There is no direct equation connecting Gravitational force and Specific gravity.
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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.
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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”_2″O”}} ## 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}}} ##
