6.1-6.3 Planetary Motion6.1-6.3 Planetary Motion

Force fieldsForce fields

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

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

For example…For example…

• A A gravitational fieldgravitational field affects objects that affects objects that have masshave mass

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

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

Why use force fields?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 pricesForce fields are like prices

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

• It’s like pricing objects in a storeIt’s like pricing objects in a store

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

• Or give a price per object?Or give a price per object?

Price Price

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

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

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

Take another look at an equation you know….Take another look at an equation you know….

How does:How does:

F = F = GmGm11mm22

dd22

Define a gravitational field?Define a gravitational field?

See notebook file for derivation of gravitational fieldSee notebook file for derivation of gravitational field

Planetary orbitsPlanetary orbits

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

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

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

Circular motion and orbitsCircular motion and orbits

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

• This situation is similar to how orbits are This situation is similar to how orbits are formedformed

Circular orbit and planetary motionCircular orbit and planetary motion

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

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

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

• See notebook file for the derivation of orbital See notebook file for the derivation of orbital velocity velocity

Change in orbitChange in orbit

• What happens if Fg was to change?What happens if Fg was to change?

• What happens if v changes?What happens if v changes?

Kepler’s LawsKepler’s Laws

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

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

Kepler’s First LawKepler’s First Law

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

Kepler’s Second LawKepler’s Second Law

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

Kepler's Second Kepler's Second Law InteractiveLaw Interactive

Kepler’s Third LawKepler’s Third Law

• The cube of the average radius , r, of a planet’s orbit The cube of the average radius , r, of a planet’s orbit is directly proportional to the square of its period, Tis directly proportional 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 Note that: Cs is based on the object that is creating the gravitational fieldthe gravitational field

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

Understanding Escape EnergyUnderstanding Escape Energy

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

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

Discuss the energyDiscuss the energy

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

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

You’re in an energy debtYou’re in an energy debt

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

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

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

Escape energyEscape energy

r

E

Radius of planet

How do you get away from the bungee cord? How do you get away from the bungee cord?

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

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

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

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

• See notebook file for derivationSee notebook file for derivation

Tied up in debtTied up in debt

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

• Think about an orbiting satellite: Think about an orbiting satellite: • It remains in orbit around the earth (therefore, it is It remains in orbit around the earth (therefore, it is

still “tied” to the earth – how do we know? What still “tied” to the earth – how do we know? What happens if it stops moving?)happens if it stops moving?)

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

The debt in orbitThe debt in orbit

r

E

Eg

r

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

Etotal

Ek

Total energy of objects in a gravitational fieldTotal energy of objects in a gravitational field

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

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

Case 1 – Object just escapes: ET = 0Case 1 – Object just escapes: ET = 0

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

• All the kinetic All the kinetic energy is used to pay energy is used to pay the energy debtthe energy debt

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

r

E

Eg

Ek

Case 2 – Object escapes with v > 0: ET > 0Case 2 – Object escapes with v > 0: ET > 0

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

r

E

Eg

Ek

Case 3 – Bound object: ET < 0Case 3 – Bound object: ET < 0

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

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

r

E

Eg

Ek