chapter 7 - gravitation key terms kepler’s laws gravitational force law of universal gravitation...

Chapter 7 - GravitationChapter 7 - Gravitation

Key Terms

Kepler’s Laws

Gravitational force

Law of Universal gravitation

Gravitational field

Inertial mass

Gravitational mass

Early AstronomyEarly Astronomy Not only did early humans navigate by means of the sky, Not only did early humans navigate by means of the sky,

but the motions of objects in the sky predicted the but the motions of objects in the sky predicted the changing of the seasons, etc.changing of the seasons, etc.

Stonehenge was built outside of Salisbury, England in Stonehenge was built outside of Salisbury, England in 1500 1500 BCBC. It was used to track the movement of the sun and mark . It was used to track the movement of the sun and mark the solstice. Only seven stones still stand today. This photo the solstice. Only seven stones still stand today. This photo shows as it would have stood when it was built.shows as it would have stood when it was built.

Early AstronomyEarly Astronomy

There were many early attempts There were many early attempts both to describe and explain the both to describe and explain the motions of stars and planets in the motions of stars and planets in the sky. sky.

All were unsatisfactory, for one All were unsatisfactory, for one reason or another.reason or another.

The Earth-Centered The Earth-Centered UniverseUniverse

A A geocentricgeocentric (Earth-centered) solar (Earth-centered) solar system is often credited to system is often credited to PtolemyPtolemy, an Alexandrian Greek, , an Alexandrian Greek, although the idea is very old.although the idea is very old.

Image from: http://abyss.uoregon.edu/~js/ast123/lectures/lec02.html

Ptolemy’s Solar SystemPtolemy’s Solar System

Ptolemy’s solar system could be Ptolemy’s solar system could be made to fit the observational data made to fit the observational data pretty well, but only by becoming pretty well, but only by becoming veryvery complicated.complicated.


Copernicus’ Solar SystemCopernicus’ Solar System

The Polish cleric The Polish cleric CopernicusCopernicus proposed a proposed a heliocentricheliocentric (Sun (Sun centered) solar system in the centered) solar system in the 1500’s.1500’s.


Objections to CopernicusObjections to Copernicus How could Earth be moving at How could Earth be moving at

enormous speeds when we don’t feel enormous speeds when we don’t feel it?it? (Copernicus didn’t know about (Copernicus didn’t know about inertiainertia.).)

Why can’t we detect Earth’s motion Why can’t we detect Earth’s motion against the background stars (stellar against the background stars (stellar parallax)?parallax)?

Copernicus’ model did Copernicus’ model did notnot fit the fit the observational data very well.observational data very well.

Galileo & CopernicusGalileo & Copernicus GalileoGalileo became became

convinced that convinced that Copernicus was correct Copernicus was correct by observationsby observations of the of the Sun, Venus, and the Sun, Venus, and the moons of Jupiter using moons of Jupiter using the newly-invented the newly-invented telescope.telescope.

Perhaps Galileo was Perhaps Galileo was motivated to motivated to understand understand inertiainertia by by his desire to understand his desire to understand and defend Copernicus’ and defend Copernicus’ ideas.ideas.

Tycho and KeplerTycho and Kepler

In the late 1500’s, a In the late 1500’s, a Danish nobleman Danish nobleman named named Tycho BraheTycho Brahe set out to make the set out to make the most accurate most accurate measurementsmeasurements of of planetary motions to planetary motions to date, in order to date, in order to validate his own ideas validate his own ideas of planetary motion.of planetary motion.

Tycho and KeplerTycho and Kepler

Tycho’s data was Tycho’s data was successfully successfully interpreted by the interpreted by the German German mathematician mathematician and scientist and scientist Johannes KeplerJohannes Kepler in the early in the early 1600’s.1600’s.

Kepler’s LawsKepler’s Laws

KeplerKepler determined that the orbits determined that the orbits of the planets were not perfect of the planets were not perfect circles, but circles, but ellipsesellipses, with the Sun , with the Sun at one focus.at one focus.

Sun

Planet

Kepler’s Second LawKepler’s Second Law

KeplerKepler determined that a planet determined that a planet moves moves faster when nearfaster when near the Sun, the Sun, and and slower when farslower when far from the Sun. from the Sun.

Sun

Planet

Faster Slower

Zoom Putt-Putt

Why?Why?

Kepler’s Laws provided a complete Kepler’s Laws provided a complete kinematical kinematical description of description of planetary motion (including the planetary motion (including the motion of planetary satellites, like motion of planetary satellites, like

the Moon) - but the Moon) - but whywhy did the did the planets move like that?planets move like that?

The Apple & the MoonThe Apple & the Moon

Isaac Newton realized that the Isaac Newton realized that the motion of a falling apple and the motion of a falling apple and the motion of the Moon were both motion of the Moon were both actually the actually the same motionsame motion, caused , caused by the by the same forcesame force - the - the gravitational forcegravitational force..

