1. Circular Motion Gravity and Planets
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2. Observations and explanations of planetary motion
Newton was among the first to hypothesize that the Moon moves in a circular orbit around Earth because Earth pulls on it, continuously changing the direction of the Moon's velocity.
He wondered if the force exerted by Earth on the Moon was the same type of force that Earth exerted on falling objects, such as an apple falling from a tree.
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3. Tip
You might wonder why if Earth pulls on the Moon, the Moon does not come closer to Earth in the same way that an apple falls from a tree.
The difference in these two cases is the speed of the objects. The apple is at rest with respect to Earth before it leaves the tree, and the Moon is moving tangentially.
If Earth stopped pulling on the Moon, it would fly away along a straight line.
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4. Dependence of gravitational force on mass
Newton deduced by recognizing that acceleration would equal g only if the gravitational force was directly proportional to the system object's mass.
Newton deduced by applying the third law of motion.
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5. The law of universal gravitation
Newton deduced but did not know the proportionality constant; in fact, at that time the mass of the Moon and Earth were unknown.
Later scientists determined the proportionality:
G is the universal gravitational constant, indicating that this law works anywhere in the universe for any two masses.
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6. The universal gravitational constant
G is very small.
For two objects of mass 1 kg that are separated by 1 m, the gravitational force they exert on each other equals 6.67 x 10–11 N.
The gravitational force between everyday objects is small enough to ignore in most calculations.
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7. Newton's law of universal gravitation
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8. Law of Universal Gravitation Obeys an Inverse-Square Law
Gravity is a universal force that affects all objects in the universe.
Newton proposed that the force of gravity has the following properties:
The force is inversely proportional to the square of the distance between the objects.
The force is directly proportional to the product of the masses of the two objects.
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9. Inverse-Square Law Newton’s law of gravity is an inverse- square law.
Doubling the distance between two masses causes the force between them to decrease by a factor of 4.
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10. Gravity on Other Worlds
If we use values for the mass and the radius of the moon, we compute gmoon = 1.62 m/s2.
A 70-kg astronaut wearing an 80-kg spacesuit would weigh more than 330 lb on the earth but only 54 lb on the moon.
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11. Gravity on Other Worlds
If you traveled to another planet, your mass would be the same but your weight would vary. The weight of a mass m on the moon is given by
Using Newton’s law of gravity the weight is given by:
Since these are two expressions for the same force, they are equal and
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12. Making sense of the gravitational force that everyday objects exert on Earth
It might seem counterintuitive that objects exert a gravitational force on Earth; Earth does not seem to react every time someone drops something.
Acceleration is force divided by mass, and the mass of Earth is very large, so Earth's acceleration is very small.
For example, the acceleration caused by the gravitational force a tennis ball exerts on Earth is 9.2 x 10–26 m/s2.
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13. Astronauts are NOT weightless in the International Space Station
Earth exerts a gravitational force on them!
This force causes the astronaut and space station to fall toward Earth at the same rate while they fly forward, staying on the same circular path.
The astronaut is in free fall (as is the station).
If the astronaut stood on a scale in the space station, the scale would read zero even though the gravitational force is nonzero.
Weight is a way of referring to the gravitational force, not the reading of a scale.
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14. Newton’s Law of Gravity and Orbits!!!
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15. Orbital Motion
The force of gravity on a projectile is directed toward the center of the earth.
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16. Orbital Motion (Show Newtons Cannon Video)
If the launch speed of a projectile is sufficiently large, there comes a point at which the curve of the trajectory and the curve of the earth are parallel.
Such a closed trajectory is called an orbit.
An orbiting projectile is in free fall.
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17. Orbital Motion
The force of gravity is the force that causes the centripetal acceleration of an orbiting object:
An object moving in a circle of radius r at speed vorbit will have this centripetal acceleration if
That is, if object moves parallel (in orbit) to the surface of an object with speed
(This is also the speed at which apparent weightlessness will occur n=mg)
v= 𝑔𝑟 𝑜𝑟 … 𝑖𝑛 𝑎 𝑚𝑖𝑛𝑢𝑡𝑒 𝐼 𝑤𝑖𝑙𝑙 𝑠ℎ𝑜𝑤 𝑦𝑜𝑢 𝑖𝑡 𝑖𝑠 𝑎𝑙𝑠𝑜= 𝑣= 𝐺 𝑀 𝑃𝑙𝑎𝑛𝑒𝑡 𝑟
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18. Orbital Motion: A couple of Definitions and Reminders
Apparent Weight: the upward force (normal force) that opposes a supported object from falling (What a scale reads)
Object True Weight: Force exerted by gravity or mg.
Wapparent = Actual Weight except:
Object has acceleration with a vertical component (i.e y direction)
Some force other than earth’s gravity is acting on the object: Normal, Magnetic, buoyant, centripetal or gravitational force of another body
Free Fall: A free falling object is an object that is falling at g. There are two important motion characteristics that are true of free-falling objects……….

19. Orbital Motion: A couple of Definitions and Reminders
So objects in orbit…are in free fall and have no apparent weight (normal force =0). If object apparent weight does not equal 0 (or 𝑣= 𝑔𝑟 ) it will either
fly off into space (𝑁>𝑚𝑔 𝑣> 𝑔𝑟 ) or
be pulled by gravity (𝑚𝑔>𝑁 𝑣< 𝑔𝑟 )

20. Orbital Motion
The orbital speed of a projectile just skimming the surface of a smooth, airless earth is
Note: Radius of earth is 6.37 x 106m
We can use vorbit to calculate the period of the satellite’s orbit:
Substitude 𝑣 𝑜𝑟𝑏𝑖𝑡 = 𝑔𝑟 do some alg 2 with exponential fractions
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21. Here Comes Newton Gravitational Law and UCM
Newton’s second law tells us that FM on m = ma, where FM on m is the gravitational force of the large body on the satellite and a is the satellite’s acceleration.
