Ch 9: Gravity; Ex 2, 12, 16, 17, 22, 39, 40 (Ch10.17); Pb 3

**Ex 9.2 Gravitational force acts on all bodies in proportion to their masses.
Why, then does not a heavy body fall faster than a light body?**

F= maWhile the gravitational force

is, indeed, proportional to the mass of a falling object, that ispreciselywhat we require to have aconstantacceleration due to Newton's Second Law,F= ma.

**Ex 9.12 The 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
the Earth?**

**(With a rubber band stretched between your thumb and forefinger, which is
pulled more strongly by the band, your thumb or your forefinger?)**

Remember Newton's Third Law, Action and Reaction,F_{12}= -F_{21}?The force the Earth exerts on the Moon is

exactly the sameas the force the Moon exerts on the Earth (but they act in opposite directions).

** **

**Ex 9.16 A small light source located 1 m in front of a 1-m ^{2} opening
illuminates a wall behind. If the wall is 1 m behind the opening (2 m from the
light source), the illuminated area covers 4 m^{2}. How many square
metters will be illuminated if the wall is 3 m from the light source? 5m? 10m?**

At 10 m, the area illuminated will be (10 m)^{2} = 100 m^{2}

** **

**Ex 9.17 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 scarcely weigh three times
more on the surface of Jupiter as it would on the surface of Earth. Can you
think of an explanation for why this is so?**

**( Hint: Let the terms in the equation for gravitational force guide
your thinking.)**

F = G M m / R ^{2}M

_{J}= 300 M_{E}F

_{J}= 3 F_{E}F

_{J}= G M_{J}m / R_{J}^{2}= 3 [ G M_{E}m / R_{E}^{2}] = 3 F_{E}F

_{J}= G (300 M_{E}) m / R_{J}^{2}= 3 [ G M_{E}m / R_{E}^{2}] = 3 F_{E}G (300 M

_{E}) m / R_{J}^{2}= 3 [ G M_{E}m / R_{E}^{2}](300 M

_{E}) / R_{J}^{2}= 3 [ M_{E}/ R_{E}^{2}](300 ) / R

_{J}^{2}= 3 [ 1 / R_{E}^{2}]100 / R

_{J}^{2}= 1 / R_{E}^{2}R

_{J}^{2}/ 100 = R_{E}^{2}R

_{J}^{2}= 100 R_{E}^{2}R

_{J}= 10 R_{E}The radius of Jupiter is about

ten timesas large as the radius of Earth!

** **

**Ex 9.22 If you were in a freely falling elevator and you dropped a pencil,
you would see the pencil hovering. Is the pencil falling? Explain. Think of
a glass elevator on the outside of a building -- so it can be seen easily and
seen into easily.**

It depends upon your "frame of reference". Ifyouare falling along with the elevator,youwill see the pencil hovering -- so it certainly isn't falling asyouobserve it. However, someone standing at rest on the Earthwatchingfrom the outside will see you, the elevator,and the pencilall to be falling.

** **

**Ex 9.39 If the Earth were of uniform composition, would your weight increase
or decrease at the bottom of a deep mine shaft? Why?**

For auniform Earth, the force of gravity woulddecreaseas we go below the surface. Therefore, your weight should belessat the bottom of a deep mine shaft.However, Earth does

nothave uniform density. The center of Earth ismore densethan Earth's crust or outer layers. Therefore, as you go down in a deep mine, your weight actuallyincreases!

**Ex 9.40 It so happens that an actual increase in weight is found
even in the deepest mine shafts. What does this tell us about the density of
the Earth's composition?**

However, . . . , your weight will actually

increase!That means you are getting closer tomoremass. That is, the outer layer of Earth hasless density(less mass per volume) than the center. The center might be made of something like lead while the outer layers are of something like rock and soil.

From Chapter 10

**Ex 10.17 Which planets, those closer to the Sun than the Earth or those
farther from the Sun than the Earth, have a period greater than 1 Earth year?**

Kepler's Third Law of Planetary Motion

T ^{2}/ R^{3}= constantmeans that the period increases as the radius increases. For the Earth, the period is exactly one Earth year! Planets with orbital radii

greater than Earth'swill haveperiodsthat are alsogreater than Earth's. Mars, Jupiter, and Saturn have periodsgreaterthan one Earth year. Planets with orbital radiismaller than Earth'swill haveperiodsthat are alsosmaller than Earth's. Mercury and Venus have periodssmallerthan one Earth year.

Extra Exercise

Ex 9.X If the mass of the Earth somehow increased, with all other factors remaining the same, would your weight also increase?

(Hint:Let the equation for gravitational force guide your thinking.)Increasing the mass of the Earth would increase the force of gravity on me and "the force of gravity on me" is my weight. That is, my weight would

increase.

Calculational Problem

Pb 9.3 The value of g at Earth's surface is about 9.8 m/s^{2}. What is the value of g at a distance from Earth's center that is 4 times Earth's radius?The force of gravity is an inverse-square force. Earth's surface is one Earth radius away from Earth's center. So being 4 times Earth's radius away from the center means the force of gravity -- and the acceleration due to gravity -- will be^{1}/_{16}as great as it is on Earth's surface.g' = ( ^{1}/_{16}) (9.8 m/s^{2})g' = 0.61 m/s

^{2}

Extra Problem

Pb 9. X Saturn is approximately ten times as far from the Sun as the Earth's distance from the Sun. By way of Kepler's Third Law, estimate the approximate time in Earth years for Saturn to take to make one revolution about the Sun.R _{S}= 10 R_{E }T

^{2}/ R^{3}= constT

_{S}^{2}/ R_{S}^{3}= T_{E}^{2}/ R_{E}^{3}T

_{S}^{2}/ (10 R_{E})^{3}= T_{E}^{2}/ R_{E}^{3}T

_{S}^{2}/ (1,000 R_{E}^{3}) = T_{E}^{2}/ R_{E}^{3}T

_{S}^{2}/ (1,000) = T_{E}^{2}T

_{S}^{2}= 1,000 T_{E}^{2}T

_{S}= [SQRT (1,000) ] T_{E}T

_{S}= [31.6] T_{E}The period for Saturn is about 31.6 Earth years.

