Hw 4.11 Answer gravity questions posted at www.bpi.edu

The Motion of Planets

History of Astronomy

•Ptolemy•Copernicus•Galileo•Brahe

•Kepler•Newton•Cavendish•Einstein

Assumptions of Early Models

Geocentric - Earth in the middle Everything orbits the Earth Everything moves in uniform circular

motions

PROBLEM: retrograde motion of the planets!

Claudius Ptolemy (87-165)

Helped explain non-uniform motions

Very accurate models!

At least for a while….

Alexandria Egypt - access to information

Model based upon previous ideas

Nicolaus Copernicus (1473-1543)

Errors building up

Must be a better way!

Let’s try a Heliocentric system!

Explains –

phases of Moon

Retrograde motion

Not any more accurate though

Tycho Brahe (1546-1601)

Observations of comet – beyond the Moon

Observation of supernova – far away

Naked eye observations of planets

Accuracy through repetition

Best observations of planetary positions

Atmospheric refraction (bending of light)

Castle, dwarf, nose, bladder

Johannes Kepler (1571-1630)

Worked for Brahe and had access to his data

Took data after his death

Spent years figuring out the motions of the planets

Came up with…

Three Laws of Planetary Motion

KEPLER’S FIRST LAW: Planets move in elliptical orbits with the Sun at one focus.

Sun

Foci (sing. Focus)

Perihelion Aphelion

Average distance from earth to Sun = 1 Astronomical Unit (1 A.U.)

KEPLER’S SECOND LAW: Planets sweep out equal areas in equal amounts of time.

1 Month1 Month

(This is because they speed up as they get closer to the sun)

KEPLER’S THIRD LAW:The ratio of orbital period squared divided by orbital radius cubed is a constant!

T = period of the orbit (time to go around the sun)

r = orbital radius (average distance from sun

T2 / r3 = constantExample: Mercury Earth JupiterT = 1 yearr = 1 A.U.T2/r3= 1 !!

Kepler’s Third Law

Tm = 1.91 yrs Te=1.00 yrs

rm = 2.3x1011m re=1.5x1011 m

= 1.53 AU =1.00 AU

Tm2/rm

3= Te2/re

3=

Kepler’s three laws apply to ANY orbital system, but the constant depends on what you are orbiting

For example, all of the planets are orbiting the sun so T2/r3 is the same for all of them.

All of Jupiters moons are orbiting Jupiter so T2/r3 is the same for all of them (but not the same as for the planets).

Galileo Galilei (1564-1642)

Knew of Copernicus’s & Kepler’s work

Used a telescope to look at the sky

What did he see?

The Moon was an imperfect object

Venus has phases

Jupiter has objects around it

Saturn is imperfect

The Sun is imperfect

Isaac Newton (1642-1727)

How can I explain Kepler’s laws

Planets must obey my Three Laws of Motion

What force makes planets curve out of a straight line?

Apple falls on his head….aha!

221

d

MGMF

F=force of gravityF=force of gravity

G=constant=6.67 x 10G=constant=6.67 x 10-11-11 N-m N-m22/kg/kg22

MM11, M, M22 = masses = masses

d=distance between objects (center to d=distance between objects (center to

center)center)

F=force of gravityF=force of gravity

G=constant=6.67 x 10G=constant=6.67 x 10-11-11 N-m N-m22/kg/kg22

MM11, M, M22 = masses = masses

d=distance between objects (center to d=distance between objects (center to

center)center)

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

Henry Cavendish…

Measured force of attraction between two massive objects (1797)

Used to determine the value of ‘G’

“Weighed” the earth

Albert Einstein

Theory of general relativity can explain gravity by looking at the geometry of space-time.

Massive objects cause space itself to become distorted.

Theory explains things that Newton’s law can’t (precession of the perihelion of Mercury)

Makes very precise predictions that are later tested by experiment.

Using the Law of Gravity

1. A bowling ball (m=5.0 kg) and a tennis ball (m=0.30 kg) are separated by 0.40 meters. What is the force of gravity they exert on one another?

2. The radius of Mercury is 2.44x106m and the mass is 3.3x1023 kg. What is the value of ‘g’ on Mercury?

3. Earth has a mass of 6.0 x 1024 kg and it’s radius is 6.4x106 m. What is the orbital speed of a satellite that is 1,200 km above the earth?

