You are trying to determine the specific gravity of a solid object that floats in water. If m is the mass of your object, mS is the apparent mass of the combination of 2 masses with one (the sinker) submerged, and mOS is the apparent mass of the combination of 2 masses with both submerged, what is the formula for specific gravity

Specific Gravity = m/[m (s) - m (os) ]

Explanation:

Specific gravity, also called relative density, is the ratio of the density of a substance to the density of a reference substance. By this definition we need to find out the ratio of density of the object of mass m to the density of the surrounding liquid.

m = mass of the object

Weight in air

W (air) = mg, where g is the gravitational acceleration

Weight with submerged with only one mass

m (s) g + Fb = mg + m (b) g, consider this to be equation 1

where Fb is the buoyancy force

Weight with submerged with both masses

m (os) g + Fb' = mg + m (b) g, consider this to be equation 2

equation 1 - equation 2 would give us

m (s) g - m (os) g = Fb' - Fb

where Fb = D x V x g, where D is the density of the liquid the object is submerged in, g is the force of gravity and V is the submerged volume of the object

m (s) g - m (os) g = D (l) x V x g

m (s) - m (os) = D (l) x V

we know that Mass = Density x V, which in our case would be, D (b) x V, which also means

V = Mass/D (b), where D (b) is the density of the mass

Substituting V into the above equation we get

m (s) - m (os) = [D (l) x m) / D (b) ]

Rearranging to get the ratio of density of object to the density of liquid

D (b) / D (l) = m/[m (s) - m (os) ], where D (b) / D (l) denotes the specific gravity