PURDUE UNIVERSITY

EAS 105-THE PLANETS

Prof. Robert L. Nowack

 

Lecture 5

 

 

Isaac Newton (1643-1727)

 

 

 

 

Newton revolutionized science and the way we look at the world when he developed his Three Laws of Motion.  He was able to apply these laws to celestial motions and show that the same force, i.e., gravity, which causes an apple to fall from a tree, also controls the orbits of the Moon and the planets.  1987 was the 300th anniversary of the publication of Newton's famous book "Principia" published in 1687.

 

 

Newton's First Law

 

1)   Every body continues in its state of rest or uniform motion in a straight line unless it is compelled to change by the action of some force [i.e. in absence of forces, velocity stays constant.]

 

The first part of this law is obvious.  The second part is not so obvious since everything that starts moving ultimately stops.  Newton realized that this was because the force of friction would slow things down.  If friction is reduced, motion in a straight line continues for longer and longer times.

 

The measure of how fast an object is moving is called velocity.

 

Ex.)   A car is moving at v = 60 miles/1 hour

 

However, this depends on units:

 

v = (60 miles)/(1 hour) x (1 hour/60 min.)

> (1 mile)/(1 minute)

 

Velocity has both a magnitude (speed) and a direction.

 

Ex.)   v = (60 miles)/(hour) (a magnitude) toward the Northeast (a direction).

 

Any change in the velocity (either in magnitude or direction) is called an acceleration.  Starting, stopping, speeding up, slowing down or changing direction are all accelerations (changes in the motion).

 

Ex.)   A car going from v = 0 to v = 60 miles/hour in 10 seconds has a greater acceleration than a car going from v = 0 to v = 60 mph in 20 seconds.

 

 

 

Newton's Second Law

 

2)   The change of motion (acceleration) is proportional to the force acting on the body, and inversely proportional to the mass of the body.

 

The acceleration is equal to (Applied Force)/(Mass):  a = F/M, or, force equals mass times acceleration.

 

In metric units, force is measured in Newtons (after guess who).

 

1 Newton is the force required to accelerate 1 kg to v = 1 meter/sec in 1 second

1 Newton = 1 kg x m/second per second

 

Force = Mass x Acceleration

 

Ex.)   In order to save a small infant, Superman must stop a 10 ton bus moving at 5 mph (2.2 m/sec) in 1 sec.  How much force must be applied?

 

F = ma       F = 10,000 kilograms x 2.2 meters/second per second

                  F = 22,000 Newtons

 

Ex.)   To stop a 100 kilogram person moving at 2.2 m/s in 1 sec.

 

F = 100 kilograms x 2.2 meters/second per second

F = 220 Newtons

 

 

 

Newton's Third Law

 

3)   To every action there is an equal and opposite reaction.

 

Ex.)   When hitting a baseball, the recoil of the bat shows that the ball exerts a force on the bat, just as the bat does on the ball.

 

Ex.)   A rocket works by this law:  Hot gases are ejected from behind and shoves the rocket forward.  These exhaust gases need not push against air or the Earth.  A rocket works best in a vacuum!

 

Recall acceleration results from either changes in speed or changes in direction.

 

Ex.)   A ball revolving on a string is constantly changing direction and, therefore, accelerating.  According to Newton's 2nd Law, a force must be acting on it.

 

 

 

 

If the string is let go (the force removed), the ball will depart in a straight line.  Also, if the string is pulled in shorter, the ball revolves faster.

 

Newton found that planetary bodies moving in circular orbits must be accelerating since they are not moving in a straight line.  Therefore, a force must be acting on each of the planets (by Newton's 2nd Law).  But, obviously the Sun and the planets have no string attached to them!

 

Newton hypothesized the existence of a force acting at a distance between all bodies.  He called this force gravity.

 

 

 

 

The force of gravity will slightly pull the plumb bob toward a nearby mountain.  Geological prospectors use this principle to search for heavy ore bodies in the Earth.

 

 

 

Newton's Universal Law of Gravitation

 

Between any two bodies in the universe there exists a force of attraction acting at a distance which is proportional to the mass of each body and inversely proportional to the square of the distance between them.  In mathematical form, this can be written:

 

Force of attraction due to Gravity

  = 

 Constant G

 x

  Mass of
 1st body

 x

  Mass of 2nd body

 x

 1/(distance between them)2

 

where:  G = 6.67 x 10-11

 

This law is extremely important.  It governs the motion of an apple falling to the Earth as well as a spacecraft to Jupiter.  For a person on Earth, every part of the Earth attracts the person.

 

 

 

 

However, for purposes of Newton's Gravity Law, all of Earth's mass can be put at its center (its center of mass).  Thus, objects fall in the direction of the Earth's center.

 

A person's "weight" is the gravitational force between him and the Earth (or if on another planet, the gravitational force between him and that planet).  At the Earth's surface, the acceleration due to gravity has been measured at ~ 9.81 m/sec/sec.

 

Ex.)   For a parachutist in free fall:         0 sec v = 0

                                                            1 sec v = 9.81 m/sec

                                                            2 sec v = 19.62 m/sec

                                                            3 sec v = 29.43 m/sec

                                                                          

 

 

Note that the terminal velocity due to air resistance depends on the shape of the body, but is about ~ 50 m/sec.

