First slide

Accuracy; precision of instruments and errors in measurements

Question

The mass and density of a solid sphere are measured to be $(12.4 \pm 0.1) kg$ and $(4.6 \pm 0.2) kg / m^{3}$ . Calculate the volume of the sphere with error limits.

Moderate

A

$(2.7 \pm 0.10) m^{3}$
B

$(2.7 \pm 0.1) m^{3}$
C

$(2.7 \pm 0.14) m^{3}$
D

$(7.2 \pm 0.14) m^{3}$

Solution

Here, $m \pm Δ m = (12.4 \pm 0.1) kg$
$ρ \pm Δ ρ = (4.6 \pm 0.2) kg / m^{3}$
Volume, $V = \frac{m}{ρ} = \frac{12.4}{4.6}$
$= 2.69 m^{3} = 2.7 m^{3}$ (rounding off to one decimal place)
$\frac{Δ V}{V} = \pm (\frac{Δ m}{m} + \frac{Δ ρ}{ρ})$
$Δ V = \pm (\frac{Δ m}{m} + \frac{Δ ρ}{ρ}) \times V$
$= \pm (\frac{0.1}{12.4} + \frac{0.2}{4.6}) \times 2.7$
$= \pm 0.14$
$V \pm Δ V = (2.7 \pm 0.14) m^{3}$

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