What is the specific gravity of a marble stone, which weighs 400 N in air, and 200 N in the water? (g = 10m/s²)

Concept:

The weight of the stone in the air is given by, ρ_s × V × g

Where ρ_s is the density of the stone and V is the volume of the stone.

When the stone is immersed in the water there will be buoyancy force acting upon the stone applied by water.

Then the weight of the stone in the water is given by Weight in air – Buoyancy force upon the stone

Buoyancy force = ρ_w × V × g

Where ρ_w is the density of water and V is that volume which is immersed in water

The specific gravity of stone = \(\frac{{Density\;of\;stone\;}}{{Density\;of\;water}}\)

Calculation:

Given W_s-air = 400 N

then ρ_s × V × g = 400 N

∴ ρ_s × V = 40 N      ----(1)

Now when the stone is immersed in water then W_s-water = 200 N

ρ_s × V × g - ρ_w × V × g = 200 N

(ρ_s - ρ_w) × V = 20 N      ----(2)

Now dividing eqn. (2) by eqn. (1)

\(\left( {\frac{{{\rho _s} - {\rho _w}}}{{{\rho _s}}}} \right) = \frac{{20}}{{40}} = \frac{1}{2}\)

\( \Rightarrow 1 - \frac{{{\rho _w}}}{{{\rho _s}}} = \frac{1}{2}\)

⇒ \(\frac{{{\rho _w}}}{{{\rho _s}}} = \frac{1}{2}\)

\(\frac{{{\rho _s}}}{{{\rho _w}}} = 2\)