General Physics Question 22
Question 22
- To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railway engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency $f$. The engineer finds that $d$ is proportional to $S^{1 / n}$. The value of $n$ is
(2014 Adv.)
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Answer:
Correct Answer: 30. (d)
Solution:
- Let $d=k(\rho)^{a}(S)^{b}(f)^{c}$
where, $k$ is a dimensionless. Then,
$$ [\mathrm{L}]=\frac{\mathrm{M}^{a}}{\mathrm{~L}^{3}} \frac{\mathrm{ML}^{2} \mathrm{~T}^{-2}}{\mathrm{~L}^{2} \mathrm{~T}} \quad \frac{1}{\mathrm{~T}} $$
Equating the powers of $M$ and $L$, we have
$$ \begin{aligned} & 0=a+b \ & 1=-3 a \end{aligned} $$
Solving these two equations, we get
$$ b=\frac{1}{3} \Rightarrow \therefore n=3 $$
