force of gravity is much weaker.
Astronauts float in the space shuttle because:
Astronauts in the space shuttle float because they are in “free fall” around Earth, just like a satellite or the Moon. Again, it is gravity that provides the centripetal force that keeps them in circular motion.
Follow-up: How weak is the value of g at an altitude of 300 km?

4. Last Time History of gravitation Newton’s law of universal gravitation
Kepler’s laws
Free floating in space

5. Today Orbital maneuvers Ocean tides Geophysical applications
Free floating in space
Satellites and weightlessness
Principle of Equivalence
Black holes

6. Conceptual Quiz The Moon does not crash into Earth because:
A) It’s in Earth’s gravitational field
B) The net force on it is zero
C) It is beyond the main pull of Earth’s gravity
D) It’s being pulled by the Sun as well as by Earth
E) none of the above is precise enough
The Moon does not crash into Earth because:

7. Conceptual Quiz The Moon does not crash into Earth because:
A) It is attracted to Earth
B) The net force on it is zero
C) It is beyond the main pull of Earth’s gravity
D) It’s being pulled by the Sun as well as by Earth
E) None of the above is precise enough
The Moon does not crash into Earth because:
The Moon does not crash into Earth because of its high speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity providing the centripetal force.
Follow-up: What happens to a satellite orbiting Earth as it slows?

8. The Global Positioning System
Originally were 24 satellites. Now have 30 healthy ones. Russia has a system, and China, Europe, and India are planning/developing systems. iPhone 3G has GPS.

9. Orbital Maneuvers
Move to higher orbit

10. Orbital Maneuvers
Move to lower orbit

11. Ocean Tides
Arrows denote force due to the moon relative to the force at the center of Earth.
Newton finally correctly explained tides!
The arrows indicate how the forces from the Moon at Earth’s surface differ from the average value of the force on Earth. The arrows along the plane perpendicular to the line connecting Earth and the Moon should be much smaller than are drawn here.

12. Gravitation Force Due To Ring. A mass M is ring shaped with radius r
Gravitation Force Due To Ring. A mass M is ring shaped with radius r. A small mass m is placed at a distance x along the ring’s axis as shown in the figure. Show that the gravitational force on the mass m due to the ring is directed inward along the axis and has magnitude
[Hint: Think of the ring as made up of many small point masses dM; sum over the forces due to each dM and use symmetry.]
Giancoli, 4th ed, Problem 6-18

13. Vector Form of Newton’s Universal Gravitation
If there are many particles, the total force is the vector sum of the individual forces:

14. We can relate the gravitational constant to the local acceleration of gravity. We know that on the surface of the Earth:
Solving for g gives:
g can be measured to 1 part in 109 so that mineral and oil deposits can be detected using sensitive gravitometers.

15. The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical.

16. Geosynchronous satellite.
A geosynchronous satellite stays above the same point on the Earth, which is possible only if it is above a point on the equator. Such satellites are used for TV and radio transmission, for weather forecasting, and as communication relays. They must have an orbit of precisely 24 hours. In order to do that, they must be about 22,000 miles above the Earth and have a precise speed.
Solution: a. The speed of the satellite must be the same as the speed of a point on the Earth’s equator. This gives us the centripetal force, and therefore the gravitational force required, and therefore the radius of the satellite’s orbit. Answer: it is about 36,000 km above the Earth’s surface.
b. V = 3070 m/s.
c. The speed is inversely proportional to the square root of the radius of the orbit, so v = 7780 m/s.

17. Lagrange point
The mathematician Joseph-Louis Lagrange discovered five special points in the vicinity of the Earth’s orbit about the Sun where a small satellite (mass m) can orbit the Sun with the same period T as Earth’s (= 1 year).
One of these “Lagrange Points,” called L1, lies between the Earth and Sun on the line connecting them.
Several satellites are being placed in Lagrange points. We probably will not be able to service them like we have done with the Hubble.
Lagrange ( ) was born in Italy, but had connections to France. Both claimed him. He was a mathematician.
Solution: The satellite can remain in a smaller orbit at the same orbital speed as the Earth because the net gravitational force on it is less than just the force due to the Sun. This involves quite a bit of algebra, and assumes that the satellite’s mass is negligible (fine, as the existence of the L1 point doesn’t depend on having a satellite there). Result: d = 1.5 x 106 km.