Universal GravitationUniversal Gravitation

Newton’s idea was that gravity Newton’s idea was that gravity was a was a universaluniversal force acting force acting between between any two objectsany two objects..

At the Earth’s SurfaceAt the Earth’s Surface

Newton knew that the Newton knew that the gravitational forcegravitational force on the apple on the apple equals the apple’s equals the apple’s weight, mg,weight, mg, where g = 9.8 m/swhere g = 9.8 m/s22. .

W = mg

Weight of the MoonWeight of the Moon

Newton reasoned that the Newton reasoned that the centripetal force on the moon was centripetal force on the moon was also supplied by the Earth’s also supplied by the Earth’s gravitational force.gravitational force.

Fc = mg?


Newton’s calculations showed that Newton’s calculations showed that the centripetal force needed for the centripetal force needed for the Moon’s motion was about the Moon’s motion was about 1/36001/3600thth of Mg, however, where M of Mg, however, where M is the mass of the Moon.is the mass of the Moon.


Newton also knew that the Moon Newton also knew that the Moon was about was about 60 times farther60 times farther from from the center of the Earth than the the center of the Earth than the apple.apple.

And 60And 6022 = 3600 = 3600


From this, Newton reasoned that From this, Newton reasoned that the strength of the gravitational the strength of the gravitational force is force is not constantnot constant, in fact, the , in fact, the magnitude of the force is magnitude of the force is inversely proportional to the inversely proportional to the square of the distancesquare of the distance between between the objects.the objects.


Newton concluded that the Newton concluded that the gravitational force is:gravitational force is: Directly proportionalDirectly proportional to the to the

massesmasses of of bothboth objects. objects. Inversely proportionalInversely proportional to the to the

distancedistance between the objects. between the objects.

Law of Universal Law of Universal GravitationGravitation

In symbols, In symbols, Newton’s Law of Newton’s Law of Universal GravitationUniversal Gravitation is: is:

FFgravgrav = G = G

m is the mass of each objectm is the mass of each object r is the distance between the two massesr is the distance between the two masses Where G is a constant of proportionality.Where G is a constant of proportionality. G = 6.67 x 10G = 6.67 x 10-11-11 N m N m22/kg/kg22

m1m2

r 2

Inverse Square LawInverse Square Law

Newton’s Law of Universal Newton’s Law of Universal Gravitation is often called an Gravitation is often called an inverse square lawinverse square law, since the , since the force is inversely proportional to force is inversely proportional to the square of the distance.the square of the distance.

An Inverse-An Inverse-Square ForceSquare Force

Since gravitational force is inversely proportional to the square of the distance between the two objects, more separation distance will result in weaker gravitational forces.

Distance = 1, force =1

Distance = 2, force = ¼

Distance = 3, force = 1/9

Experimental EvidenceExperimental Evidence

The Law of Universal Gravitation The Law of Universal Gravitation allowed allowed extremely accurate extremely accurate predictionspredictions of planetary orbits. of planetary orbits.

CavendishCavendish measured gravitational measured gravitational forces between human-scale forces between human-scale objectsobjects before 1800. His before 1800. His experiments were later simplified experiments were later simplified and improved by von Jolly.and improved by von Jolly.

Action at a DistanceAction at a Distance

In Newton’s time, there was much In Newton’s time, there was much discussion about HOW gravity discussion about HOW gravity worked - how does the Sun, for worked - how does the Sun, for instance, reach across empty instance, reach across empty space, with no actual contact at space, with no actual contact at all, to exert a force on the Earth?all, to exert a force on the Earth?

This spooky notion was called This spooky notion was called “action at a distance.”“action at a distance.”

The Gravitational FieldThe Gravitational Field

During the 19th century, the During the 19th century, the notion of the notion of the ““fieldfield”” entered entered physics (via Michael Faraday). physics (via Michael Faraday).

Objects with mass create an Objects with mass create an invisible disturbance in the invisible disturbance in the space around themspace around them that is felt that is felt by other massive objects - this is a by other massive objects - this is a gravitational fieldgravitational field..

The Gravitational FieldThe Gravitational Field

So, since the Sun is very massive, So, since the Sun is very massive, it creates an intense gravitational it creates an intense gravitational field around it, and the field around it, and the Earth Earth responds to the fieldresponds to the field. No more . No more “action at a distance.”“action at a distance.”

Gravitational Field Gravitational Field StrengthStrength

To measure the strength of the To measure the strength of the gravitational field at any point, gravitational field at any point, measure the gravitational force, F, measure the gravitational force, F, exerted on any “test mass”, m.exerted on any “test mass”, m.

