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Conceptual Physics

Paul G. Hewitt

Chapter 9

Gravity

Educators


Problem 1

What was the prevailing notion regarding the motion of celestial bodies before Newton?

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Problem 2

What is the Newtonian synthesis?

Mathis E.
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Problem 3

How is circular motion actually a “fall”?

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Problem 4

State Newton’s law of universal gravitation in words. Then do the same with one equation.

Mathis E.
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Problem 5

What is the magnitude of the gravitational force between two 1-kg bodies that are 1 m apart?

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Problem 6

What is the magnitude of Earth’s gravitational force on a 1-kg body at Earth’s surface?

Mathis E.
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Problem 7

When G was first measured by Henry Cavendish, news- papers of the time hailed his experiment as the “weighing Earth experiment.” Why?

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Problem 8

How does the force of gravity between two bodies change when the distance between them is tripled?

Mathis E.
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Problem 9

How does the area covered by paint sprayed on a surface change when the sprayer is held twice as far away?

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Problem 10

Where do you weigh less: at the top of a mountain or at ground level? Why?

Mathis E.
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Problem 11

Would the springs inside a bathroom scale be more compressed or less compressed if you weighed yourself in an elevator that was accelerating upward? Downward?

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Problem 12

Would the springs inside a bathroom scale be more compressed or less compressed if you weighed yourself in an elevator that was moving upward at constant velocity? Downward at constant velocity?

Mathis E.
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Problem 13

When is your weight measured as greater than mg?

Salamat A.
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Problem 14

Give an example of when your weight is less than mg. Give an example of when your weight is zero.

Mathis E.
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Problem 15

Why are International Space Station occupants weightless when they are firmly in the grip of Earth’s
gravity?

Salamat A.
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Problem 16

Do tides depend more on the strength of gravitational pull or on the difference in strengths? Explain.

Mathis E.
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Problem 17

Why do both the Sun and the Moon exert a greater gravitational force on one side of Earth than on the other?

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Problem 18

What is the alignment of the Sun, Earth, and the Moon during spring tides?

Mathis E.
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Problem 19

Do tides occur in the molten interior of Earth for the same reason that tides occur in the oceans?

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Problem 20

What is the position of the Moon during neap tides? Why do ponds never display tides?

Mathis E.
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Problem 21

Would a torque on the Moon occur if the Moon were spherical, with both its center of mass and center of gravity in the same location?

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Problem 22

Why is a gravitational field an example of a force field?

Mathis E.
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Problem 23

What happens to the acceleration of a body at the center of Earth?

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Problem 24

For a planet of uniform density, how would the magnitude of the gravitational field halfway to the center compare with the field at the surface?

Mathis E.
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Problem 25

What would be the magnitude of the gravitational field anywhere inside a hollow, spherical planet?

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Problem 26

Newton viewed the curving of the path of a planet as being caused by a force acting on the planet. How did Einstein view the curved path of a planet?

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Problem 27

If Earth shrank but there was no change in its mass, what would happen to your weight at the surface?

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Problem 28

What happens to the strength of the gravitational field at the surface of a star that shrinks?

Mathis E.
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Problem 29

How is a black hole detected?

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Problem 30

What was the cause of perturbations discovered in the orbit of the planet Uranus? What greater discovery did this lead to?

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Problem 31

Hold your hands outstretched in front of you, one twice as far from your eyes as the other,
and make a casual judgment about which hand looks bigger. Most people see them as about the same size, although many see the nearer hand as slightly bigger. Almost no one, upon casual inspection, sees the nearer hand as four times as big, but, by the inverse-square law, the nearer hand should appear to be twice as tall and twice as wide and therefore seem to occupy four times as much of your visual field as the farther hand. Your belief that your hands are the same size is so strong that you likely over- rule this information. Now, if you overlap your hands slightly and view them with one eye closed, you’ll see the nearer hand as clearly bigger. This raises an interesting question (that Frank Oppenheimer posed to me years ago when I first taught at the Exploratorium): “What other illusions do you have that are not so easily checked?”