 

By knowing this acceleration due to gravity and using Newton's Laws, the mass of Earth, which causes the acceleration, can be determined.  Using Newton's Gravity Law, we know that the acceleration due to gravity is proportional to the Earth's mass.  We can use this in reverse to estimate the mass of the Earth which is found to be

 

MEarth ~ 6 x 1024 kilograms

 

Let's define Density to be (mass)/(volume) where:

 

Mass - total amount of material in body

Volume - total space taken up by the body (its size)

Density is measured in either: (gram)/(cm3) or (kg/m3) which is (Mass)/(Volume)

 

Ex.)   The density of water (surface pressure and temperature):

(1 gram)/(cm3) or (1000 kg)/(m3)

 

The Volume of a Spherical Earth is:  volume = 4/3 x  x R3 ~ 4.19 R3, where R = radius of the Earth.  Thus:  Volume ~ 4.19 (6.371 ´ 106 m)3 = ( 1.083 x 1021 )m3

 

Density = mass/volume = (6 ´ 1024 kg)/(1.083 ´ 1021 m3)

 

The computed density of the Earth is ~ 5540 kg/m3

 

Due to the slight nonspherical shape of the Earth, the actual the average density is 5515 kg/m3

 

 

Densities of Different Substances

Object                          Density kg/m3

Air                               1.2

Wood                          600-900

Ice                               900

Water                           1000

Salt Water                    1025

Human Body                1030

Granite                         2700

Iron                              7800

Silver                            10,500

Lead                            11,300

Gold                             18,900

 

The average density of a planet can be used to guess its composition.

 

Ex.)   For the Earth, the average density is 5515 kg/m3, which is somewhere between granite and iron in the table above.

 

Problem:  Most planets are stratified.  Higher density material is deep within, so further detective work is needed to determine the composition of planets.

 

Can mass and density be found for other planetary bodies from Newton's gravity law?  Yes!  One can use the acceleration of a satellite in orbit about a planet to obtain the mass of the "parent" planet using Newton's laws.

 

 

 

Average Density

 

           

                        kg/m3

The Sun            1400

 

Densitygranite < Dplanet < Densityiron

 

Giant planets with lower density

Mercury        5430

Rocky with less iron

Jupiter      1330

Venus           5240

Moon    3360

Saturn       690 (would float in water!)

Earth             5515

Mars     3940

Uranus      1270

 

 

Neptune   1640

 

 

Recall, on the Earth's surface, the acceleration due to gravity is 9.81 m/s/s.

 

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Aside:   A person's weight is the force pulling on a person due to gravity.

            From Newton's 2nd Law, this is just Force = Mass times Acceleration.

 

Ex.)   A 75 kg man would "weigh":  F = 75 kg x 9.81 m/s/s = 736 Newtons

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Acceleration of gravity on the surface of another planet depends on its mass and radius.

 

Acceleration in (m/s2)

Moon       1.62

Earth         9.81

Venus       8.60

Mars         3.72

Pluto         0.70

 

For example, Elvis has just returned from his secret hideout.  However, he has porked out to 120 kg (~ 265 lbs.).  According to Newton's 2nd law, Elvis' weight on Earth is:

 

Force = 120 kg x 9.81 m/s/s = 1,177 Newtons

 

Elvis' agent thinks Elvis will look slimmer if he performs his next concert on the Moon.  His "weight" on the Moon will be:

 

Force = 120 kg x 1.62 m/s/s = 195 Newtons! or about 1/6 his weight on Earth

 

Does he have less mass on the Moon?  No - He will have the same mass, but he will certainly feel lighter.

 

 

 

Escape Velocity

 

If one shoots a rifle horizontally from a mountain top, the higher the muzzle velocity, the farther the bullet will go.  If the muzzle velocity reaches the critical velocity, a circular orbit around Earth would result.

 

 

 

 

The Critical Velocity, vc, is given by:

 

vc2 = Acceleration of gravity (g) x distance to planet’s center (Re)

 

vc2 = g x Re

 

Ex.)   On Earth:  g = 9.81 m/s/s            Re = 6.371 E 106 meters

 

vc2 = 9.81 x (6.371 x 106) = 62.5 x 106

 

vc = 7905 m/s = 7.905 km/sec

 

Thus for Earth:  vC = 7.9 km/sec (~ 5 miles/sec), to achieve a circular orbit about the Earth.

 

If the bullet goes even faster, elliptical orbits will develop.  Finally, at the Escape Velocity, ve, a parabolic orbit will be reached and the bullet will leave Earth's orbit, where

 

ve= 1.41 x vc

 

For Earth, the Escape Velocity is ve = 11.172 km/sec or about 7 miles/sec or about 25,000 miles/hour.

 

 

 

 

 

Escape Velocities for Different Planets

 

                     Gas Giants

                     ve (km/sec)

                     ve (km/sec)

Mercury              4.3

Jupiter                 57.5

Venus                  10.3

Saturn                 35.6

Earth                   11.2

Uranus                21.2

Mars                   5.0

Neptune              23.6

Pluto                   1.33

 

Moon                  2.0

 

 

 

These values of Escape Velocity, ve, are very important to estimate what type of atmospheric gases a planet can retain.

 

The range of velocities of molecules of a given gas are temperature dependent with lighter molecules able to achieve higher speeds.

 

Ex.)   The space shuttle uses a low Earth orbit about 300 km above the Earth’s surface.

(R = 6371 km + 300 km = 6671 km from the center of the Earth)

 

The space shuttle would have a critical orbit velocity of about 7.74 km/sec at that height.  How long will it take to make 1 orbit about the Earth?

 

Circumference (c) = 2 p  R = 41,913 kilometers

Period = c/ vC = (41,913 km)/(7.74 km/sec) = 5415 seconds (about 90 minutes)

 

 

Ex.)   Geosynchronous Orbit

 

A geosynchronous orbit is an orbit with the same period as the rotation period of the Earth itself (about 24 hours).  To get this longer orbit period, a geosynchronous orbit must be much farther away [Kepler's 3rd Law! where (Period)2 ~ (R)3 ].