Chapter 13Chapter 13

Gravitational InteractionsGravitational Interactions

13.1 Gravitational Fields13.1 Gravitational Fields

A magnetic field is a force field, because A magnetic field is a force field, because magnetic materials in it experience a forcemagnetic materials in it experience a forceThe gravitational field around Earth is The gravitational field around Earth is similarsimilarA mass in the field region experiences a A mass in the field region experiences a gravitational forcegravitational forceThe field is strongest at the surface of The field is strongest at the surface of Earth, and declines as the distance from Earth, and declines as the distance from Earth’s center increasesEarth’s center increases

g is the acceleration due to gravityg is the acceleration due to gravity

G is the universal gravitational constantG is the universal gravitational constant

gg is the gravitational field vectoris the gravitational field vector

They are all different quantities that are They are all different quantities that are relatedrelated

GG and g have the same magnitude and and g have the same magnitude and their units are equivalent: N/kg = m/stheir units are equivalent: N/kg = m/s22

gg = GM/R = GM/R22

gg = 9.8 N/kg is valid only near the planet’s = 9.8 N/kg is valid only near the planet’s surfacesurface

as R increases, as R increases, gg decreases decreases

The strength of Earth’s gravitational field is the The strength of Earth’s gravitational field is the force per unit mass exerted by Earth on any force per unit mass exerted by Earth on any objectobject

gg = = FF

mm

gg = 9.8 N/kg = 9.8 m/s = 9.8 N/kg = 9.8 m/s22

gg weakens with increasing distance from Earth weakens with increasing distance from Earth

QuestionsQuestions

If Earth had the same size but twice the If Earth had the same size but twice the mass, what would be the acceleration of mass, what would be the acceleration of freely falling objects at its surface?freely falling objects at its surface? Twice g, or nearly 20 m/sTwice g, or nearly 20 m/s22

If Earth has the same mass but half the If Earth has the same mass but half the size, what would be the acceleration of size, what would be the acceleration of freely falling objects at its surface?freely falling objects at its surface? Four times g, or nearly 40 m/sFour times g, or nearly 40 m/s22

If Earth had twice the mass, and half the If Earth had twice the mass, and half the size, what would be the acceleration of size, what would be the acceleration of freely falling objects at its surface?freely falling objects at its surface? Eight times g, or nearly 80 m/sEight times g, or nearly 80 m/s22

What evidence would you look for to tell What evidence would you look for to tell whether or not you were in a gravitational whether or not you were in a gravitational field?field? The presence of a gravitational forceThe presence of a gravitational force

Compared to its strength at Earth’s Compared to its strength at Earth’s surface, what is the strength of the surface, what is the strength of the gravitational field at a distance of two gravitational field at a distance of two Earth radii from the center of Earth?Earth radii from the center of Earth? ¼ as much¼ as much

13.2 Gravitational Field Inside a 13.2 Gravitational Field Inside a PlanetPlanet

Gravitational field on Earth exists inside Gravitational field on Earth exists inside Earth as well as outsideEarth as well as outside

Imagine a hole drilled completely through Imagine a hole drilled completely through EarthEarth

FALLING THROUGH THE FALLING THROUGH THE CENTER OF THE EARTHCENTER OF THE EARTH

Start at the North PoleStart at the North Pole

Fall & gain speed down Fall & gain speed down to the centerto the center

Lose speed all the way to Lose speed all the way to the South Polethe South Pole

Gain speed toward the Gain speed toward the centercenter

Lose speed moving away Lose speed moving away from the centerfrom the center

a=g

a=g/2

a=0

a=g/2

a=g

As you fell faster and faster into the Earth, As you fell faster and faster into the Earth, your acceleration diminishes because the your acceleration diminishes because the pull of the mass above you partly cancels pull of the mass above you partly cancels the pull belowthe pull below

At the Earth’s center the pulls cancel to At the Earth’s center the pulls cancel to zero and your acceleration is zerozero and your acceleration is zero

