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GRAVITATIONAL FIELDS

 


EG: Gravitational potential energy of the body with respect to central body (J)
G:  universal gravitation constant (G = 6.67 x 10-11 N.m2/kg2)
(same everywhere in the universe)
m:  Mass of a body influenced by the gravitational field of the central body (kg)
M:  Mass of the central body (kg)

r:  Distance between the centres of the two bodies (m)
g:  Gravitational field intensity (N/kg)

v:  Orbital speed (m/s)

Mass of sun:  mE = 1.98 x 1030 kg
Radius sun:  rE = 6.95 x 108 m
Gravitational field intensity close to the surface of the sun = 270.00 N/kg

Mass of the Earth:  mE = 5.98 x 1024 kg
Radius of the Earth:  rE = 6,38 x 106 m
Gravitational field intensity close to the surface of the earth = 9.80 N/kg

Mass of the moon:  mE = 7.34 x 1022 kg
Radius of the moon:  rE = 1.74 x 106 m
Gravitational field intensity close to the surface of the moon = 1.62 N/kg

 

FORCE OF GRAVITY BETWEEN TWO BODIES

 

 

Gravitational field intensity:

 

 

CALCULATION OF GRAVITATIONAL FIELD INTENSITY ON THE SURFACE OF THE EARTH:

g:  gravitational field intensity on the surface of the Earth
mE:  Mass of the Earth (mE = 5.98 x 1024 kg)
rE:  Radius of the Earth (rE = 6,38 x 106 m)

          

When the above given values are substituted into the formula,

gravitational field intensity on the surface of the Earth will be calculated as g = 9,8 N/kg.

 

 

GRAVITATIONAL POTENTIAL ENERGY

Gravitational potential energy is always negative.  As r approaches to infinity, EG approaches to zero.

 

CHANGE IN GRAVITATIONAL POTENTIAL ENERGY BETWEEN TWO POSITIONS

∆E:  Change in gravitational potential energy

 

APPROXIMATE FORMULA FOR THE CHANGE IN GRAVITATIONAL POTENTIAL ENERGY

Approximate formula can be used in case the difference between r1 and r2 is not much, or ∆h is not very large.

∆h:  The change in elevation (m)

 

 

GENERAL FORMULA FOR KINETIC ENERGY:

 

TOTAL ENERGY:

ET = EG + Ek

 

 

ORBITAL SPEED

v:  Orbital speed (m/s): Orbital speed is the speed is required for an object to stay in orbit above a central body.  Orbital speed of the object is undergoing uniform circular motion (m/s).  Orbital speed is a constant speed and perpendicular to the gravitational force.

 

FG = FC

 

ORBITAL KINETIC ENERGY 

 

GRAVITATIONAL POTENTIAL ENERGY

 

ORBITAL TOTAL ENERGY

ET = EG + Ek

 


 

ESCAPE FROM A GRAVITATIONAL FIELD

ESCAPE ENERGY

To escape an object from the orbit of the central body, the objects' total energy must be greater than zero.  Since the gravitational potential energy of the object is always negative, and depends on the position of the object, the kinetic energy must be increased to make total energy greater than zero.  This additional kinetic energy is called the binding energy.  when the object escape from the central body, it will no longer to be bound to the central body.

BINDING ENERGY

In general, binding energy is the energy required to disassemble a whole into separate parts.  Gravitational binding energy is the amount of additional kinetic energy needed by an orbiting object to escape from the orbit of the central body.  To escape the orbiting object from the potential well of a central body, total mechanical energy must be greater than zero (positive value).  Therefore, the formula for binding energy:

Binding Energy = 0 - total mechanical energy = - total mechanical energy

Example 1:  Finding binding energy of an object currently orbiting around a central body

Total energy of object of orbiting around the central body:

 

 

Since the orbital gravitational potential energy does not change at the given location, kinetic energy must be increased to make the orbital total energy greater than zero.

Binding Energy = G M m / 2r

 

Example 2:  Finding the binding energy of an object at rest on the surface of the Earth

Since the object at rest, kinetic energy of the object equals to zero and total energy of the object equals to gravitational potential energy of the object.

ET = - G ME m / rE

Binding Energy = 0 - total mechanical energy = - total mechanical energy = Binding Energy = G ME m / rE

 

ESCAPE VELOCITY

Escape velocity is the minimum velocity needed for an object to escape the gravitational pull of another object at a given distance.  To escape an object from the potential well of a central body, total mechanical energy must be greater than zero (positive value).  At the escape velocity, any object will escape from the central body and never come back again to the central body.

The escape speed does not depend on the direction in which a projectile is fired from a planet.  Direction of the escape speed is not important.  Unless the object hits the central body, the object will escape if its speed is greater than escape speed.  Escape speed from the earth surface is 11.2 km/s.
 

ET = EK + EG = 0

EK = - EG

 

v:  Escape speed (m/s)
MMass of the central body (kg)
m:  Mass of a body influenced by the gravitational field of the central body (kg)
rDistance between the centres of the two bodies (m)

 

 

KEPLERíS LAWS OF PLANETARY MOTION:

1. The planets move around the sun in elliptical orbits, with the sun at one focus of the ellipses.

2. The straight line joining the sun and a given planet sweeps out equal areas in equal intervals of time.
 

3.  The square of the period of revolution of a planet about the sun is proportional to the cube of its mean distance from the sun.

C:  The constant of proportionality of the central body (m3/s2)
G:  universal gravitation constant (G = 6.67 x 10-11 N.m2/kg2)
M:  Mass of the central body (kg)
r:  Distance between the centres of the two bodies (m)
T:  Period of the object undergoing uniform circular motion (s)

The constant of proportionality of the central body (general equation) 
(also called Kepler constant, K)

 

 

The constant of proportionality of the sun:  Cs = 3.355 x 1018 m3/s2