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Problem 32

Repeat the eyeballing experiment, only this time use two dollar bills—one unfolded and the other folded along its center lengthwise and then again widthwise, so it has 1/4 the area. Now hold the two in front of your eyes. Where do you hold the folded one so that it looks the same size as the unfolded one? Nice?

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Problem 33

$F=G \frac{m_{1} m_{2}}{d^{2}}$
Calculate the force of Earth's gravity on a 2 -kg mass at Earth's surface. The mass of Earth is $6.0 \times 10^{24} \mathrm{kg}$ and its radius is $6.4 \times 10^{6} \mathrm{m} .$ Does the result surprise you?

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Problem 34

Calculate the force of gravity on the same 2 -kg mass if it were $12.8 \times 10^{6} {m}$ above Earth's surface (that is, if it were three Earth radii from Earth's center).

Mathis E.
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Problem 35

Calculate the force of gravity that Earth (mass $6.0 \times$ $10^{24} {kg} )$ and the Moon (mass $7.4 \times 10^{22} {kg}$ ) exert on each other. The average Earth-Moon distance is $3.8 \times$ $10^{8} {m} .$

Salamat A.
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Problem 36

Calculate the force of gravity that Earth and the Sun exert on each other (Sun's mass is $2.0 \times 10^{30} \mathrm{kg} ;$ average Earth- Sun distance is $1.5 \times 10^{11} \mathrm{m} )$

Mathis E.
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Problem 37

Calculate the force of gravity a newborn baby $(\text { mass } 3.0 \mathrm{kg})$ and the planet Mars (mass $6.4 \times 10^{23} \mathrm{kg} )$ exert on each other when Mars is at its closest to Earth
(distance $5.6 \times 10^{10} \mathrm{m} )$

Salamat A.
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Problem 38

Calculate the force of gravity a newborn baby of mass 3.0 $\mathrm{kg}$ and the obstetrician of mass 100.0 $\mathrm{kg}$ exert on each other when the distance between them is 0.5 $\mathrm{m} .$
Which exerts more gravitational force on the baby: Mars or the obstetrician? By how much?

Mathis E.
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Problem 39

Suppose you stood atop a ladder so tall that you were 2 times as far from Earth’s center as you presently are. How would your weight vary from its present value?

Salamat A.
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Problem 40

Show that the gravitational force between two planets is quadrupled if the masses of both planets are doubled but the distance between them stays the same.

Mathis E.
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Problem 41

Show that there is no change in the force of gravity between two objects when their masses are doubled and the distance between them is also doubled.

Salamat A.
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Problem 42

Find the change in the force of gravity between two planets when the distance between them becomes 5 times smaller.

Mathis E.
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Problem 43

Many people mistakenly believe that astronauts who orbit Earth are “above gravity.” Calculate g for space shuttle territory, 200 km above Earth’s surface. Earth’s mass is $6.0 \times 10^{24} \mathrm{kg},$ and its radius is $6.38 \times 10^{6} \mathrm{m}$ $(6380 \mathrm{km}) .$ Your answer is what percentage of 9.8 $\mathrm{m} / \mathrm{s}^{2}$ ?

Salamat A.
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Problem 44

Newton’s universal law of gravity tells us that
$F=G \frac{m_{1} m_{2}}{d^{2}}$
Newton's second law tells us that $a=\frac{F_{n e t}}{m}$
a. With a bit of algebraic reasoning, show that your gravitational acceleration toward any planet of mass M a distance $d$ from its center is $a=\frac{G M}{d^{2}}$
b. How does this equation tell you whether or not your gravitational acceleration depends on your mass?

Mathis E.
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Problem 45

The planet and its moon gravitationally attract each other. Rank the forces of attraction between each pair, from greatest to least.

Salamat A.
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Problem 46

Consider the light of multiple candle flames, each of the same brightness. Rank the light that enters your eye from brightest to dimmest for the following situations:
a. Three candles seen from a distance of 3 m
b. Two candles seen from a distance of 2 m
c. One candle seen from a distance of 1 m

Mathis E.
Numerade Educator

Problem 47

Pretend you fall into a hole bored completely through the Earth. Discounting friction and rotational effects, rank from greatest to least positions A, B, C, and D for your
a. speed.
b. acceleration toward Earth’s center.