Momentum carries you against a growing Momentum carries you against a growing acceleration past the center to the opposite acceleration past the center to the opposite side where it is again, gside where it is again, g

STUCK IN A CAVITYSTUCK IN A CAVITYIn a cavity at the center of the Earth, your In a cavity at the center of the Earth, your weight would be zero, because you would weight would be zero, because you would be pulled equally by gravity in all directionsbe pulled equally by gravity in all directions

You are pulled in every direction equallyYou are pulled in every direction equally

The gravitational field at Earth’s center is The gravitational field at Earth’s center is zerozero

QuestionsQuestions

If you dropped a rock into a tunnel through If you dropped a rock into a tunnel through Earth, what would happen?Earth, what would happen? It would gain speed until it reached Earth’s It would gain speed until it reached Earth’s

center, and then lose speed the rest of the center, and then lose speed the rest of the way. Its speed at the far end of the tunnel way. Its speed at the far end of the tunnel would be the same as its initial speed. It would be the same as its initial speed. It would then fall back and repeat the motion in would then fall back and repeat the motion in cyclic fashion.cyclic fashion.

As the velocity of the rock increases when As the velocity of the rock increases when falling into the Earth tunnel, what happens falling into the Earth tunnel, what happens to the acceleration?to the acceleration? It decreases as the gravitational field It decreases as the gravitational field

decreases, and is zero at Earth’s center. The decreases, and is zero at Earth’s center. The falling body has its maximum velocity at falling body has its maximum velocity at Earth’s center, where both the field and Earth’s center, where both the field and acceleration are zeroacceleration are zero

13.3 Weight and Weightlessness13.3 Weight and Weightlessness

Weight: support forceWeight: support force We are as heavy as we feelWe are as heavy as we feel

Weightlessness: the absence of a support Weightlessness: the absence of a support forceforce The queasy feeling you experience when in a The queasy feeling you experience when in a

car that speeds over the top of a hillcar that speeds over the top of a hill

FOR EXAMPLE: A SCALEFOR EXAMPLE: A SCALEStand on a bathroom scale on the floorStand on a bathroom scale on the floor

The gravitational force between you and The gravitational force between you and Earth pulls you against the supporting floor Earth pulls you against the supporting floor and scaleand scale

By Newton’s 3By Newton’s 3rdrd law, the floor and scale in law, the floor and scale in turn push upward on youturn push upward on you

NOW…WEIGHT IN AN NOW…WEIGHT IN AN ELEVATORELEVATOR

UP, UP, AND AWAYUP, UP, AND AWAYIf the elevator If the elevator accelerates accelerates upward, the upward, the bathroom scale bathroom scale and floor would and floor would push harder push harder against your feetagainst your feetThe scale would The scale would show an increase show an increase in your weightin your weight

GROUND FLOOR PLEASEGROUND FLOOR PLEASEIf the elevator If the elevator accelerated accelerated downward, the downward, the scale would show scale would show a decrease in your a decrease in your weightweightThe support force The support force of the floor would of the floor would now be lessnow be less

ELEVATOR CABLE BROKEELEVATOR CABLE BROKEElevator falls freely, Elevator falls freely, the scale would read the scale would read zerozeroAccording to the According to the scale, you would be scale, you would be weightless and feel weightless and feel weightlessweightlessYour insides would no Your insides would no longer be supported longer be supported by your legs and hipsby your legs and hips

QuestionQuestion

Why would you feel weightless in an Why would you feel weightless in an elevator with a broken cable?elevator with a broken cable? There would be no support force-the floor There would be no support force-the floor

would fall as fast as you would fall as fast as you

13.4 Ocean Tides13.4 Ocean Tides

Think of someone pulling on your coatThink of someone pulling on your coatIf some one pulled only on the sleeve, the If some one pulled only on the sleeve, the coat could tearcoat could tearBut if every part of your coat were pulled But if every part of your coat were pulled equally, it and you would accelerate, but equally, it and you would accelerate, but the coat wouldn’t tearthe coat wouldn’t tearIt tears when one part is pulled harder It tears when one part is pulled harder than another because of a difference in than another because of a difference in forces acting on the coatforces acting on the coat

