Gravitation

Newton's First Law of Gravitation
Newton's Second Law of Gravitation Escape Velocity Keplar's Law of Planetary Motion


Newton's Law of Gravitation

Apples had a significant contribution to the discovery of gravitation. The English physicist Isaac Newton (1642-1727) introduced the term "gravity" after he saw an apple falling onto the ground in his garden. "Gravity" is the force of attraction exerted by the earth on an object. The moon orbits around the earth because of gravity too. Newton later proposed that gravity was just a particular case of gravitation. Every mass in the universe attracts every other mass. This is the main idea of Newton's Law of Universal Gravitation.


Thus, the law states that every particle in the universe exerts a force on every other particle along the line joining their centers. The magnitude of the force is directly proportional to the product of the masses of the two particles, and inversely proportional to the square of the distances between them.
In the figure, the gravitational force and the external force are equal in magnitude and opposite in direction. 


m1 exerts a force on m2 .
m2 exerts a force on m1 .



By Newton's third law:
F12 = -F21
The magnitude of the gravitational force is: F12 = G(m1*m2/r2).


G is Newton's constant:
G = 6.67 x 10- 11 N m 2 /kg 2


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Newton's Second Law of Gravitation

"Change of motion is proportional to the force applied, and take place along the straight line the force acts."

Newton's Second Law for the gravity force weight can be expressed as
F = m g
where F = force (weight), m = mass and g = acceleration of gravity.


The force caused by gravity g is called weight.
Note: Mass m is a property.


The acceleration of gravity can be observed by measuring the change of velocity of a free falling object:
g = dv / dt
where dv = change in velocity and dt = change in time.


A dropped object accelerate to a speed of 9.81 m/s or 32.174 ft/s in one second.
Acceleration of Gravity in SI Units g = 9.81 m/s2.
Acceleration of Gravity in Imperial Units g = 32.174 ft/s2.



Velocity and Distance travelled of a Free Falling Object

The velocity of a free falling object can be expressed as: v = g t
where v = velocity
The distance traveled by a free falling object can be expressed as: s = 1/2 g t2
where s = distance traveled by the object 

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Escape Velocity

If an object moves fast enough it can escape a massive object's gravity and not be drawn back toward the massive object. The critical speed needed to do this is the Escape Velocity. More specifically, this is the initial speed something needs to escape the object's gravity and assumes that there is no other force acting on the object besides gravity after the initial boost. Rockets leaving the Earth do not have the escape velocity at the beginning but the engines provide thrust for an extended period of time, so the rockets can eventually escape. The concept of escape velocity applies to anything gravitationally attracted to anything else (gas particles in planet atmospheres, comets orbiting the Sun, light trying to escape from black holes, galaxies orbiting each other, etc.).
Using Newton's laws of motion and law of gravity, you can find that the escape velocity vesc looks very similar to the orbital speed:

vesc = Sqrt[(2 G M)/r]
.

This is a factor Sqrt[2] larger than the circular orbital speed. Since the mass M is on top of the fraction, the escape velocity increases as the mass increases. More massive bodies exert greater gravity force, so escaping objects have to move faster to overcome the greater gravity. Also, the distance from the center of the object r is in the bottom of the fraction, so the escape velocity DEcreases as the distance increases. Gravity decreases with greater distance, so objects farther from a massive body do not need to move as quickly to escape it than those closer to it.


Formulae:







Orbital speed = Sqrt[G × Mass / distance].
Escape velocity = Sqrt[2G × Mass / distance]. 
 
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Kepler's Law of Planetary Motion

In the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor - Tycho Brahe - with three statements which described the motion of planets in a sun-centered solar system. Kepler's efforts to explain the underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves are still considered an accurate description of the motion of any planet and any satellite.

Kepler's three laws of planetary motion can be described as follows:

Kepler's first law - sometimes referred to as the Law of Ellipses - explains that planets are orbiting the sun in a path described as an ellipse. The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus.


Kepler's second law - sometimes referred to as the Law of Equal Areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time.

Kepler's third law - sometimes referred to as the Law of Harmonies - compares the orbital period and radius of orbit of a planet to those of other planets.The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. 



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