Salamat A.
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Problem 48

Rank the average gravitational forces from greatest to least between the
a. Sun and Mars.
b. Sun and the Moon.
c. Sun and Earth.

Mathis E.
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Problem 49

Rank the microtidal forces on your own body, from greatest to least, produced by the
a. Moon.
b. Earth.
c. Sun.

Salamat A.
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Problem 50

Comment on whether or not the following label on a consumer product should be cause for concern:
CAUTION: The mass of this product pulls on every other mass in the universe, with an attracting force that is proportional to the product of the masses and inversely proportional to the square of the distance between their centers.

Mathis E.
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Problem 51

Gravitational force acts on all bodies in proportion to their masses. Why, then, doesn’t a heavy body fall faster than a light body?

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Problem 52

What would be the path of the Moon if somehow all gravitational forces on it vanished to zero?

Mathis E.
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Problem 53

Is the force of gravity stronger on a piece of iron than on a piece of wood of the same mass? Defend your answer.

Salamat A.
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Problem 54

Is the force of gravity stronger on a crumpled piece of paper than on an identical piece of paper that has not been crumpled? Defend your answer.

Mathis E.
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Problem 55

What is the relationship between force and distance in an inverse-square law?

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Problem 56

An apple falls because of the gravitational attraction to Earth. How does the gravitational attraction of Earth to the apple compare? (Does force change when you interchange $m_{1}$ and $m_{2}$ in the equation for gravity $-m_{2} m_{1}$ instead of $m_{1} m_{2} ? )$

Mathis E.
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Problem 57

Larry weighs 300 N at the surface of Earth. What is the weight of Earth in the gravitational field of Larry?

Salamat A.
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Problem 58

Is the acceleration due to gravity more or less atop Mt. Everest than at sea level? Defend your answer.

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Problem 59

An astronaut lands on a planet that has the same mass as Earth but twice the diameter. How does the astronaut’s weight differ from that on Earth?

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Problem 60

An astronaut lands on a planet that has twice the mass as Earth and twice the diameter. How does the astronaut’s weight differ from that on Earth?

Mathis E.
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Problem 61

If Earth somehow expanded to a larger radius, with no change in mass, how would your weight be affected? How would it be affected if Earth instead shrunk? (Hint: Let the equation for gravitational force guide your thinking.)

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Problem 62

Why does a person in free fall experience weightlessness, while a person falling at terminal velocity does not?

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Problem 63

Why do the passengers in high-altitude jet planes feel the sensation of weight while passengers in an orbiting space vehicle, such as a space shuttle, do not?

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Problem 64

Is gravitational force acting on a person who falls off a cliff? On an astronaut inside an orbiting space shuttle?

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Problem 65

If you were in a car that drove off the edge of a cliff, why would you be momentarily weightless? Would gravity still be acting on you?

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Problem 66

What two forces act on you while you are in a moving elevator? When are these forces of equal magnitude and when are they not?

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Problem 67

If you were in a freely falling elevator and you dropped a pencil, it would hover in front of you. Is there a force of gravity acting on the pencil? Defend your answer.

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Problem 68

Why does a bungee jumper feel weightless during the jump?

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Problem 69

Your friend says that the primary reason astronauts in orbit feel weightless is that they are beyond the main pull of Earth’s gravity. Why do you agree or disagree?

Salamat A.
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Problem 70

An astronaut in the International Space Station cannot stand on a weighing scale. But an astronaut inside a rotating space station (not yet built) can stand on a weighing scale. Explain.

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Problem 71

The force due to gravity on you is mg. Under what condition is mg also your weight?

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Problem 72

Stand on a bathroom scale on a level floor, and the reading on the scale shows the gravitational force on you, mg. If the floor is slanted at an angle, the scale reading will be less than mg. Discuss why this is so, and why it is a good idea to measure your weight when the scale is on a horizontal surface.

