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?

Weight of the MoonWeight of the Moon 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, where m is the of mg, where m is the mass of the Moon.mass of the Moon.

Weight of the MoonWeight 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

Universal GravitationUniversal Gravitation 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 = = GG 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 G = 6.67 x 10G = 6.67 x 10-11-11 N m N m22/kg/kg22

m1m2

r 2

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 = = GG 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

scale

150 lbs

scale

25.6 lbs

scale

406 lbs

g = 9.81 m/s2 g = 1.67 m/s2 g = 26.6 m/s2

An Inverse-An Inverse-Square ForceSquare ForceSince 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 =1Distance = 2, force = ¼Distance = 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.

Inertial vs. Gravitational Mass

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!

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

A satellite is a projectile shot from a very high elevationand is in free fall about the Earth.

VelocityVectors

AccelerationVectors

ForceVectors

Centripetal Acceleration and Centripetal Acceleration and ForceForce

The time it takes to make one revolution is The time it takes to make one revolution is the period (T), and the displacement is 2the period (T), and the displacement is 2rr

The velocity is v = d = The velocity is v = d = 22rr t t TT

acceleration aacceleration acc = v = v22/r = 4/r = 422r\Tr\T22 Force FForce Fcc = ma = macc

ExampleExample A 13 g rubber stopper is attached to a 0.93 m A 13 g rubber stopper is attached to a 0.93 m

string. The stopper is swung in a horizontal string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the circle, making one revolution in 1.18 s. Find the centripetal force exerted by the string on the centripetal force exerted by the string on the stopper.stopper.

aacc = v = v22 = 4 = 422rr = 4= 422(0.93m )(0.93m )= 26 m/s= 26 m/s22

r Tr T22 (1.18s)(1.18s)22

FFcc= ma= ma = (.013 g) 26 m/s= (.013 g) 26 m/s22

= 0.338 N= 0.338 N

Planet

Force of gravityFg

Centripetal forceFc

Orbi

tal

path

Fg = Fc

Fg = G m1 m2

r2

Fc = m v2

r

G m1 m2 = m1 v2

r2 rCanceling m1 & r on both sides

V2 = G m2

r

Gravitation & Satellite ProblemsGravitation & Satellite Problems(a)What is the velocity of a satellite 5000 (a)What is the velocity of a satellite 5000 kmkm aboveabove the Earth? (b)What is it period of the Earth? (b)What is it period of rotation ?rotation ?

r = re + hme = 6 x 10 24 kgRe = 6.4 x 10 6 m

(a) V2 = G m2 , v = G me 1/2

r rV = (6.67 x 10-11 )(6 x 1024) 1/2

(6.4 x 106 + 5 x 106)V = 5900 m/s

(b) V = 2π r / T, T = 2π r/ v T = 2π (11.4 x 106) / 5900 = 1.2 x 104 sec = 3.4 hr.

( )( )

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?

Why the Why the MoonMoon?? 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).

Why the Why the MoonMoon?? 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.