Because it’s moving in a circular orbit, Newton’s second law gives
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22. Gravity and Orbits
A satellite must have this specific speed in order to maintain a circular orbit of radius r about the larger mass M.
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23. Gravity and Orbits
For a planet orbiting the sun, the period T is the time to complete one full orbit. The relationship among speed, radius, and period is the same as for any circular motion:
v = 2πr/T
Combining this with the value of v for a circular orbit.
𝐺𝑀𝑚 𝑟 2 = 𝑚 𝑣 2 𝑟
If we square both sides and rearrange, we find the period of a satellite:
Here is “Keplers 3rd Law” now proven!!!!
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24. Question
Astronauts on the International Space Station are weightless because
There’s no gravity in outer space.
The net force on them is zero.
The centrifugal force balances the gravitational force.
g is very small, although not zero.
They are in free fall.
Answer: E
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25. Question
Astronauts on the International Space Station are weightless because
There’s no gravity in outer space.
The net force on them is zero.
The centrifugal force balances the gravitational force.
g is very small, although not zero.
They are in free fall.
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26. Question 1 Gravitational force between two people
You are seated in your physics class next to another student m away. Estimate the magnitude of the gravitational force between you. Assume that you each have a mass of 65 kg.
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27. Question 1: Gravitational force between two people (cont.)
solve The gravitational force is given by: assess The force is quite small, roughly the weight of one hair on your head. This seems reasonable; you don’t normally sense this attractive force!
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28. Question 2 The force of Planet Y on Planet X is ___ the magnitude of .
One quarter
One half
The same as
Twice
Four times 2M M
Planet X
Planet Y
Answer: C
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29. Question 2 The force of Planet Y on Planet X is ___ the magnitude of .
One quarter
One half
The same as
Twice
Four times 2M M
Newton’s third law
Planet X
Planet Y
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30. Question 3
The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled?
250,000 N
500,000 N
1,000,000 N
2,000,000 N
4,000,000 N
Answer: A
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31. Question 3
The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled?
250,000 N
500,000 N
1,000,000 N
2,000,000 N
4,000,000 N
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32. Question 4
Planet X has free-fall acceleration 8 m/s2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y
g = 2 m/s2
g = 4 m/s2
g = 8 m/s2
g = 16 m/s2
g = 32 m/s2
Answer: B
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33. Question 4
Planet X has free-fall acceleration 8 m/s2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y
g = 2 m/s2
g = 4 m/s2
g = 8 m/s2
g = 16 m/s2
g = 32 m/s2
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34. Question 5
A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest?
Answer: A
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35. Question 5
A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest? A
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36. Question 7
Two satellites have circular orbits with the same radius. Which has a higher speed?
The one with more mass.
The one with less mass.
They have the same speed.
Answer: C
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37. Question 7
Two satellites have circular orbits with the same radius. Which has a higher speed?
The one with more mass.
The one with less mass.
They have the same speed.
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38. Question 8
Two identical satellites have different circular orbits. Which has a higher speed?
The one in the larger orbit
The one in the smaller orbit
They have the same speed.
Answer: B
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39. Question 8
Two identical satellites have different circular orbits. Which has a higher speed?
The one in the larger orbit
The one in the smaller orbit
They have the same speed.
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40. Question 9
A satellite orbits the earth. A Space Shuttle crew is sent to boost the satellite into a higher orbit. Which of these quantities increases?
Speed
Angular speed
Period
Centripetal acceleration
Gravitational force of the earth
Answer: C
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41. Question 9
A satellite orbits the earth. A Space Shuttle crew is sent to boost the satellite into a higher orbit. Which of these quantities increases?
Speed
Angular speed
Period
Centripetal acceleration
Gravitational force of the earth
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42. Gravity on a Grand Scale
No matter how far apart two objects may be, there is a gravitational attraction between them.
Galaxies are held together by gravity.
All of the stars in a galaxy are different distances from the galaxy’s center, and so orbit with different periods.
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43. 2nd Day Notability Discusion
Formula review
Gravitational Field
Inertial vs Gravitational Mass
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44. Satellites and astronauts: Putting it all together
Geostationary satellites stay at the same location in the sky. This is why a satellite dish always points in the same direction.
These satellites must be placed at a specific altitude that allows the satellite to travel once around Earth in exactly 24 hours while remaining above the equator.
An array of such satellites can provide communications to all parts of Earth.
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45. Example 4.8: Geostationary satellite
You are in charge of launching a geostationary satellite into orbit.
At which altitude above the equator must the satellite orbit be to provide continuous communication to a stationary dish antenna on Earth?
The mass of Earth is 5.98 x 1024 kg. 24 hrs to orbit or and a 24 hr period is 86400sec.
This is left for the instructor to solve as an example
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46. Example 4.8: Geostationary satellite
This is left for the instructor to solve as an example
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47. Example 4.8 Locating a geostationary satellite (cont.)
assess This is a high orbit, and the radius is about 7 times the radius of the earth. (26,221 miles) Radius of the International Space Station’s orbit is only about 5% larger than that of the earth. (About 50 km altitude or 5.0 x 105 m) What is the speed of the Space Station. Will it be more or less than the speed of a satellite at geostationary orbit.
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48. Quantitative Exercise 4
Quantitative Exercise 4.9: Are astronauts weightless in the International Space Station?
The International Space Station orbits approximately 5.0 x 105 m above Earth's surface, or 6.78 x 106 m from Earth's center.
Compare the force that Earth exerts on an astronaut in the station to the force that Earth exerts on the same astronaut when he is on Earth's surface. (calculate using the F=mac)
This is left for the instructor to solve as an example
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