18. Satellites and “Weightlessness”
Objects in orbit are said to experience “weightlessness”. They do have a gravitational force acting on them, though! The satellite and all its contents are in free fall, so there is no normal force. This is what leads to the experience of weightlessness.
Figure Caption: (a) An object in an elevator at rest exerts a force on a spring scale equal to its weight. (b) In an elevator accelerating upward at ½ g, the object’s apparent weight is 1 ½ times larger than its true weight. (c) In a freely falling elevator, the object experiences “weightlessness”: the scale reads zero.

19. More properly, this effect is called apparent weightlessness, because the gravitational force still exists. It can be experienced on Earth as well, but only briefly:
Figure Caption: Experiencing “weightlessness” on Earth.

20. Various airplanes have been used for vomit comet
Various airplanes have been used for vomit comet. Occupants are weightless for about 25 s.

21. Gravitational Field
The gravitational field is the gravitational force per unit mass:
The gravitational field due to a single mass M is given by:

22. Principle of Equivalence
Inertial mass: the mass that appears in Newton’s second law.
Gravitational mass: the mass that appears in the universal law of gravitation.
Principle of equivalence: inertial mass and gravitational mass are the same.
We can do no experiment to tell the difference between gravitational and inertial mass. Fundamental tenet of the General Theory of Relativity.
Eotvos did precision experiments. Now known to 1 part in 10^12.

23. One way to visualize the curvature of space (a two-dimensional analogy):
If the gravitational field is strong enough, even light cannot escape, and we have a black hole. Einstein predicted in 1915 that light should be attracted by gravity to mass.
Do demo

24. At rest
Light should be deflected by a massive object. On the right side, the person can not tell whether acceleration caused the light to bend or whether gravity did it.
Figure Caption: (a) Light beam goes straight across an elevator that is not accelerating. (b) The light beam bends (exaggerated) in an elevator accelerating in an upward direction.

25. This bending has been measured during total solar eclipses.
Figure Caption: (a) Three stars in the sky. (b) If the light from one of these stars passes very near the Sun, whose gravity bends the light beam, the star will appear higher than it actually is. In 1919 Eddington looked for two stars behind the sun. With the total eclipse (moon blocking the sun) , the two stars could be observed in different positions than without the sun. The light was bent while going by the sun.

26. Gravitational Lensing
Massive stars can collapse under the gravitational force. They can become black holes, and nothing can escape even light.
Einstein showed gravity even bends light!

27. Gravity Assist

28. Milky Way Galaxy. The Sun rotates about the center of the Milky Way Galaxy (see figure) at a distance of about 30,000 light-years from the center (1 ly = 9.5 x 1015 m). If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun (2 x 1030 kg), how many stars would there be in our Galaxy?
Giancoli, 4th ed, Problem 6-60

29. Conceptual Quiz A) B) C)
D) it’s the same
E) 2
Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A?
Answer: 2

30. Conceptual Quiz A) B) C)
D) it’s the same
E) 2
Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A?
Using the Law of Gravitation:
we find that the ratio is .
Note the
1/r2 factor

31. Conceptual Quiz
A planet of mass m is a distance d from Earth. Another planet of mass 2m is a distance 2d from Earth. Which force vector best represents the direction of the total gravitation force on Earth? A B C D E 2d d 2m m
Earth
Answer: 2

32. Conceptual Quiz 2d 2m E D d C B A m F2m = GME(2m) / (2d)2 = GMm / d 2
A planet of mass m is a distance d from Earth. Another planet of mass 2m is a distance 2d from Earth. Which force vector best represents the direction of the total gravitation force on Earth? A B C D E 2d d 2m m
The force of gravity on the Earth due to m is greater than the force due to 2m, which means that the force component pointing down in the figure is greater than the component pointing to the right.
F2m = GME(2m) / (2d)2 = GMm / d 2
Fm = GME m / d 2 = GMm / d 2