Deriving the law of gravity F=ma Fg=mv2/r

Fg=m(2r/T)2/r

Fg=42rm/T2

Fg=(42rm)/Cr3

Fg=GMm/r2

T2/r3 = C T2=Cr3

v=2r/T C=42/GM

Earth

Mars

deferent epicycle

•Epicycles helped a little

•Later models included an off-center Earth

•Still not very accurate though

Wed Dec 2 Unit 6 Test Monday (Chapter 8) HW worksheet (14,15,16,20,22,23)

Warmup: Name the person who…

1. Explained gravity as the curvature of space2. Said that planets followed elliptical orbits

around the sun.3. Spent 20-30 years making extremely

precise observations of planetary positions.4. Found the formula to calculate the

gravitational attraction between any 2 spherical masses.

5. Measured the attraction of masses on Earth and determined the value of ‘G’

6. Proposed a heliocentric model of the solar system with circular orbits and epicycles.

The recently demoted “planet” Pluto orbits the sun at a distance of 39.5 AU’s. How long does it take to go around the sun once?

The moon’s orbital radius is 3.9x108m and its orbital period is 27 days.

1. Use Kepler’s Third Law to determine the orbital radius of a geosynchronous satellite (T=1 day).

2. What is the orbital speed of a geosynchronous satellite?

3. Use this information with Newton’s Law of Gravity to calculate the mass of the earth.

Practice

Determine the orbital speed of a satellite that orbits the earth at an altitude of 2,000 km.

Mercury has a radius of 2.44x106 m and a mass of 3.30x1023 kg. If an astronaut on Mercury’s surface threw a rock upward at a speed of 15 m/s, how high would it go?

Halley’s comet orbits the sun once every 74 years. Find its average orbital distance in AU.

Wed Dec 17

Hw> (p174) 2-4 (p181)14 (p192) 61,67Warmup: Mars has a mass of 6.4x1023 kg and

orbits the sun at a distance of 2.3x1011 m. 1. If the sun exerts a force of 1.6x1021 N on

Mars, what is the mass of the sun?2. At what speed does Mars orbit the sun?

(F=ma, a=v2/r)3. How long does it take Mars to orbit the

sun? Express your answer in Earth years.

F=ma a=v2/r v=2r/T Fg=mv2/r GMm/r2=mv2/r GM/r=v2 v=(GM/r)1/2

GM/r=42r2/T2

T2/r3=42/GM

41.No, the force the moon exerts on the earth is the SAME as the force the earth exerts on the moon! (Newton’s Third Law)

46.g=GM/r2

gJ/ge=(GMJ/rJ2)/(GMe/re

2)

= (MJre2)/(MerJ

2)

= (300Mere2)/[Me(10re)2]

=300/100=3 gJ = 3ge = 30 m/s2

53. m1=5.9 kg

m2=0.47 kg

r=0.055 mFg=?

Fg=GM1M2/r2 = 6.1x10-9 N

55.m1=70kg m2=50kg r=20 m Fg=?

Fg=GM1M2/r2=5.8x10-10 N

72. m=7x1020 kg r=5x105 m Fg=?

mg=GMm/r2

g=GM/r2=0.187 N/kgFg=mg=16.8 N

80. r=3.9x108 m Fg=1.9x1020 N mE=6.0x1024 kg mmoon=?

Fg=GMEMm/r2 Mm =(Fgr2)/(GME)

=7.2x1022kg

Monday, December 15

HW> (p191-2) 41, 46, 53, 55, 72, 80 Quiz on Friday

Warmup

A merry-go-round makes one revolution every 20 seconds. What is its angular speed in degrees/s? In radians/s?

If the radius of the ride is 4.0 meters, what is the centripetal acceleration of a person on the edge?

How many revolutions will the merry-go-round make as it decelerates uniformly to rest in 1 minute?

Warmup

P81 1-5 Today:

Circuit schematics Power in electric circuits

Physics 12/9/04

THUR: Discuss lab results Kepler’s Laws Newton’s Universal Law of Gravitation HW 2.12 (195) 32-36

Monday December 11Test tomorrow!!Today: Circular motion continuedWarmup:1. Is a car going around a corner accelerating? Explain.2. Is a planet going around the sun accelerating? Explain3. According to Newton’s Second Law of Motion, what causes an object to accelerate?4. What force(s) cause the object’s in 1&2 to accelerate?

Wednesday December 13 Hw 2.14 Read 8.1 (p184)1.1,1.2,1.3 (193) 1-8 Today: Kepler’s Laws Warmup: 1. A 50 kg kid is riding a merry-go-round. Her horse is 4.0

meters from the center and is traveling at 2.8 m/s. What is her acceleration? In what direction does the net force on the girl act?

2. A go cart navigates a circular track with a radius of 60 meters. It completes one lap every 45 seconds. How fast is it traveling. What is the magnitude of its acceleration?