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Problem 73

Suppose you jounce up and down while weighing yourself on your bathroom scale. The weight reading likewise “jounces up and down.” Does this mean that the force of gravity, mg, varies when you jounce?

Salamat A.
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Problem 74

If somebody tugged hard on your shirt sleeve, it would likely tear. But if all parts of your shirt were tugged equally, no tearing would occur. How does this relate to tidal forces?

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Problem 75

Most people today know that the ocean tides are caused principally by the gravitational influence of the Moon, and most people therefore think that the gravitational pull of the Moon on Earth is greater than the gravitational pull of the Sun on Earth. What do you think?

Salamat A.
Numerade Educator

Problem 76

Is there a torque about the Moon’s center of mass when the Moon’s long axis is aligned with Earth’s gravitational field? Explain how this compares with a magnetic compass.

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Problem 77

Would ocean tides exist if the gravitational pull of the Moon (and the Sun) were somehow equal on all parts of the world? Explain.

Salamat A.
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Problem 78

Why aren’t high ocean tides exactly 12 hours apart?

Mathis E.
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Problem 79

With respect to spring and neap ocean tides, when are the tides lowest? That is, which tide is best for digging clams?

Salamat A.
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Problem 80

Whenever the ocean tide is unusually high, will the following low tide be unusually low? Defend your answer in terms of “conservation of water.” (If you slosh water in a tub so that it is extra deep at one end, will the other end be extra shallow?)

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Problem 81

The Mediterranean Sea has very little sediment churned up and suspended in its waters, mainly because of the absence of any substantial ocean tides. Why do you suppose the Mediterranean Sea has practically no tides? Similarly, are there tides in the Black Sea? In the Great Salt Lake? Your county reservoir? A glass of water? Explain.

Salamat A.
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Problem 82

The human body is composed mostly of water. Why does the Moon overhead cause appreciably less tidal effect in the fluid compartment of your body than a 1-kg melon held over your head does?

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Problem 83

The value of $g$ at Earth's surface is about 10 $\mathrm{m} / \mathrm{s}^{2}$ . What is the value of $g$ at a distance of twice Earth's radius?

Salamat A.
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Problem 84

If Earth were of uniform density (same mass/volume throughout), what would the value of g be inside Earth at half its radius?

Mathis E.
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Problem 85

If Earth were of uniform density, would your weight increase or decrease at the bottom of a deep mine shaft? Defend your answer.

Salamat A.
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Problem 86

It so happens that an actual increase in weight is found even in the deepest mine shafts. What does this tell us about how Earth’s density changes with depth?

Mathis E.
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Problem 87

Make up two multiple-choice questions—one to evaluate a classmate’s understanding of the inverse-square law, and another to check the distinction between weight and weightlessness.

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Problem 88

A friend says that above the atmosphere, in space shuttle territory, Earth’s gravitational field is zero. Discuss your friend’s misconception by using the equation for gravitational force in your explanation.

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Problem 89

A friend says that, since Earth’s gravity is so much stronger than the Moon’s gravity, rocks on the Moon
could be dropped to Earth. Discuss the wrongness of this assumption.

Salamat A.
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Problem 90

Another friend says that the Moon’s gravity would prevent rocks dropping from the Moon to Earth, but that if the Moon’s gravity somehow no longer pulled on its own rocks, then rocks on the Moon would fall to Earth. Discuss the wrongness of this assumption.

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Problem 91

Somewhere between Earth and the Moon, gravity from these two bodies on a space pod would cancel. Is this location nearer Earth or the Moon?

Salamat A.
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Problem 92

Earth and the Moon are attracted to each other by gravitational force. Does the more massive Earth attract the less massive Moon with a force that is greater, smaller, or the same as the force with which the Moon attracts Earth? (With an elastic band stretched between your thumb and forefinger, which is pulled more strongly by the band: your thumb or your forefinger?)

