1. Gravitational Interactions
Chapter 13
Gravitational Interactions

2. 13.1 Gravitational Fields
A magnetic field is a force field, because magnetic materials in it experience a force
The gravitational field around Earth is similar
A mass in the field region experiences a gravitational force
The field is strongest at the surface of Earth, and declines as the distance from Earth’s center increases

3. g is the acceleration due to gravity
G is the universal gravitational constant
g is the gravitational field vector
They are all different quantities that are related
G and g have the same magnitude and their units are equivalent: N/kg = m/s2

4. g = GM/R2
g = 9.8 N/kg is valid only near the planet’s surface
as R increases, g decreases
The strength of Earth’s gravitational field is the force per unit mass exerted by Earth on any object
g = F m
g = 9.8 N/kg = 9.8 m/s2
g weakens with increasing distance from Earth

5. Questions
If Earth had the same size but twice the mass, what would be the acceleration of freely falling objects at its surface?
Twice g, or nearly 20 m/s2
If Earth has the same mass but half the size, what would be the acceleration of freely falling objects at its surface?
Four times g, or nearly 40 m/s2

6. If Earth had twice the mass, and half the size, what would be the acceleration of freely falling objects at its surface?
Eight times g, or nearly 80 m/s2
What evidence would you look for to tell whether or not you were in a gravitational field?
The presence of a gravitational force

7. Compared to its strength at Earth’s surface, what is the strength of the gravitational field at a distance of two Earth radii from the center of Earth?
¼ as much

8. 13.2 Gravitational Field Inside a Planet
Gravitational field on Earth exists inside Earth as well as outside
Imagine a hole drilled completely through Earth

9. FALLING THROUGH THE CENTER OF THE EARTH
Start at the North Pole
Fall & gain speed down to the center
Lose speed all the way to the South Pole
Gain speed toward the center
Lose speed moving away from the center
a=g
a=g/2
a=0
a=g/2
a=g

10. As you fell faster and faster into the Earth, your acceleration diminishes because the pull of the mass above you partly cancels the pull below
At the Earth’s center the pulls cancel to zero and your acceleration is zero
Momentum carries you against a growing acceleration past the center to the opposite side where it is again, g

11. STUCK IN A CAVITY
In a cavity at the center of the Earth, your weight would be zero, because you would be pulled equally by gravity in all directions
You are pulled in every direction equally
The gravitational field at Earth’s center is zero

12. Questions
If you dropped a rock into a tunnel through Earth, what would happen?
It would gain speed until it reached Earth’s center, and then lose speed the rest of the way. Its speed at the far end of the tunnel would be the same as its initial speed. It would then fall back and repeat the motion in cyclic fashion.

13. As the velocity of the rock increases when falling into the Earth tunnel, what happens to the acceleration?
It decreases as the gravitational field decreases, and is zero at Earth’s center. The falling body has its maximum velocity at Earth’s center, where both the field and acceleration are zero

14. 13.3 Weight and Weightlessness
Weight: support force
We are as heavy as we feel
Weightlessness: the absence of a support force
The queasy feeling you experience when in a car that speeds over the top of a hill

15. FOR EXAMPLE: A SCALE Stand on a bathroom scale on the floor
The gravitational force between you and Earth pulls you against the supporting floor and scale
By Newton’s 3rd law, the floor and scale in turn push upward on you

16. NOW…WEIGHT IN AN ELEVATOR

17. UP, UP, AND AWAY
If the elevator accelerates upward, the bathroom scale and floor would push harder against your feet
The scale would show an increase in your weight

18. GROUND FLOOR PLEASE
If the elevator accelerated downward, the scale would show a decrease in your weight
The support force of the floor would now be less

19. ELEVATOR CABLE BROKE Elevator falls freely, the scale would read zero
According to the scale, you would be weightless and feel weightless
Your insides would no longer be supported by your legs and hips

20. Question
Why would you feel weightless in an elevator with a broken cable?
There would be no support force-the floor would fall as fast as you

21. 13.4 Ocean Tides Think of someone pulling on your coat
If some one pulled only on the sleeve, the coat could tear
But if every part of your coat were pulled equally, it and you would accelerate, but the coat wouldn’t tear
It tears when one part is pulled harder than another because of a difference in forces acting on the coat

