Newton’s Law of Universal GravitationNewton’s Law of Universal Gravitation

GravitationGravitation

Every object with mass attracts Every object with mass attracts every other object with mass.every other object with mass. Newton realized that the force of Newton realized that the force of

attraction between two massive objects:attraction between two massive objects: Increases as the mass of the objects Increases as the mass of the objects

increases.increases. Decreases as the distance between the Decreases as the distance between the

objects increases.objects increases.

Law of Universal GravitationLaw of Universal Gravitation

FFGG = G = G

G = Gravitational ConstantG = Gravitational Constant G = 6.67x10G = 6.67x10-11-11 N*m N*m22/kg/kg22

MM11 and M and M22 = the mass of two bodies = the mass of two bodies r = the distance between themr = the distance between them

MM11MM22

rr22

Law of Universal GravitationLaw of Universal Gravitation

The LoUG is an inverse-square law:The LoUG is an inverse-square law: If the distance doubles, the force drops to If the distance doubles, the force drops to

1/4.1/4. If the distance triples, the force drops to If the distance triples, the force drops to

1/9.1/9. Distance x 10 = FDistance x 10 = FGG / 100. / 100.

Law of Universal GravitationLaw of Universal Gravitation

M1 and M2 = 10 kg

01E-112E-113E-114E-115E-116E-117E-118E-11

0 20 40 60 80 100

Distance (m)

Gra

vita

tio

nal

Fo

rce

(N)

Law of Universal GravitationLaw of Universal Gravitation

Jimmy is attracted to Betty. Jimmy’s mass is 90.0 kg Jimmy is attracted to Betty. Jimmy’s mass is 90.0 kg and Betty’s mass is 57.0 kg. If Jim is standing 10.0 and Betty’s mass is 57.0 kg. If Jim is standing 10.0 meters away from Betty, what is the gravitational meters away from Betty, what is the gravitational force between them?force between them? FFGG = GM = GM11MM22 / r / r22

FFGG = (6.67x10 = (6.67x10-11 -11 NmNm22/kg/kg22)(90.0 kg)(57.0 kg) / (10.0 )(90.0 kg)(57.0 kg) / (10.0 m)m)22

FFGG = (3.42x10 = (3.42x10-7-7 Nm Nm22) / (100. m) / (100. m22)) FFGG = 3.42x10 = 3.42x10-9-9 N = 3.42 nN N = 3.42 nN

In standard terms, that’s 7.6 ten-billionths of a pound of In standard terms, that’s 7.6 ten-billionths of a pound of force.force.

Law of Universal GravitationLaw of Universal Gravitation The Moon is attracted to the Earth. The Moon is attracted to the Earth.

The mass of the Earth is 6.0x10The mass of the Earth is 6.0x102424 kg kg and the mass of the Moon is 7.4x10and the mass of the Moon is 7.4x102222 kg. If the Earth and Moon are kg. If the Earth and Moon are 345,000 km apart, what is the 345,000 km apart, what is the gravitational force between them?gravitational force between them? FFGG = GM = GM11MM22 / r / r22

FFGG = (6.67x10 = (6.67x10-11-11 Nm Nm22/kg/kg22))

FFGG = 2.49x10 = 2.49x102020 N N

(6.0x10(6.0x102424 kg)(7.4x10 kg)(7.4x102222 kg)kg)

(3.45x10(3.45x1088 m) m)22

Gravitational FieldGravitational Field

Gravitational field – an area of Gravitational field – an area of influence surrounding a massive body.influence surrounding a massive body. Field strength = acceleration due to Field strength = acceleration due to

gravity gravity (g(g)).. gg = GM / r = GM / r22

Notice that field strength does not depend Notice that field strength does not depend on the mass of a second object.on the mass of a second object.

GMGM11MM22/r/r22 = M = M22gg = F = FGG = F = Fww

So gravity causes mass to have weight.So gravity causes mass to have weight.

Gravitational Field StrengthGravitational Field Strength The mass of the Earth is 6.0x10The mass of the Earth is 6.0x102424 kg and its radius is kg and its radius is

6378 km. What is the gravitational field strength at 6378 km. What is the gravitational field strength at Earth’s surface?Earth’s surface? gg = GM/r = GM/r22

gg = (6.67x10 = (6.67x10-11-11 Nm Nm22/kg/kg22)(6.0x10)(6.0x102424 kg) / (6.378x10 kg) / (6.378x1066 m)m)22

gg = 9.8 m/s = 9.8 m/s22

A planet has a radius of 3500 km and a surface A planet has a radius of 3500 km and a surface gravity of 3.8 m/sgravity of 3.8 m/s22. What is the mass of the planet?. What is the mass of the planet? (3.8 m/s(3.8 m/s22) = (6.67x10) = (6.67x10-11-11 Nm Nm22/kg/kg22)(M) / (3.5x10)(M) / (3.5x1066 m) m)22

(3.8 m/s(3.8 m/s22) = (6.67x10) = (6.67x10-11-11 Nm Nm22/kg/kg22)(M) / (1.2x10)(M) / (1.2x101313 m m22)) (4.6x10(4.6x101313 m m33/s/s22) = (6.67x10) = (6.67x10-11-11 Nm Nm22/kg/kg22)(M))(M) M = 6.9x10M = 6.9x102323 kg kg

Variations in Gravitational Field StrengthVariations in Gravitational Field Strength

Things Newton Didn’t KnowThings Newton Didn’t Know

Newton didn’t know what Newton didn’t know what causedcaused gravity, although he knew that all gravity, although he knew that all objects with mass objects with mass havehave gravity and gravity and respond torespond to gravity. gravity.

