There is no direct equation connecting Gravitational force and Specific gravity.

Gravitational force is the force with which two bodies of mass ##m_1## and ##m_2## attract each other. If one of the bodies is earth the expression reduces to

##F_Gravity=G(M_e m)/R_e^2##

where ##G## is Universal Gravitational Constant ##M_e and R_e## are the mass and radius of earth respectively and ##m## is the mass of other object.

This equation can be written as first three are constants

##F_Gravity=mgN##

here ##g## is the local due to gravity.

##N## newton being the unit of force. ##F_Gravity## is the weight of body.

Specific gravity is the ratio of the density of a substance to the density of a reference substance.

For solids and liquids reference substance is water at ##4^@C## and at one atmospheric .

For gases it is air at room temperature ##21^@C## and at one atmospheric pressure.

We know that the density ##rho-=mass per unit volume## of the substance under test.

We can therefore write

##Specific Gravity = frac {rho_text{sample}}{rho_{ H_2O}} ##

We see that specific gravity is a dimensionless quantity as it is a ratio of two densities.

Indirectly both are related as follows

Specific gravity can be computed from the expression for weights ##W## of sample and water both of equal volume ##V##

##SG = frac {rho_text{sample}}{rho_(H_2O}} = frac {(m_text{sample}//V)}{(m_{ H_2O}//V)}##

## = frac {m_text{sample}}{m_{ H_2O}} ##

Multiplying and dividing with ##g##

## = frac {m_text{sample}}{m_{ H_2O}} g/g##

or ##SG= frac {W_{V_text{sample}}}{W_{V_{ H_2O}}} ##