Question

differentiate between acceleration due to gravity and universal gravitation constant derive a relationship between G and g​

Answer 1

Gravitational force is given by

F = GMm/r^2

Here F = ma = mg

Therefore,

mg = GMm/r^2

Divide both sides by m

g = GM / r^2

In above g and G relation

g = acceleration due to gravity of planet

G = universal gravitational constant

M = mass of planet

r = radius of planet

Universal gravitational constant(G) and acceleration due to gravity(g) are two different things having different meaning, units, values and uses(the only similarity can be that both are related to Gravitation). G is a constant, the value of which was obtained by Henry Cavendish, whereas g is an acceleration that is involved whenever an object is falling or ascending in the earth’s atmosphere. Moreover, what G and g are and how these value are obtained can be understood as :

Universal gravitational constant(G) : All the objects that exist in universe attract other objects. The attractive force that exists between any two objects(say A and B, as in picture below) is directly proportional to the product of their masses and inversely proportional to the square of distance between their center(center of gravity or centroid). The value of G was obtained by sensitive balance in laboratory, firstly by Cavendish.

Answer 2

Difference Between Gravity and Gravitation :-

Gravitation refers to the attractive force between two masses in general.

Gravity refers to the resultant force between a mass and Earth. The resultant force also accounts for other forces present in the system, for instance, the force on the mass due to the Earth’s rotation.

Relationship between g and G:-

Explanation:

Suppose Earth is a sphere of radius

r

It has mass M

Gravitational Force on the object of mass m which is situated at a distance r

from the center of Earth is

F=GMm/r^2 . . .[A]

If the object is free falling from the height h from the surface of earth (or at a distance r from the center of earth) it experience the acceleration g

According to Newton's second law

Force on the object due to acceleration g is

F=mg . . .[B]

Comparing Equation [A] and [B]

mg = GMm/r^2

(where m is cut in both side)

gr^2 = GM

Where

r = h+R

R=Radius of Earth

h=height of object from the surface of earth

If h<< R

then we can write

r=R+h≈R

or

gR^2 =GM . . .[C]

(When the object is near the surface of earth so we can neglect the height of object comparing with Radius of Earth)

If average density of earth is ρ

then mass of earth

M=Volume×density

M

=

4/3πR^3ρ

In equation [c]

gR^2=G4/3πR^3ρ

g=G4/3πR^3ρR^2

g=4/3πGRρ

HOPE IT HELP YOU AND MARK THE BRAINLIEST.