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# In the following experiment, find the value of the density in g cm$^{-3}$ up to the appropriate significant figures, stating the uncertainty in the value of the density.

While calculating the density of a rectangular block, a student makes the following observations:

Mass of the block (m) = 39.3 g

Length of the block (l) = 5.12 cm

Breadth of the block (b) = 2.56 cm

Thickness of the block (t) = 0.37 cm

The uncertainty of measurement of m is $\pm$ 0.1g, and in the measurement of l, b and t it is $\pm$ 0.01 cm.

Density $\rho = \large\frac{\text{Mass}}{\text{ Volume}}$$= \large\frac{\text{m}}{\text{ l x b x t }}$$ = \large\frac{39.3}{5.12\times2.56\times\0.37}$$= 8.1037 \;g\;cm^{-3} Now, \large\frac{\Delta \rho}{\rho}$$ = \large\frac{\Delta m}{m} $$+ \large\frac{\Delta l}{l}$$+ \large\frac{\Delta b}{b} $$+ \large\frac{\Delta t}{t} \quad \quad = \large\frac{ 0.1}{39.2}$$+ \large\frac{0.01}{5.12} $$+ \large\frac{0.01}{2.56}$$+ \large\frac{0.01}{0.37}$$=0.0353$
$\Rightarrow \Delta \rho = 0.0353 \times \rho = 0.0353 \times 8.1037 = 0.286\; g \;cm^{-3}$
Rounding off to the first significant digit, $\Delta \rho = 0.3 \; g\; cm^{-3}$
$\Rightarrow \rho = 8.1037\;g\;cm^{-3}$ is not accurate to the fourth decimal space and needs to be rounded off to the first decimal point to be accurate.
Hence, $\rho = (8.1 \pm 0.3) \;g \;cm^{-3}$

edited Jul 12, 2014