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 $\Delta a(\Delta a / a \ll 1)$, then what is the error $\Delta r$ in determining $r$ ? (a) $\frac{\Delta a}{(1+a)^{2}}$ (b) $\frac{-2 \Delta a}{(1+a)^{2}}$ (c) $\frac{2 \Delta a}{(1-a)^{2}}$ (d) $\frac{2 a \Delta a}{\left(1-a^{2}\right)}$
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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
or
$$ \begin{aligned} \frac{\Delta r}{r} & =-\frac{\Delta a}{1-a}+\frac{\Delta a}{1+a} \ \frac{\Delta r}{r} & =\frac{-2 \Delta a}{1-a^{2}} \ \Delta r & =-\frac{2 \Delta a}{1-a^{2}} \quad(r)=\frac{-2 \Delta a}{(1+a)^{2}} \end{aligned} $$
