First slide

Accuracy; precision of instruments and errors in measurements

Question

You measure two quantities as $A = 1.0 m \pm 0.2 m, B = 0.2 m \pm 0.2 m .$ We should report correct value for $\sqrt{A B}$ is

Moderate

A

$1.4 m \pm 0.4 m$
B

$1.41 m \pm 0.15 m$
C

$1.4 m \pm 0.3 m$
D

$1.4 m \pm 0.2 m$

Solution

Given $A = 1.0 m \pm 0.2 m$
$B = 2.0 m \pm 0.2 m$
$x = \sqrt{A B} = \sqrt{1.0 \times 2.0} = 1.414 m$
Rounding-off to two significant digits,
$x = \sqrt{A B} = 1.4 m$
Now, $\frac{Δ x}{x} = \frac{1}{2} [\frac{Δ A}{A} + \frac{Δ B}{B}]$
$= \frac{1}{2} [\frac{0.2}{1.0} + \frac{0.2}{0.2}] = \frac{0.6}{2 \times 2.0}$
$Δ x = \frac{0.6 \times x}{2 \times 2.0} = 0.15 \times 1.414 = 0.2121$
Rounding-off to one significant digits, $Δ x = 0.2 m$
Thus, $\sqrt{A B} = 1.4 m \pm 0.2 m$

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