6.1-6.3 PLANETARY MOTION

FORCE FIELDS

Not the kind you find in Star Trek (the coolest show ever)

Force fields are used to describe the amount of a given type of force generated by an object on other objects that are near it per a unit measurement

FOR EXAMPLE…

A gravitational field affects objects that have mass

Therefore, any object that possesses mass that is within a gravitational field will experience a gravitational force acting on it

How much force is acting on it will depend on where this object is – and how much mass it has

WHY USE FORCE FIELDS?

Is the amount of gravity acting on two objects of different mass the same, if their distances from the earth are the same?

What happens if objects of the same mass move further or closer to the object creating the field?

FORCE FIELDS ARE LIKE PRICES

What force fields allow you to do is to calculate the value of a force acting on an object depending on a set of conditions

It’s like pricing objects in a store

Is it easier to give the price of one, two or three objects…

Or give a price per object?

PRICE

A price gives you the cost per item – so you can predict the cost of a purchase based on how many items you purchase

Mathematically, a simple way of viewing a field is looking at the measurement as a price

The strength of a field at one point tells you how much the total value will be based on the amount of a particular quality that the field affects

TAKE ANOTHER LOOK AT AN EQUATION YOU

KNOW….

How does:

F = Gm1m2

d2

Define a gravitational field?

See notebook file for derivation of gravitational field

PLANETARY ORBITS

The orbits of planets are not circular; they are actually ellipses:

http://mistupid.com/astronomy/orbits.htm

However, in order to derive the velocity of orbits based on the gravitational pull between two bodies can be dealt with by assuming that the orbit is circular

CIRCULAR MOTION AND ORBITS

Recall that when looking at circular motion, an object maintains a constant velocity in a circular path if there is a constant force that pulls the circulating object towards the center of its rotation

This situation is similar to how orbits are formed

CIRCULAR ORBIT AND PLANETARY MOTION

Compare the relationship between an orbiting planet and the motion of an object on a string

http://www.physclips.unsw.edu.au/jw/circular.htm

Therefore, equations related to circular motion can be used to approximate the velocity of objects in orbit

See notebook file for the derivation of orbital velocity

CHANGE IN ORBIT

What happens if Fg was to change?

What happens if v changes?

KEPLER’S LAWS

Johannes Kepler a German mathematician, astrologer and astronomer determined 3 laws that govern planetary motion

Though Kepler finally deduced the real motion of the planets, he did so by analyzing data gathered by another scientist by the name of Tycho Brae

KEPLER’S FIRST LAW

Planetary orbits are elliptical, with the sun at one focus of the ellipse

KEPLER’S SECOND LAW

The straight line The straight line connecting the connecting the planet and the planet and the sun sweeps out sun sweeps out equal areas in equal areas in the same amount the same amount of timeof time

Kepler's Second Law Kepler's Second Law InteractiveInteractive

KEPLER’S THIRD LAW

The cube of the average radius , r, of a planet’s orbit is directly The cube of the average radius , r, of a planet’s orbit is directly proportional to the square of its period, Tproportional to the square of its period, T

Namely: rNamely: r33 αα T T22

Therefore: Therefore: rr33 = CsT= CsT22

Where: Cs = constant of proportionality for the sunWhere: Cs = constant of proportionality for the sun

Note that: Cs is based on the object that is creating the gravitational Note that: Cs is based on the object that is creating the gravitational fieldfield

Kepler's Third Law InteractiveKepler's Third Law Interactive See derivation of Cs in notebook fileSee derivation of Cs in notebook file

UNDERSTANDING ESCAPE ENERGY

Objects on planets are “bound” to the planet in a situation very similar to the following

Imagine being tied to a bungee cord to another object, and the only method of escape that you have is to run as fast as you can in order to try to “break” the cord

DISCUSS THE ENERGY

What must you do in order break free? Discuss your energy expenditure

What is the relationship between your energy expenditure and the distance between you and the object?

YOU’RE IN AN ENERGY DEBT

You can view the energy of this system in terms of how much you “owe” the bungee cord in order to get free

All the effort that you put in in order to break free of the bungee cord doesn’t increase your speed – it’s all put into stretching or trying to break the cord

If the cord wasn’t there – you would be going at a much faster speed for the same distance travelled!

ESCAPE ENERGY

r

E

Radius of planet

HOW DO YOU GET AWAY FROM THE BUNGEE

CORD?

An object exiting a gravitational field must do so by paying the energy “debt” with kinetic energy

In order to escape the pull of the planet, the total kinetic energy of the rocket must EQUAL OR EXCEED the energy debt owed to the planet

ESCAPE VELOCITY refers to the minimum velocity required for an object to just escape the gravitational pull of the planet,

ESCAPE ENERGY refers to the energy associated with the kinetic energy required to repay the energy “debt”

See notebook file for derivation

TIED UP IN DEBT

Any object that is “bound” to earth remains so because its total energy does not exceed the energy debt owed to the planet

Think about an orbiting satellite:

It remains in orbit around the earth (therefore, it is still “tied” to the earth – how do we know? What happens if it stops moving?)

But it also has a kinetic energy See notebook file for derivation

THE DEBT IN ORBIT

r

E

Eg

r

•Total energy for objects in orbit will equal to HALF of the potential energy owed at that particular radius

Etotal

Ek

TOTAL ENERGY OF OBJECTS IN A

GRAVITATIONAL FIELD

Therefore, total energy of any Therefore, total energy of any object in a gravitational field is object in a gravitational field is therefore equal to the sum of its therefore equal to the sum of its gravitational potential energy and gravitational potential energy and the object’s kinetic energythe object’s kinetic energy

Therefore, there are 3 cases that Therefore, there are 3 cases that can be set up due to thiscan be set up due to this

CASE 1 – OBJECT JUST ESCAPES: ET = 0

In this situation, the kinetic energy is just enough for the object to escape

All the kinetic energy is used to pay the energy debt

What will the object’s motion be like when it escapes the field?

r

E

Eg

Ek

CASE 2 – OBJECT ESCAPES WITH V > 0: ET

> 0

In this situation, there is enough Ek to pay the debt and provide the object with enough Ek to continue onwards with a constant velocity as r ∞

r

E

Eg

Ek

CASE 3 – BOUND OBJECT: ET < 0

In this situation, there is enough Ek is not great enough, so total energy is still negative

Since there is still an energy debt, the object remains bound to the planet and cannot escape

r

E

Eg

Ek