3. The earth is 1.5x1011m from the sun. Calculate the speed our planet is traveling. (Assume a circular orbit)

Thursday December 14

HW 15> (180-1) 1-4 (194-5) 17,18, 31,34,37

Today: Planetary motion and gravity (cont.)

Remember…Law #1. Law of Inertia - Objects do whatever they are currently doing unless something messes around with them.

Conclusion: since planets don’t move in straight lines, something is messing with them!

SECOND LAW: Accelerations are caused by forces.

Fnet=ma

For circular motion a=v2/r

Conclusion: Whatever physical agent keeps planets in orbit must obey this law.

THIRD LAW: For every action there is an equal and opposite reaction.

Conclusion: If the sun pulls on the earth, the earth must exert an equal force on the sun!

Question: Could the force that pulls objects down to the earth be the same force that keeps planets in their orbit?

Answer: yes, but only if the force gets weaker the further away you go.

Newton figured out just how this gravitational force would need to behave in order to comply with Kepler’s Laws and his own laws of motion.

Because this force could describe gravity on earth and in heaven (space) he called it….

Monday December 18 HW 16> (195) 41,42,45,47,48 (196)51,52 Today: Using the Law of gravity Warmup:1. What is the weight of a 110 kg astronaut

standing on the moon (The moon’s mass is 7.3x1022kg, and its radius is 1.74x106 m).

2. Halley’s comet takes 76 years to orbit the sun. What is its average orbital distance?

3. Describe Cavendish’s experiment. What was the significance of his results?

Practicing circular motion Tarzan (mass=80kg) swings from a vine that

is 6.0 meters long. At the bottom of his swing the tension in the vine is 1100 N. How fast is he moving?

What is the gravitational attraction between Tarzan and Jane (mass=40 kg) when they are 0.5 meters apart?

Mercury has a mass of 3.2 x 1023kg and a radius of 2.4x106m. How much would a 0.6 kg hammer weigh on Mercury’s surface. If you dropped the hammer, how long would it take to fall 3.0 meters?

Tuesday December 19Hw 2.17> (194) 11,12, 21-26,29 (196) 54-56

Warmup: The dwarf planet Pluto has a mass of 1.5x1022 kg. It’s radius is 1.15x106m and it orbits the sun at an average distance of 5.91x1012m. A)Find the strength of the gravitational field (g) on the surface of Pluto.B)If a projectile was launched horizontally at a speed of 15m/s from a Plutonian cliff that was 18.0 m tall, how far from the base of the cliff would it land?C)Find the orbital period of Pluto in its orbit around the sun WITHOUT using Kepler’s third law. (The mass of the sun is 2.0x1030 kg). State your answer in both seconds and years.

More!!! According to the table on page 204, Kepler’s constant

for our solar system is approx. 3x10-19 s2/m3. Use this information to “weigh” the sun.

A 600 kg car goes around a curve with a 1000 meter radius at a speed of 12 m/s. What is it’s centripetal acceleration? What is the minimum frictional coefficient that will keep it from sliding off of the road?

The earth’s moon makes one orbit every 27 days and orbits at a distance of 3.84 x108 meters. What is the orbital radius of a “geosynchronous” satellite which makes one orbit every 24 hours? How fast is the satellite moving?

Wednesday December 20

TEST TOMORROW!! (Ch 7.2, 8.1, 8.2) Do you remember:

ac=v2/r v=2r/TF=mv2/r

Contributions of: Ptolemy, Copernicus, Galileo, Brahe, Kepler, Newton, Cavendish, Einstein

Kepler’s 3 Law’s of Planetary Motion Fg=GM1m2/d2 (universal law of gravity)

Practice & Review1. Briefly list the contributions of each person to our

understanding of gravity and planetary motion:A) Kepler B) Cavendish C) GalileoD) Newton E) Einstein F) Brahe

2. A cave man swings a 2.0 kg rock around his head on the end of an 80cm rope. The tension in the rope is 60.0 N. Find the following:A) The centripetal accelerationB) The speed of the rockC) How long does it take him to swing the rock around 12 times?

3. Find the maximum speed a car can go around an unbanked curve with a radius of curvature of 90 meters and a coefficient of friction of 0.25.

Practice & Review NASA decides to place a satellite in orbit

300 km above the surface of Mars. (Mmars=6.4x1023kg, rmars=3.40x106m)

A) What is the strength of the gravitational field (g) at this location?

B) What is the orbital speed and period of the satellite at this location?

C) A martian moon is discovered in orbit at 3 times the orbital radius of of the satellite. Use Kepler’s third law to determine its period.