`implies` One day in 1665, Newton observing an apple falling from a tree, he was inspired to arrive at an universal law of gravitation.
`implies` Newton.s reasoning was that the moon revolving in an orbit of radius R Moon was subject to a centripetal acceleration due to earth.s gravity of magnitude.
`a_m=v^2/R=(4pi^2R_m)/T^2" "[:.v=Romega=Rxx(2pi)/T]`
`R_m = 3.84 xx 10^(8)m`
`implies` The time period of revolution of moon around earth T = 27.3 days `= 27.3 xx 86,400` S
`implies` Speed of moon
`v=("circumference orbit of rotation")/("orbital periodic time")`
`= (2piR_m)/T`
`=(2pixx(3.84xx10^8m))/(27.3 xx 86,400 S )`
`= 1.02 xx 10^(3) ` m/s
`implies` Centripetal acceleration of moon,
`a_m =v^2/R_m=((1.02 xx10^3)^2)/(3.84 xx10^8)`
`= 2.70 xx 10^(-3) m//s^2" "...(1)`
`implies` Acceleration due to gravity on the surface of earth,
` g = 9.8 m//s^(2) " "...(2)`
`implies` From equation (1) and (2) , `a_m lt lt g`
`implies` It is clearly shows that gravitational force of earth decreases with distance. (The distance of moon from earth is large and hence gravitational acceleration due to earth is small).
`implies` From the magnitude of `a_m and g`, Newton shows that gravitational force is inversely proportional to the square of distance
so `a_m prop 1/(Rm^2) " "..(3)`
`implies` and acceleration due to gravity on the surface of earth,
`g propR_(E^2)" "....(4)`
where `R_m` = distance of moon from the earth
`R_E` = radius of earth
`implies` Taking ratio of equation (3) and (4),
`a_m/g=(1/R_m^2)/(1/R_E^2)`
`:.a_m/g= ((R_E)/R_m)^(2) " ".....(4)`
Newton know thaat `R_E/R_m = 1/60`
`:. a_m = g((R_E)/R_m)^2`
`:. a_m=(9.8)(1/60)^2`
`a_m = 2.72 xx 10^(-3)m//s^2 " "...(5)`
`implies` Value of `a_m` is obtained from inverse of square of distance. This value is similar to the value obtain in equation (1). From this Newton declared that gravitational force between two bodies is inverse proportional to the square of distance between them.
Gravitational force `F prop 1/r^2 " "..(6)`
and distance remains constant, the gravitation force directly proportional to the product of masses of two bodies.
`F prop m_(1) m_(2) " "...(6)`
`implies` From this Newton.s universal law of gravitation is as below.
"Every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them". [The direction of this force is along the line joining the two bodies).