To Newton, gravity was simply a To Newton, gravity was simply a property of objects with mass.property of objects with mass.

Newton also couldn’t explain how Newton also couldn’t explain how gravity was able to span between gravity was able to span between objects that weren’t touching.objects that weren’t touching. He didn’t like the idea of “action-at-a-He didn’t like the idea of “action-at-a-

distance”.distance”.

Discovery of NeptuneDiscovery of Neptune

Newton’s Law of Universal Gravitation did a Newton’s Law of Universal Gravitation did a very good job of predicting the orbits of very good job of predicting the orbits of planets.planets. In fact, the LoUG was used to predict the In fact, the LoUG was used to predict the

existence of Neptune.existence of Neptune. The planet Uranus was not moving as expected.The planet Uranus was not moving as expected. The gravity of the known planets wasn’t sufficient The gravity of the known planets wasn’t sufficient

to explain the disturbance.to explain the disturbance. Urbain LeVerrier (and others) predicted the Urbain LeVerrier (and others) predicted the

existence of an eighth planet and worked out the existence of an eighth planet and worked out the details of its orbit.details of its orbit.

Neptune was discovered on September 23, 1846 Neptune was discovered on September 23, 1846 by Johann Gottfried Galle, only 1º away from by Johann Gottfried Galle, only 1º away from where LeVerrier predicted it would be.where LeVerrier predicted it would be.

Precession of PerihelionPrecession of Perihelion

Newton’s Law has some flaws, however:Newton’s Law has some flaws, however: It does It does notnot predict the precession of predict the precession of

Mercury’s perihelion, nor does it explain it.Mercury’s perihelion, nor does it explain it. All planets orbit the Sun in slightly elliptical orbits.All planets orbit the Sun in slightly elliptical orbits. Mercury has the most elliptical orbit of any planet Mercury has the most elliptical orbit of any planet

in the solar system (if you don’t count Pluto).in the solar system (if you don’t count Pluto). The closest point in Mercury’s orbit to the Sun is The closest point in Mercury’s orbit to the Sun is

called the perihelion.called the perihelion. Over long periods of time, Mercury’s perihelion Over long periods of time, Mercury’s perihelion

precedes around the Sun.precedes around the Sun. This effect is not explainable using only Newtonian This effect is not explainable using only Newtonian

mechanics.mechanics.

Precession of Mercury’s PerihelionPrecession of Mercury’s Perihelion

The eccentricity of Mercury’s orbit has

been exaggerated for effect in this

diagram.

Other Things Newton Didn’t KnowOther Things Newton Didn’t Know

Newton didn’t know that gravity bends Newton didn’t know that gravity bends light.light. This was verified by the solar eclipse This was verified by the solar eclipse

experiment you read about earlier this experiment you read about earlier this year.year.

He also didn’t know that gravity slows He also didn’t know that gravity slows down time.down time. Clocks near the surface of Earth run slightly Clocks near the surface of Earth run slightly

slower than clocks higher up.slower than clocks higher up. This effect must be accounted for by GPS This effect must be accounted for by GPS

satellites, which rely on accurate time satellites, which rely on accurate time measurements to calculate your position.measurements to calculate your position.

Einstein and RelativityEinstein and Relativity

Einstein’s theory of relativity explains Einstein’s theory of relativity explains many of the things that Newtonian many of the things that Newtonian mechanics cannot explain.mechanics cannot explain. According to Einstein, massive bodies According to Einstein, massive bodies

cause a curvature in space-time.cause a curvature in space-time. Objects moving through this Objects moving through this

curvature move in curvature move in locallylocally straight straight paths through curved space-time.paths through curved space-time. To any observer To any observer insideinside this curved this curved

space-time, the object’s motion would space-time, the object’s motion would appear to be curved by gravity.appear to be curved by gravity.

Curvature of Space-TimeCurvature of Space-Time

Curvature of Space-TimeCurvature of Space-Time

Gravity: Not So Simple AnymoreGravity: Not So Simple Anymore

According to Einstein, gravity isn’t According to Einstein, gravity isn’t technically a force.technically a force. It’s an effect caused by the curvature of space-It’s an effect caused by the curvature of space-

time by massive bodies.time by massive bodies. Why treat it as a force if it isn’t one?Why treat it as a force if it isn’t one?

Because in Because in normal situationsnormal situations, Newton’s LoUG , Newton’s LoUG provides an excellent approximation of the provides an excellent approximation of the behavior of massive bodies.behavior of massive bodies.

And besides, using the LoUG is a lot simpler And besides, using the LoUG is a lot simpler than using the theory of relativity, and than using the theory of relativity, and provides results that are almost as good provides results that are almost as good in in most casesmost cases..

So there.So there.