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

Ocean tides are caused by differences in Ocean tides are caused by differences in the gravitational pull of the moon on the gravitational pull of the moon on opposite sides of Earthopposite sides of Earth

The moon’s attraction is stronger on The moon’s attraction is stronger on Earth’s oceans closer to the moon, and Earth’s oceans closer to the moon, and weaker on the oceans farther from the weaker on the oceans farther from the moon moon gravitational forces weakened with increased gravitational forces weakened with increased

distancedistance

Why are there two tides a day?Why are there two tides a day?There would only be one tide per day, if the Earth were There would only be one tide per day, if the Earth were “nailed down” in one place and held stationary, except “nailed down” in one place and held stationary, except for a daily rotationfor a daily rotation

But the Earth and moon are in orbit around each otherBut the Earth and moon are in orbit around each other

The ocean nearest the moon is pulled upward toward The ocean nearest the moon is pulled upward toward the moon, while the main body of Earth is pulled the moon, while the main body of Earth is pulled toward the moontoward the moon

away from the ocean on the far sideaway from the ocean on the far side

Earth is closer to the moon than the far-side ocean is, Earth is closer to the moon than the far-side ocean is, so Earth’s waters get slightly elongated-at both ends so Earth’s waters get slightly elongated-at both ends creating ocean bulgescreating ocean bulges

Earth makes one complete turn per day beneath Earth makes one complete turn per day beneath these ocean bulges creating two sets of ocean these ocean bulges creating two sets of ocean tides per daytides per day

The part of Earth that passes beneath one of the The part of Earth that passes beneath one of the bulges has a high tidebulges has a high tide

The tide cycle is every 24 hours and 50 minutesThe tide cycle is every 24 hours and 50 minutes

The sun also contributes to ocean tides, but not The sun also contributes to ocean tides, but not as much as the moon.as much as the moon.

The tilt of Earth’s axis is another factorThe tilt of Earth’s axis is another factor

QuestionsQuestions

Which pulls harder on the oceans of Earth, Which pulls harder on the oceans of Earth, the sun or the moon?the sun or the moon? The sunThe sun

Which is most effective in raising tides?Which is most effective in raising tides? The moonThe moon

The difference in pulls decreases as the The difference in pulls decreases as the cube of the distance between the centers cube of the distance between the centers of the bodiesof the bodies Twice as far away produces 1/8 the tideTwice as far away produces 1/8 the tide 3 times as far, only 1/27 the tide3 times as far, only 1/27 the tide

Only relatively close distances result in Only relatively close distances result in appreciable tides, and so the nearby moon appreciable tides, and so the nearby moon “out-tides” the enormously more massive “out-tides” the enormously more massive but far away sunbut far away sun

When the sun, Earth, and the moon are When the sun, Earth, and the moon are lined up, the tides due to the sun and the lined up, the tides due to the sun and the moon coincidemoon coincide

There are higher high tides and lower low tidesThere are higher high tides and lower low tides

These are called These are called spring tidesspring tides They have nothing to do with springThey have nothing to do with spring

If the alignment is perfect, there is an If the alignment is perfect, there is an eclipseeclipse

ECLIPSEECLIPSE

A A lunar eclipselunar eclipse is produced when Earth is is produced when Earth is directly between the sun and moondirectly between the sun and moon

A A solar eclipsesolar eclipse is produced when the is produced when the moon is directly between the sun and moon is directly between the sun and EarthEarth

PHASES OF THE MOONPHASES OF THE MOONWhen the Earth is between the sun and the moon –When the Earth is between the sun and the moon –full moonfull moon