Gravitational Field Strength, g Gravitational Field Strength, g = F/m= F/m

Gravitational Field Gravitational Field StrengthStrength

Near the surface of the Earth, g = Near the surface of the Earth, g = F/m = 9.8 N/kg = 9.8 m/sF/m = 9.8 N/kg = 9.8 m/s22..

In general, In general, g = GM/rg = GM/r22, where M is , where M is the mass of the object creating the the mass of the object creating the field, r is the distance from the field, r is the distance from the object’s center, and G = 6.67 x10object’s center, and G = 6.67 x10--

1111 Nm Nm22/kg/kg22..

Gravitational ForceGravitational Force

If g is the strength of the If g is the strength of the gravitational field at some point, then gravitational field at some point, then the gravitational force on an object of the gravitational force on an object of mass m at that point is mass m at that point is FFgrav grav = mg.= mg.

If g is the gravitational field strength If g is the gravitational field strength at some point (in N/kg), then the free at some point (in N/kg), then the free fall acceleration at that point is also g fall acceleration at that point is also g (in m/s(in m/s22).).

Gravitational Field Inside a Gravitational Field Inside a PlanetPlanet

If you are located a distance r from If you are located a distance r from the center of a planet:the center of a planet: all of the planet’s mass all of the planet’s mass inside inside a a

sphere of radius r sphere of radius r pulls you pulls you toward toward the centerthe center of the planet. of the planet.

All of the planet’s mass All of the planet’s mass outsideoutside a a sphere of radius r exerts sphere of radius r exerts nono net net gravitational forcegravitational force on you. on you.


TheThe blue-shadedblue-shaded part partof the planet pulls youof the planet pulls youtoward point C.toward point C.

The The grey-shadedgrey-shaded part partof the planet of the planet does does not pull younot pull you at all. at all.


Half way to the center Half way to the center of the of the planet, g hasplanet, g has one-half of its surface one-half of its surface value.value.

At the center At the center of the planet,of the planet, g = 0 g = 0 N/kg.N/kg.

Black HolesBlack Holes

When a very massive star gets old When a very massive star gets old and runs out of fusionable and runs out of fusionable material, gravitational forces may material, gravitational forces may cause it to collapse to a cause it to collapse to a mathematical point - a mathematical point - a singularity.singularity. All normal matter is crushed out of All normal matter is crushed out of existence. This is a existence. This is a black hole.black hole.

Black Hole Gravitational Black Hole Gravitational ForceForce

Black Hole Gravitational Black Hole Gravitational ForceForce

The black hole’s gravity is the The black hole’s gravity is the same same as the original star’sas the original star’s at distances at distances greater than the star’s original greater than the star’s original radius.radius.

Black hole’s Black hole’s don’tdon’t magically “suck magically “suck things in.”things in.”

The black hole’s gravity is intense The black hole’s gravity is intense because you can get really, really because you can get really, really close to it!close to it!

Earth’s TidesEarth’s Tides

There are 2 high tides and 2 low There are 2 high tides and 2 low tides per day.tides per day.

The tides follow the Moon.The tides follow the Moon.

Why Why TwoTwo Tides? Tides?

Tides are caused by the Tides are caused by the stretchingstretching of a of a planet.planet.

Stretching is caused by a Stretching is caused by a difference in difference in forcesforces on the two sides of an object. on the two sides of an object.

Since Since gravitational force depends on gravitational force depends on distancedistance, there is more gravitational , there is more gravitational force on the side of Earth closest to the force on the side of Earth closest to the Moon and less gravitational force on the Moon and less gravitational force on the side of Earth farther from the Moon.side of Earth farther from the Moon.

Why Why TwoTwo Tides? Tides?

Remember that Remember that

Why the Why the MoonMoon??

The Sun’s gravitational pull on The Sun’s gravitational pull on Earth is Earth is much largermuch larger than the than the Moon’s gravitational pull on Earth. Moon’s gravitational pull on Earth. So why do the tides follow the So why do the tides follow the Moon and not the Sun?Moon and not the Sun?


Since the Sun is Since the Sun is much farthermuch farther from from Earth than the Moon, the difference Earth than the Moon, the difference in distance across Earth is much in distance across Earth is much less significant for the Sun than the less significant for the Sun than the Moon, therefore the Moon, therefore the difference in difference in gravitational force on the two sides gravitational force on the two sides of Earth is less for the Sunof Earth is less for the Sun than for than for the Moon (even though the Sun’s the Moon (even though the Sun’s force on Earth is more).force on Earth is more).


The The SunSun does have a does have a small effectsmall effect on Earth’s tides, but the major on Earth’s tides, but the major effect is due to the Moon.effect is due to the Moon.

chapter 7 - gravitation key terms kepler’s laws gravitational force law of universal gravitation...

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