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Gravitational Force

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The motion of stars and planets is in important ways simpler than other mechanical phenomena, because there is no friction to worry about. These massive bodies interact through the gravitational force, which is always an attractive force. The gravitational force is always pointing towards the other body. Gravitational forces can have significant impacts on the human body. As the gravitational force increases, breathing becomes labored. Let us discuss the more details about the gravitational force in the following sections.

Definition

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Gravitational force is a natural force by which all objects in the universe attract each other. This force is responsible for the weight of an object. Because of the attractive force every object falls to the ground when it is dropped.

Equation

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We can formulate the equation for gravitational force from the above statement. The equation can be written as,

F = $\frac{Gm_{1}m_{2}}{d^{2}}$

Where F is the gravitational force
          G is the universal gravitation constant (6.67$\times10^{-11}$Nm2/kg2)
          $m_{1}$ and $m_{2}$ are masses of two objects
          d is the distance between two centers of the masses

Gravitational Force Equation

Gravitational Force Inside an Object

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According to the standard theory, all the material inside the earth uniformly attracts a body which is outside of the earth. It is possible for a local gravity variations exist because of the terrain or the material in the ground. For a strong attractive force to exist inside the earth in a particular place, the material that is causing the strong gravity has to be enormous in size as well as in mass according to the prevailing theories. The distance of the material has to be close to the surface, so that the gravity exerted by the material will be strong enough to cause the gravitational effect on the objects on the surface of the earth.

Gravity of Earth

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Rain falls from the clouds, skydivers plunge towards the earth and balls thrown into the air return to the ground. All objects are pulled towards the center of the earth by the force of gravity. The earth is not the only body that exerts gravitational force. The sun's gravitational force holds the planets in their regular orbits. In fact, every mass exerts a gravitational attraction on all other masses. The existence of gravitational force between any two bodies is a fundamental law of the universe. The gravitational force decreases as objects get farther apart. Studies of planetary motion show that this decrease in force depends on the square of the separation between objects. Gravitational force also depends on the masses of the interacting bodies.

Gravity of Earth

Newton's Law of Universal Gravitation

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Sir Isaac Newton invented that there must be an attractive force associated with a the gravitational interaction between two objects. Gravitational force acting between two objects have a linear connection with the product of the masses and inverse relation to the square of of the distance between the centers of the two objects. The gravitational force exerted on object 2 by object 1 is expressed by gravitational force equation:

F = $\frac{Gm_{1}m_{2}}{r^{2}}$

The exact statement of the law is:

Every particle of matter in the universe attracts every other particle of matter with a force whose magnitude is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Magnitude 

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Suppose a body of mass m is in free fall with the free fall acceleration of magnitude g. Then if we neglect the effects of the air, the only force acting on the body is the gravitational force.We can relate this downward force and a downward acceleration with Newton's second law (F = ma), in this case F = mg
In words, the magnitude of the gravitational force is equal to the product mg.
This same gravitational force, with the same magnitude, still acts on the body even when the body is not in free fall but is, at rest on a pool table or moving across the table.

Work done 

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The work done on an object by a particular type of constant force namely, the gravitational force. Suppose a particle like object of mass m, is thrown upward with an initial speed of v1. As it rises, it is slowed by a gravitational force that acts downward in the direction opposite the object's motion. We expect that Fgrav does negative work on the object as it rises because the force is in the direction opposite the motion.

The work done by the gravitational force is positive when the force and the displacement of the object are in the same direction and the work done by the force is negative when the force and displacement are in opposite directions.

Problems

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Let us discuss the problems related to the gravitational force.

Solved Examples

Question 1: Calculate the gravitational force between 2.5$\times10^{-5}$kg and 3$\times10^{-2}$ objects and the distance between the centers are given as 1.5$\times10^{-3}$m?

Solution:
 
Given that,
$m_{1}$ = 2.5$\times10^{-5}$kg, $m_{2}$ = 3$\times10^{-2}$, d = 1.5$\times10^{-3}$m and G = 6.67$\times10^{-11}$Nm2/kg2
Formula for gravitational force is,

F = $\frac{Gm_{1}m_{2}}{d^{2}}$

F = $\frac{6.67\times10^{-11}\times2.5\times10^{-5}\times3\times10^{-2}}{(1.5\times10^{-3})^{2}}$ = 2.22$\times10^{-11}$N

 

Question 2: Calculate the gravitational force between 5$\times10^{-5}$kg and 10$\times10^{-3}$ objects and the distance between the centers are given as 3$\times10^{-6}$m?

Solution:
 
Given that,
$m_{1}$ = 5$\times10^{-5}$kg, $m_{2}$ = 10$\times10^{-3}$, d = 3$\times10^{-6}$m and G = 6.67$\times10^{-11}$Nm2/kg2
Formula for gravitational force is,

F = $\frac{Gm_{1}m_{2}}{d^{2}}$

F = $\frac{6.67\times10^{-11}\times5\times10^{-5}\times10\times10^{-3}}{(3\times10^{-6})^{2}}$ = 3.75$\times10^{-6}$N