UniversalUniversalGravitationGravitation

Space Station linkSpace Station link

NasaNasa

Gravitational ForceGravitational Force

An attractive force that exists between An attractive force that exists between allall objectsobjects

The weakest of the four basic kinds of The weakest of the four basic kinds of forces (EM, strong nuclear and weak forces (EM, strong nuclear and weak nuclear are the others)nuclear are the others)

We know how things fall but not whyWe know how things fall but not why

GravityGravity

Galileo and Galileo and Newton gave the Newton gave the name gravity to name gravity to the force that the force that exists between exists between Earth and objects.Earth and objects.

The force of the The force of the earth on the earth on the moon holds the moon holds the moon in its orbitmoon in its orbit

TidesTides

The periodic The periodic change in the change in the surface level of surface level of the oceans due to the oceans due to the gravitational the gravitational force of the sun force of the sun and moon on the and moon on the EarthEarth

Spring tide: greater than normal, since the moon,Earth and sun are alignedSpring tide: greater than normal, since the moon,Earth and sun are aligned

Neap tide:lower than normal due to the sun, Earth and moon at right anglesNeap tide:lower than normal due to the sun, Earth and moon at right angles

Inverse Square LawInverse Square Law

The net force The net force between objects is between objects is inversely proportional inversely proportional to the square of the to the square of the distance between distance between their centerstheir centers

The net force is The net force is directly proportional directly proportional to the product of the to the product of the masses.masses.

Cavendish BalanceCavendish Balance

Henry Cavendish 1798Henry Cavendish 1798

Used his torsion balance to measure the Used his torsion balance to measure the forces needed to rotate the rods through forces needed to rotate the rods through given anglesgiven angles

Calculated the attractive forcesCalculated the attractive forces Calculated G = 6.67 x 10Calculated G = 6.67 x 10-11-11 Nm Nm22/kg/kg22

Gravitation FormulaGravitation Formula

FFgg = Gmm/d = Gmm/d22

d is the distance between the center of the d is the distance between the center of the objects or the average radius of orbit in objects or the average radius of orbit in metersmeters

m is the mass in kgm is the mass in kg G is 6.67 x 10G is 6.67 x 10-11-11 Nm Nm22/kg/kg22

Problems P. 191 # 53 to 59Problems P. 191 # 53 to 59 (64, 75, 78 to 80, 82, 83)(64, 75, 78 to 80, 82, 83)

Using gravitationUsing gravitation

F = ma F = ma Weight = FWeight = Fgg = mg = mg

Force of gravitational Force of gravitational attraction = Gmmattraction = Gmmee/d/d22

Calculate acceleration Calculate acceleration due to gravity.due to gravity.

FFgravitation = gravitation = FFgg

Substitute in Substitute in gravitation formulagravitation formula GmmGmm = mg = mg

dd22

Cancel mass of object being acceleratedCancel mass of object being accelerated

g = Gm/dg = Gm/d22

p. 192 # 72 to 74, 81, 84p. 192 # 72 to 74, 81, 84

Gravitational FieldGravitational Field

Anything that has mass has a Anything that has mass has a gravitational fieldgravitational field

Acts on a body resulting in attractionActs on a body resulting in attraction g is the field strengthg is the field strength g is F/mg is F/m Field is numerically equal Field is numerically equal

to gravityto gravity

Mass calculationsMass calculations

Use FUse Fgg = F = Fcc

Gravitation = CentripetalGravitation = Centripetal GmGmeemmss = = mmss4422RR

RR22 T T22

Cancel the mass of the “satellite”Cancel the mass of the “satellite”

MMee = = 4422RR33

GTGT22

You can also rearrange the formula to You can also rearrange the formula to find the mass of a planetfind the mass of a planet

Ex:Ex: MMee = gr = gr22/G where r is the radius of the /G where r is the radius of the

earthearth Substitute in the values for G, d and gSubstitute in the values for G, d and g MMee = 9.8(6.37 x 10 = 9.8(6.37 x 1066)/6.67 x 10)/6.67 x 10-11-11

p. 193 # 80p. 193 # 80

Kepler’s Laws of Kepler’s Laws of Planetary MotionPlanetary Motion

Brahe – the Earth s the center of the Brahe – the Earth s the center of the universeuniverse

Kepler – Brahe’s assistant, performed Kepler – Brahe’s assistant, performed careful mathematical analysis of Brahe’s careful mathematical analysis of Brahe’s datadata

Kepler’s LawsKepler’s Laws

Describe the behavior of planets and Describe the behavior of planets and satellites.satellites.

