Gravitation

Gravitation

Newton's law of universal gravitation Derivation of Newton's law of universal gravitation

      

Newton's law of gravitation, states that every particle of matter in the universe attracts every other particle in the universe with a force varying directly as the product of their masses and inversely as the square of the distance between the particles.
Let there are two objects of mass m₁ and m₂ and the distance between the objects is r, then according to law of universal attraction, the attraction between the objects
    F  Eq-1
          Eq-2
By combining Eq-1 and Eq-2 we get
       
    
Where G is constant which is called Universal Gravitational constant & the value is 6.67 × 10^-11.

Relation Between Universal Law Of Gravitation (G) and Acceleration due to gravity (g):

Let us consider the earth to be made of large number of concentric spherical shells. The total mass of the shells is just the mass of the earth.
 The gravitational force due to any shell on a point as situated outside is just as if the entire mass of the shell is concentrated at the centre of the shell.But the force of attraction due to any shell on a point mass situated inside it is zero.
    
Consider a point mass ‘m’ situated inside the earth,at a distance ‘r’ from the centre.The point p lies outside the sphere of radius’r’.For certain shells,whose radius greater than ‘r’,the point p is inside poin and these shells exert no gravitational force on the mass ‘m’ at p.But,the shells with radius are less than or equal to ‘r’ are assumed to be a sphere of radius’r’,for which the point p lies on the surface.Hence a gravitational force exert on the mass ‘m’ at p by the earth of radius ‘r’.
If M_r is the mass of the assumed sphere then,the force on the mass ‘m’ at p is given by F=   
If  R_E (or) R is the radius of the earth and mass of the earth is M_E (or)M,then density  is given by =  
Similarly the mass of the sphere of ‘r’ is given by
       
 

If the mass ‘m’ is situated on the surface of earth then r = R_E then the gravitational force on the mass ‘m’ placed on the surface of the earth
          
The force acting on the body due to gravitational pull of the earth is F=mg_E from (4) and (5) mg_E =
          
This relation is valid for any planet including the earth

a)In terms of Radius (R) and Mean Density    
we know that   
         

b)In terms of Mass (M) and Density \(\rho\) 
As \( g = \frac{{Gm}} {{R^2 }} \)  
\( \begin{gathered} M = \left[ {\frac{4} {3}\Pi R^3 } \right]\rho \Rightarrow R^2 = \left[ {\frac{{3M}} {{4\Pi \rho }}} \right]^{\frac{2} {3}} \hfill \\ \Rightarrow g = \frac{{GM}} {{\left( {\frac{3} {{4\Pi }}} \right)^{\frac{2} {3}} .\frac{{M^{\frac{2} {3}} }} {{\rho ^{\frac{2} {3}} }}}} \hfill \\ \Rightarrow g = \left[ {\frac{{4\Pi }} {3}} \right]^{\frac{2} {3}} G.M^{\frac{1} {3}} \rho ^{\frac{2} {3}} \hfill \\ \end{gathered} \)