Gravitation and Weightlessness - Part I

We have studied about gravitation in the chapter on Force Fields. In this chapter we will study more about gravitation and its effects, difference between mass and weight and what it means to be weightless. Gravitational force has been explained well by Newton’s law of gravitation or universal law of gravitation. This is an inverse squared law that is the gravitational force is inversely proportional to the square of the distance between the two objects attracting each other. Gravitational force is action at a distance and can be thought of as a force field where each and every mass attracts each and every other mass.

What we will study in this chapter :
1. Application of Newton’s law of gravitation
2. Acceleration due to gravity (g)
3. Difference between mass and weight
4. Weightlessness
5. Projectiles

1. Application of Newton’s law of gravitation
Newton’s inverse square law of gravitation states that every body in the universe attracts every other body. The force of attraction is directly proportional to the product of the masses and inversely proportional to the square of the distance between the two masses. The force is known as the gravitational force and is always attractive.

Consider two masses m₁ and m₂ at points A and B. Let the distance between A and B be given by r.  

To write the Newton’s law in an equation form, we have

                                                                        1
F     m₁ m₂                            and       F     
                                                                        r²

Thus

             m₁ m₂
F     
               r²

                                 m₁ m₂
or   F   =    G     
                            r²

where G is the gravitational constant.  It is a universal constant.

Value of G is 6.673 x 10^-11  N.m²/kg². The gravitational constant G is the force of gravitation, which exists between two bodies that have unit masses (1kg each) kept at a unit distance of 1 meter.

Since force is a vector, the above equation can be written in a vector form as

                     m₁ m₂          
   F   =   - G                 r
                             r³

The negative sign indicates that the force is attractive.

Both the masses m₁ and m₂ are pulling each other. The sun is pulling the earth and the earth is pulling the sun with the same force. If the earth is attracting the apple towards it, the apple is also attracting the earth towards itself; but because the mass of the apple is very small as compared to the earth, the movement of the earth is negligible. Also the gravitational pull between objects on the earth on each other is extremely small, therefore the objects are not caused to move towards each other! For most general calculation the negative sign is dropped because one is calculating the magnitude of the force and not its direction.

Example 1:  Calculate the force of gravitation between a mother weighing 40 kg and her child weighing 5 kg. They are seated 1 m apart from each other. (G = 6.673 x 10^-11   N.m²/kg²).

The force of gravitational pull between the mother and her child is given by:

                                 m₁ m₂
   F   =    G      
                            r²

m₁ = 40 kg,      m₂ = 5 kg,        r = 1m

  6.673 x 10^-11   (40) x (5)
F  =         Newton
                        1 x 1

F =  1.33 x 10^-8 N

Thus the force exerted between the mother and child is extremely small, almost insignificant.

Example 2 :  Calculate the gravitational pull by the earth on the child and mother, given in example 1,  separately. The mass of the earth is 6 x 10²⁴ kg and the radius of the earth is 6.4 x 10³ km. (G = 6.673 x 10^-11   N.m²/kg²).

When calculating earth’s gravitational pull, it is conventionally accepted that all the mass of the earth is concentrated at its center. The second mass, in this case either the child or the mother, are placed at a distance R = radius of the earth, from the center of mass of the earth.

The force of gravity on the child is given by

                                m_child m_earth
F_child   =    G   
                                r²

m_earth = 6 x 10²⁴ kg,      m_child = 5 kg,     r = 6.4 x 10³ km = 6.4 x 10⁶ m

                 6.673 x 10^-11   (6 x 10²⁴) x (5)
F_child   =          Newton  
                         6.4 x 10⁶   x  6.4 x 10⁶

            =   48.8 N

The force of gravity on the mother is given by

                                          m_mother m_earth
   F_mother  =    G   
                                          r²

m_earth = 6 x 10²⁴ kg,      m_mother = 40 kg,             r = 6.4 x 10³ km = 6.4 x 10⁶ m

                       6.673 x 10^-11   (6 x 10²⁴) x (40)
    F_mother   =            Newton
                                 6.4 x 10⁶   x  6.4 x 10⁶

            =   390.9 N

Comparing the results from example 1 and 2 we see that the gravitational pull between the mother and child is extremely small when compared to the gravitational pull of the earth on the child and mother.

Example 3 : Calculate the gravitational force between the earth and the moon, given that the mass of the earth is 6 x 10²⁴ kg and the mass of the moon is 7.3 x 10²² kg. The distance between the earth and the moon is 3.84 x 10⁵ km. (G = 6.673 x 10^-11   N.m²/kg²).

The force of gravity between the earth and the moon is given by :

                                  m_moon m_earth
    F   =    G   
                                 r²

m_earth = 6 x 10²⁴ kg,      m_moon = 7.3x 10²²  kg,      r =  3.84 x 10⁵ km = is 3.84 x 10⁸ m

              6.673 x 10^-11   (6 x 10²⁴) x (7.3x 10²²)
  F        =           Newton  
                              3.84 x 10⁸ x  3.84 x 10⁸

F =  1.98 x 10²⁰ N

Thus the force exerted by the earth on the moon and vice versa is extremely large. This force keeps the moon in motion around the earth.

From comparing the results of examples given above, we can appreciate the nature of gravitational forces between masses. The magnitude of the gravitational force is large or enormous when the masses are large.  

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