22. In a similar way, the spherical Earth is “torn” into an elliptical shape by differences in gravitational forces exerted by the moon

23. Ocean tides are caused by differences in the gravitational pull of the moon on opposite sides of Earth
The moon’s attraction is stronger on Earth’s oceans closer to the moon, and weaker on the oceans farther from the moon
gravitational forces weakened with increased distance

24. Why are there two tides a day?
There would only be one tide per day, if the Earth were “nailed down” in one place and held stationary, except for a daily rotation
But the Earth and moon are in orbit around each other
The ocean nearest the moon is pulled upward toward the moon, while the main body of Earth is pulled toward the moon
away from the ocean on the far side
Earth is closer to the moon than the far-side ocean is, so Earth’s waters get slightly elongated-at both ends creating ocean bulges

25. Earth makes one complete turn per day beneath these ocean bulges creating two sets of ocean tides per day
The part of Earth that passes beneath one of the bulges has a high tide
The tide cycle is every 24 hours and 50 minutes
The sun also contributes to ocean tides, but not as much as the moon.
The tilt of Earth’s axis is another factor

26. Questions
Which pulls harder on the oceans of Earth, the sun or the moon?
The sun
Which is most effective in raising tides?
The moon

27. The difference in pulls decreases as the cube of the distance between the centers of the bodies
Twice as far away produces 1/8 the tide
3 times as far, only 1/27 the tide
Only relatively close distances result in appreciable tides, and so the nearby moon “out-tides” the enormously more massive but far away sun

28. These are called spring tides
When the sun, Earth, and the moon are lined up, the tides due to the sun and the moon coincide
There are higher high tides and lower low tides
These are called spring tides
They have nothing to do with spring
If the alignment is perfect, there is an eclipse

29. ECLIPSE
A lunar eclipse is produced when Earth is directly between the sun and moon
A solar eclipse is produced when the moon is directly between the sun and Earth

30. PHASES OF THE MOON
When the Earth is between the sun and the moon –full moon
When the moon is between the sun and Earth – new moon
Spring tides occur during a new moon and full moon
During half-moons, the tides due to the sun and the moon partly cancel each other
High tides are lower than average
Low tides are higher than average
These are called neap tides

31. Question
At the time of extra high tides, will extra low tides follow in the same day?
Yes, by the “conservation of water”. There is only so much water on Earth-extra high tides in one part of the world means extra low tides in another

32. 13.5 Tides in Earth and the Atmosphere
There are tides within Earth, which is mostly molten lava
There is a greater probability of earthquakes and volcanoes when there is an eclipse of the sun or the moon
This is when Earth experiences spring tides-greater stresses on Earth’s crust

33. If the moon were closer…
Ocean tides would be higher and so the tidal forces on the moon’s crust would be greater
If the moon were too close, Earth’s tidal forces would tear the moon into a billion pieces, forming a ring around Earth similar to those around Saturn

34. 13.6 BLACK HOLES
Occurs with stars that are at least two to three times more massive than our sun
Once the flame of thermonuclear fusion is extinguished, gravitational collapse takes over-and it doesn’t stop
Gravitation is so enormous that nothing can get back out-even light

35. A collapsed star represents condensed mass and therefore condensed gravity.
The mass of a black hole is no more than the mass of the star that collapsed to form it
Hence the gravitational field of the star and the black hole are the same at distances greater than the original radius of the star
It is only at closer distances that the enormous field occurs

36. Question
Consider a satellite companion to a star that collapses to become a black hole. How will the orbit of the companion satellite be affected by the star’s transformation to a black hole?
Not at all. No terms in the gravitational equation change

37. Summary
Earth can be thought of as being surrounded by a gravitational field that interacts with objects and causes them to experience gravitational forces.
The gravitational field, g, is equal to the acceleration of a freely falling object.
Objects in orbit around Earth have a gravitational force acting on them even though they may appear to be weightless.

38. Ocean tides (and even tides within the solid Earth and within the atmosphere) are caused by differences in the gravitational pull of the moon (and sun) on opposite sides of Earth.
When a star runs out of fuel for fusion, it collapses under gravitational forces. Sufficiently massive stars collapse to form black holes.