First slide

Accuracy; precision of instruments and errors in measurements

Question

The focal length of a spherical mirror from the following observations : object distance $u = (50.1 \pm 0.5)$ cm and image distance $v = (20.1 \pm 0.2) cm$ is

Moderate

A

$(14.3 \pm 0.2) c m$
B

$(14.3 \pm 0.1) c m$
C

$(14.3 \pm 0.4) c m$
D

$(14.3 \pm 0.3) c m$

Solution

Given
Object distance $u = (u \pm Δ u) = (50.1 \pm 0.5) cm$
Image distance $v = (v \pm Δ v) = (20.1 \pm 0.2) cm$
From mirror formula
$\frac{1}{F} = \frac{1}{v} + \frac{1}{u} \Rightarrow \frac{1}{F} = \frac{1}{50.1} + \frac{1}{20.1}$
$∴ F = 14.34 cm$
and $|\frac{Δ F}{F^{2}}| = |\frac{Δ v}{v^{2}}| + |\frac{Δ u}{u^{2}}|$
$\Rightarrow Δ F = Δ u {(\frac{F}{u})}^{2} + Δ v {(\frac{F}{v})}^{2}$
$= 0.5 {(\frac{14.34}{50.1})}^{2} + 0.2 {(\frac{14.34}{20.1})}^{2}$
$= 0.04 + 0.10 = 0.14 cm$
Hence, $F = (14.3 \pm 0.1) cm$

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