"SQRT" simply means "square root of"

but it is

fareasier for me to type!

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Typical or possible multiple-choice questions:1. According to Kepler's laws, the paths of planets about the Sun are

A) straight linesB) parabolas

C) ellipses

D) hyperbolas

2. According to Newton, the greater the masses of interacting objects, the

A) greater the force of gravity, by the product of the massesB) less the force of gravity

C) greater the force of gravity, by the square of the masses

D) less the force of gravity, inversely as the square of the masses

3. According to Newton, the greater the distance between masses of interacting objects, the

A) greater the force of gravity, proportional to the distanceB) less the force of gravity, inversely as the distance

C) greater the force of gravity, proportional to the square of the distance

D) less the force of gravity, inversely as the square of the distance

4. What is the force of gravity on a 500-newton woman standing on the earth's surface?

A) 9.8 NB) 50 N

C) 500 N

D) 5,000 N

5. If the mass of the earth somehow increased with no change in radius, your weight would

A) increaseB) decrease

C) stay the same

D)

6. If the radius of Earth somehow decreased with no change in mass, your weight would

A) increaseB) decrease

C) stay the same

D)

7. If Earth's mass decreased to one-half its original mass with no change in radius, then your weight would

A) decrease to one-quarter its original valueB) decrease to one-half its original value

C) remain the same

D) increase to twice its original value

8. The force of gravity acting on the Space Shuttle in orbit is nearly

A) zeroB) equal to the weight of the Space Shuttle at Earth's surface

C) about one-tenth its weight at Earth's surface

D) about one-one hundredth its weight at Earth's surface

9. A woman who normally weighs 400 N stands on top of a very tall ladder so she is one earth radius above the earth's surface. How much would she weigh there?

A) zeroB) 100 N

C) 200 N

D) 400 N

10. The force of gravity acts on all apples on an apple tree. Some apples are twice as far from the ground as others. These twice- as-high apples, for the same mass, have practically

A) one-fourth the weightB) one-half the weight

C) the same weight

D) twice the weight

11. The planet Jupiter is about 300 times as massive as Earth, yet on its surface you would weigh only about 3 times as much. This is because

A) your mass is 100 times less on Jupiter.B) Jupiter is significantly farther from the sun.

C) Jupiter's radius is 10 times Earth's radius.

D) you are 100 times more weightless there.

|Back to Calendar|ToC, Ch 9|Ch 10, Projectile Motion|

Answers to typical or possible multiple-choice questions:1. According to Kepler's laws, the paths of planets about the Sun are

A) straight linesB) parabolas

C) ellipsesD) hyperbolas

2. According to Newton, the greater the masses of interacting objects, the

A) greater the force of gravity, by the product of the massesF = G M m / d^{2}B) less the force of gravity

C) greater the force of gravity, by the square of the masses

D) less the force of gravity, inversely as the square of the masses

3. According to Newton, the greater the distance between masses of interacting objects, the

A) greater the force of gravity, proportional to the distanceB) less the force of gravity, inversely as the distance

C) greater the force of gravity, proportional to the square of the distance

D) less the force of gravity, inversely as the square of the distanceF = G M m / d^{2}

4. What is the force of gravity on a 500-newton woman standing on the earth's surface?

A) 9.8 NB) 50 N

C) 500 ND) 5,000 N

5. If the mass of the earth somehow increased with no change in radius, your weight would

A) increaseF = G M m / d^{2}B) decrease

C) stay the same

D)

6. If the radius of Earth somehow decreased with no change in mass, your weight would

A) increaseF = G M m / d^{2}B) decrease

C) stay the same

D)

7. If Earth's mass decreased to one-half its original mass with no change in radius, then your weight would

A) decrease to one-quarter its original value

B) decrease to one-half its original valueF = G M m / d^{2}C) remain the same

D) increase to twice its original value

8. The force of gravity acting on the Space Shuttle in orbit is nearly

A) zero

B) equal to the weight of the Space Shuttle at Earth's surfaceF = G M m / d^{2}

In a low-Earth orbit, the Space Shuttle's distance from the center of Earth is nearly the same as it was at Earth's surface.C) about one-tenth its weight at Earth's surface

D) about one-one hundredth its weight at Earth's surface

9. A woman who normally weighs 400 N stands on top of a very tall ladder so she is one earth radius above the earth's surface. How much would she weigh there?

A) zero

B) 100 NF = G M m / d^{2}

If d is made twice as large, then d^{2}will be four times as large. Since we are dividing by d^{2}, this means the Force -- her weight -- will be only one-fourth as much.C) 200 N

D) 400 N

10. The force of gravity acts on all apples on an apple tree. Some apples are twice as far from the ground as others. These twice- as-high apples, for the same mass, have practically

A) one-fourth the weightB) one-half the weight

C) the same weightF = G M m / d^{2}

Being twice as far above the ground has hardly any effect on d, the distance from the center of Earth to the apple.D) twice the weight

11. The planet Jupiter is about 300 times as massive as Earth, yet on its surface you would weigh only about 3 times as much. This is because

A) your mass is 100 times less on Jupiter.B) Jupiter is significantly farther from the sun.

C) Jupiter's radius is 10 times Earth's radius.D) you are 100 times more weightless there.

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(C) 2003, Doug Davis; all rights reserved