His explanations are not considered His explanations are not considered correct today.correct today.

http://animate

Law #1Law #1

The paths of the The paths of the planets are planets are ellipses with the ellipses with the center of the center of the sun at one sun at one focus.focus.

Law #2Law #2

An imaginary line An imaginary line from the sun to a from the sun to a planet sweeps out planet sweeps out equal areas in equal areas in equal time equal time intervals.intervals.

Speeds at different Speeds at different pointspoints

Law #3Law #3The square of the period of revolution of a The square of the period of revolution of a

planet about the sun is proportional to the planet about the sun is proportional to the cube of its mean distance from the sun.cube of its mean distance from the sun.

(T(Taa/T/Tbb))22 = (r = (raa/r/rbb))33

Law #3 con’t.Law #3 con’t. RR33/T/T22 = K where K is Kepler’s constant = K where K is Kepler’s constant

K = 3.35 x 10 K = 3.35 x 10 18 18 mm33/s/s22

Can also be used to determine distances Can also be used to determine distances and periods of two objects around earthand periods of two objects around earth

P. 174 # 1 to 5 PracticeP. 174 # 1 to 5 Practice P. 192 # 61 to 63, 67 to 69, 71, 85, 88P. 192 # 61 to 63, 67 to 69, 71, 85, 88

Review of Kepler’s Laws

The squares of the orbital periods of the The squares of the orbital periods of the planets around the Sun are proportional planets around the Sun are proportional to the cubes of the orbital semi-major to the cubes of the orbital semi-major axes. axes.

This means that if you know either how This means that if you know either how much time a planet's orbit around the much time a planet's orbit around the Sun takes you can easily know it's Sun takes you can easily know it's average distance from the Sun, or vice-average distance from the Sun, or vice-versa! versa!

Major Axis definedMajor Axis defined

Kepler's 1st Law: An ellipse has a long axis Kepler's 1st Law: An ellipse has a long axis (called the major axis). It also has two foci (called the major axis). It also has two foci (focuses). One focus is where the Sun is (focuses). One focus is where the Sun is located, the other focus is empty. As the two located, the other focus is empty. As the two foci are brought together, the ellipse looks foci are brought together, the ellipse looks more and more like a circle. In fact, a circle is more and more like a circle. In fact, a circle is just a special case of an ellipse with the two just a special case of an ellipse with the two foci at the same place (the center of the circle), foci at the same place (the center of the circle), in which case the major axis is the diameter of in which case the major axis is the diameter of the circle. Half the major axis is called the the circle. Half the major axis is called the semi-major axis (semi-major axis (aa in the upper figure, in the upper figure, rr in the in the lower figure).lower figure).

Notes:Notes:

R = d (radius is the distance between the R = d (radius is the distance between the center of the objects)center of the objects)

Satellite can mean any orbiting objectSatellite can mean any orbiting object ““planet” is the center of motionplanet” is the center of motion If an object is at some distance above a If an object is at some distance above a

planet, add the radius and the altitude of planet, add the radius and the altitude of orbitorbit

Mass of the “satellite” cancels outMass of the “satellite” cancels out

Modeling the Orbits of Modeling the Orbits of Planets and SatellitesPlanets and Satellites

P. 186P. 186 A – aphelion is the furthest distance from the A – aphelion is the furthest distance from the

sun along the major axissun along the major axis P – perihelion is the closest distance along the P – perihelion is the closest distance along the

major axismajor axis

ee is the eccentricity – ratio of the distance is the eccentricity – ratio of the distance between the foci to the length of the major orbitbetween the foci to the length of the major orbit

LabLab

PP22=a=a33 Where P is the orbital period in Earth Where P is the orbital period in Earth

years and a is the length of the years and a is the length of the semimajor axis (average distance from semimajor axis (average distance from the Sun) in Astronomical Units. the Sun) in Astronomical Units.

