Question

The specific gravity of a substance is given by G = DS/DW, where DS is the density of the substance in kg/m3 and DW is the density of water, which is known to be 1000 kg/m3. The density of a particular substance is measured to be DS = 500±5 kg/m3. Estimate the specific gravity, and find the uncertainty in the estimate

Answer 1

Answer:

$G=0.5\pm0.005$

Explanation:

The specific gravity is given by,

$G=\frac{D_S}{D_W}=\frac{500}{1000} =0.5$

Now, in order to calculate the uncertainty (relative error) in G, we must first take log (base e) on both sides of the equation,

$lnG=ln(\frac{D_S}{D_W} )=lnD_S-lnD_W$

Differentiating the above equation,

$\frac{dG}{G}=\frac{dD_S}{D_S}$

The second term is zero because it is known that $D_W=1000kg/m^3$ and hence a constant.

Putting the appropriate values, we get,

$\frac{dG}{G}=\frac{dD_S}{D_S}=\frac{5}{500} =0.01$

Therefore, uncertainty in G = $0.01\times0.5=0.005$