SATHEE: General Physics Question 2

General Physics Question 2

Question 2

Consider the ratio $r = \frac{(1 - a)}{(1 + a)}$ to be determined by measuring a dimensionless quantity $a$ . If the error in the measurement of $a$ is $Δ a (Δ a / a ≪ 1)$ , then what is the error $Δ r$ in determining $r$ ? (a) $\frac{Δ a}{(1 + a)^{2}}$ (b) $\frac{- 2 Δ a}{(1 + a)^{2}}$ (c) $\frac{2 Δ a}{(1 - a)^{2}}$ (d) $\frac{2 a Δ a}{(1 - a^{2})}$

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Answer:

Correct Answer: 3. (b)

Solution:

$\begin{aligned} r & = \frac{1 - a}{1 + a} r & = \ln (1 - a) - \ln (1 + a) \end{aligned}$

Differentiating, we get

$\frac{d r}{r} = - \frac{d a}{1 - a} - \frac{d a}{1 + a}$

or, we can write

$\begin{aligned} \frac{Δ r}{r} & = - \frac{Δ a}{1 - a} + \frac{Δ a}{1 + a} \frac{Δ r}{r} & = \frac{- 2 Δ a}{1 - a^{2}} Δ r & = - \frac{2 Δ a}{1 - a^{2}} (r) = \frac{- 2 Δ a}{(1 + a)^{2}} \end{aligned}$

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General Physics Question 2

Question 2

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