CalculationsCalculations

See procedure 4. Use the data table to See procedure 4. Use the data table to find find ee

D = 2D = 2ee(10 cm) / (10 cm) / e e + 1+ 1

d/2 C d/2d/2 C d/2 Use a different color for each planetUse a different color for each planet

Motions of SatellitesMotions of Satellites

FFcc = F = Fgg

FFcc is directed towards is directed towards

the center of the the center of the earthearth

The weight (fThe weight (fgg) of the ) of the

object keeps it in its object keeps it in its pathpath

Critical speedCritical speed

The speed at which an object is The speed at which an object is perpetually falling towards the earth but perpetually falling towards the earth but never landing.never landing.

The rate of fall is equal to the curvature The rate of fall is equal to the curvature of the earth.of the earth.

FFgg = F = Fcc

mg = mvmg = mv22/r so v = /r so v = √gr√gr

Period of a Satellite Period of a Satellite GmGmeemmss = = mmss4422RR

RR22 T T22

mmss cancels cancels Cross multiply GmCross multiply Gmee T T22 = 4 = 422RR33

Divide to solve for T Divide to solve for T TT22 = = 4422RR33

GmGmee

Simplify to Simplify to T = 2T = 2√R√R33/Gm/Gmee

(R = radius of the planet + altitude of the satellite)(R = radius of the planet + altitude of the satellite)(m(mpp = mass of planet) = mass of planet)Practice p. 178 # 6Practice p. 178 # 6

Velocity of a satelliteVelocity of a satellite

GmGmeemmss = = mmssvv22

RR22 R R

Cancel mass of the satelliteCancel mass of the satellite

GmGmee = = vv22

RR22 R R

Simplify and solve for vSimplify and solve for v

V = V = √√Gm/RGm/R

p. 181 # 12 to 14p. 181 # 12 to 14

Example : T = 2Example : T = 2√R√R33/Gm/Gme e V = V = √√Gm/R Gm/R Engineers are planning to place the International Engineers are planning to place the International

Space Station into orbit at an altitude of 450 km above Space Station into orbit at an altitude of 450 km above Earth’s surface. What would be the orbital speed and Earth’s surface. What would be the orbital speed and period of the ISS?period of the ISS?

Re = 6.38 x 10Re = 6.38 x 1066 m m Me = 5.97 x 10Me = 5.97 x 102424kgkg G = 6.67 x 10G = 6.67 x 10-11-11 N N..mm22/kg/kg22

P. 191 # 71, 85, 88, 90 to 92, 94, 96P. 191 # 71, 85, 88, 90 to 92, 94, 96

Geosynchronous satelliteGeosynchronous satellite

From Earth, a satellite in geosynchronous orbit appears to "hover" over one spot on the Equator. That means a receiving dish on the Earth can point at the satellite at one spot in the sky and not have to "track" its motion.The satellite isn't motionless, though. It's in a very high orbit where it circles the Earth once a day, matching the Earth's rotation on its axis. Weather and Cable TV are examples.

Escape speedEscape speed

The minimum speed The minimum speed an object must an object must possess in order to possess in order to escape the escape the gravitational pull of a gravitational pull of a body.body.

Calculate escape speedCalculate escape speed

V = V = √2Gm/d√2Gm/d M = mass of the celestial bodyM = mass of the celestial body D = radiusD = radius Mass of the escaping object does not Mass of the escaping object does not

mattermatter

Voyager SatellitesVoyager Satellites

Update linkUpdate link

Einstein’s Theory of Einstein’s Theory of GravityGravity

Gravity is not a force Gravity is not a force but an effect of but an effect of space itself.space itself.

Mass causes space Mass causes space to be curved.to be curved.

Bodies accelerate as Bodies accelerate as they move in curved they move in curved space.space.

AnswersAnswers

R = 6.83 x 10R = 6.83 x 1033 m m V = 7.63 x 10V = 7.63 x 1033 m/s m/s T = 5.62 x 10 T = 5.62 x 10 33 s s