When the moon is between the sun and Earth – new When the moon is between the sun and Earth – new moonmoon

Spring tides occur during a new moon and full moonSpring tides occur during a new moon and full moon

During half-moons, the tides due to the sun and the During half-moons, the tides due to the sun and the moon partly cancel each othermoon partly cancel each other High tides are lower than averageHigh tides are lower than average Low tides are higher than averageLow tides are higher than average

These are called neap tidesThese are called neap tides

QuestionQuestion

At the time of extra high tides, will extra At the time of extra high tides, will extra low tides follow in the same day?low tides follow in the same day? Yes, by the “conservation of water”. There is Yes, by the “conservation of water”. There is

only so much water on Earth-extra high tides only so much water on Earth-extra high tides in one part of the world means extra low tides in one part of the world means extra low tides in anotherin another

13.5 Tides in Earth and the 13.5 Tides in Earth and the AtmosphereAtmosphere

There are tides within Earth, which is There are tides within Earth, which is mostly molten lavamostly molten lava

There is a greater probability of There is a greater probability of earthquakes and volcanoes when there is earthquakes and volcanoes when there is an eclipse of the sun or the moonan eclipse of the sun or the moon

This is when Earth experiences spring This is when Earth experiences spring tides-greater stresses on Earth’s crusttides-greater stresses on Earth’s crust

If the moon were closer…If the moon were closer…

Ocean tides would be higher and so the Ocean tides would be higher and so the tidal forces on the moon’s crust would be tidal forces on the moon’s crust would be greatergreater

If the moon were too close, Earth’s tidal If the moon were too close, Earth’s tidal forces would tear the moon into a billion forces would tear the moon into a billion pieces, forming a ring around Earth similar pieces, forming a ring around Earth similar to those around Saturnto those around Saturn

13.6 BLACK HOLES13.6 BLACK HOLESOccurs with stars that are at least two to Occurs with stars that are at least two to three times more massive than our sunthree times more massive than our sun

Once the flame of thermonuclear fusion Once the flame of thermonuclear fusion is extinguished, gravitational collapse is extinguished, gravitational collapse takes over-and it doesn’t stoptakes over-and it doesn’t stop

Gravitation is so enormous that nothing Gravitation is so enormous that nothing can get back out-even lightcan get back out-even light

A collapsed star represents condensed mass A collapsed star represents condensed mass and therefore condensed gravity.and therefore condensed gravity.

The mass of a black hole is no more than the The mass of a black hole is no more than the mass of the star that collapsed to form itmass of the star that collapsed to form it

Hence the gravitational field of the star and the Hence the gravitational field of the star and the black hole are the same at distances greater black hole are the same at distances greater than the original radius of the starthan the original radius of the star

It is only at closer distances that the enormous It is only at closer distances that the enormous field occursfield occurs

QuestionQuestion

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

equation changeequation change

SummarySummaryEarth can be thought of as being surrounded by Earth can be thought of as being surrounded by a gravitational field that interacts with objects and a gravitational field that interacts with objects and causes them to experience gravitational forces.causes them to experience gravitational forces.

The gravitational field, g, is equal to the The gravitational field, g, is equal to the acceleration of a freely falling object.acceleration of a freely falling object.

Objects in orbit around Earth have a gravitational Objects in orbit around Earth have a gravitational force acting on them even though they may force acting on them even though they may appear to be weightless.appear to be weightless.

Ocean tides (and even tides within the Ocean tides (and even tides within the solid Earth and within the atmosphere) are solid Earth and within the atmosphere) are caused by differences in the gravitational caused by differences in the gravitational pull of the moon (and sun) on opposite pull of the moon (and sun) on opposite sides of Earth.sides of Earth.

When a star runs out of fuel for fusion, it When a star runs out of fuel for fusion, it collapses under gravitational forces. collapses under gravitational forces. Sufficiently massive stars collapse to form Sufficiently massive stars collapse to form black holes.black holes.