GATE 2024 ME – Question 51
A liquid fills a horizontal capillary tube whose one end is dipped in a large pool of the liquid. Experiments show that the distance $L$ travelled by the liquid meniscus inside the capillary in time $t$ is given by
$$L = k\gamma^aR^b\mu^c\sqrt{t},$$
where $\gamma$ is the surface tension, $R$ is the inner radius of the capillary, and $\mu$ is the dynamic viscosity of the liquid. If $k$ is a dimensionless constant, then the exponent $a$ is __________ (*rounded off to 1 decimal place*).
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Show answer and explanation
Correct answer: 0.49 to 0.51
Explanation
In terms of mass $M$, length $L$ and time $T$: $\gamma = MT^{-2}$, $R = L$, $\mu = ML^{-1}T^{-1}$ and $\sqrt{t} = T^{1/2}$, and the left side is $L$. Matching powers of $M$: $a + c = 0$. Of $L$: $b - c = 1$. Of $T$: $-2a - c + \frac{1}{2} = 0$. From the first, $c = -a$, so the third becomes $-2a + a + \frac{1}{2} = 0$ and $a = \frac{1}{2}$ (with $c = -\frac{1}{2}$ and $b = \frac{1}{2}$).