First slide

Accuracy; precision of instruments and errors in measurements

Question

The relative density of a material is found by weighing the body first in air and then in water. If the weight in air is (10.0 $\pm$ 0.1) gf and the weight in water is (5.0 $\pm$ 0.1) gf, then the maximum permissible percentage error in relative density is

Moderate

A

1
B

2
C

3
D

5

Solution

$Relative density = \frac{W_{a}}{W_{a} - W_{w}}, ρ = \frac{W_{a}}{w}$
where $ρ$ is relative density, W_a is weight in air, and w is loss in weight.
$\frac{Δ ρ}{ρ} = \frac{Δ W_{a}}{W_{a}} - \frac{Δ w}{w}$
$For maximum error, \frac{Δ ρ}{ρ} = \frac{Δ W_{a}}{W_{a}} + \frac{Δ w}{w}$
For maximum percentage error,
$\frac{Δ ρ}{ρ} \times 100 = \frac{Δ W_{a}}{W_{a}} \times 100 + \frac{Δ w}{w} \times 100$
$Given {ΔW}_{a} = 0.1 gf and W_{a} = 10.0 gf$
$\begin{aligned} \begin{array}{lc} w = 10.0 - 5.0 = 5.0 gf \\ Δ w = Δ W_{a} + Δ W_{w} = 0.1 + 0.1 = 0.2 gf \\ \frac{Δ ρ}{ρ} \times 100 = (\frac{0.1}{10.0}) \times 100 + (\frac{0.2}{5.0}) \times 100 \\ = 1 + 4 = 5 \end{array} \end{aligned}$

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