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Problem 93

If the Moon pulls Earth as strongly as Earth pulls the Moon, why doesn’t Earth rotate around the Moon, or
why don’t both rotate around a point midway between them?

Salamat A.
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Problem 94

The intensity of light from a central source varies inversely as the square of the distance. If you lived on a planet only half as far from the Sun as our Earth, how would the Sun’s light intensity compare with that on Earth? How about a planet 10 times farther away than Earth?

Mathis E.
Numerade Educator

Problem 95

A small light source located 1 $\mathrm{m}$ in front of a $1-\mathrm{m}^{2}$ opening illuminates a wall behind. If the wall is 1 $\mathrm{m}$ behind the opening $(2 \mathrm{m} \text { from the light source), the }$ illuminated area covers 4 $\mathrm{m}^{2} .$ How many square meters will be illuminated if the wall is 3 $\mathrm{m}$ from the light source? 5 $\mathrm{m}^{2} 10 \mathrm{m}$ ?

Salamat A.
Numerade Educator

Problem 96

The planet Jupiter is more than 300 times as massive as Earth, so it might seem that a body on the surface of Jupiter would weigh 300 times as much as on Earth. But it so happens that a body would weigh scarcely 3 times as much on the surface of Jupiter as on the surface of Earth. Discuss why this is so, using the terms in the equation for gravitational force to guide your thinking.

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Problem 97

Since your weight when standing on Earth equals the gravitational attraction between you and Earth, discuss whether or not your weight would be greater if Earth gained mass. If the Sun gained mass.

Salamat A.
Numerade Educator

Problem 98

Discuss and explain why this reasoning is wrong: “The Sun attracts all bodies on Earth. At midnight, when the Sun is directly below, it pulls on you in the same direction as Earth pulls on you; at noon, when the Sun is directly overhead, it pulls on you in a direction opposite to Earth’s pull on you. Therefore, you should be somewhat heavier at midnight and somewhat lighter at noon.”

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Problem 99

When will the gravitational force between you and the Sun be greater: today at noon or tomorrow at midnight? Discuss and defend your answer.

Salamat A.
Numerade Educator

Problem 100

If the mass of Earth increased, your weight would correspondingly increase. But, if the mass of the Sun
increased, your weight would not be affected at all. Discuss why this is so.

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Problem 101

Does the fact that one side of the Moon always faces Earth mean that the Moon rotates about its axis (like a top) or that it doesn’t rotate about its axis? Discuss and defend your answer.

Salamat A.
Numerade Educator

Problem 102

What would be the effect on Earth’s tides if the diameter of Earth were very much larger than it is? If Earth were at its present size but the Moon were very much larger and had the same mass?

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Problem 103

Which would produce the largest microtides in your body: Earth, the Moon, or the Sun? Discuss and defend your answer.

Salamat A.
Numerade Educator

Problem 104

Discuss why tides occur in Earth’s crust and in Earth’s atmosphere.

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Problem 105

Discuss which requires more fuel: a rocket going from Earth to the Moon or a rocket returning from the Moon to Earth.

Salamat A.
Numerade Educator

Problem 106

If you could somehow tunnel inside a uniform-density star, would your weight increase or decrease? If, instead, you somehow stood on the surface of a shrinking star, would your weight increase or decrease? Discuss why your answers differ.

Mathis E.
Numerade Educator

Problem 107

If our Sun shrank in size to become a black hole, discuss and show from the gravitational force equation that Earth’s orbit would not be affected.

Salamat A.
Numerade Educator

Problem 108

If Earth were hollow but still had the same mass and radius, would your weight in your present location be greater than, less than, or the same as it is now? Discuss and explain.

Mathis E.
Numerade Educator

Problem 109

A new discussion partner says that the International Space Station is beyond the pull of Earth’s gravity, as evidenced by inhabitants in a weightless condition. Correct this misconception.

Salamat A.
Numerade Educator

Problem 110

Some people dismiss the validity of scientific theories by saying that they are “only theories.” The law of universal gravitation is a theory. Does this mean that scientists still doubt its validity? Discuss and explain.

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