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Q.

A U-tube filled with liquid is being rotated about an axis with angular speed ω as shown in the figure. The difference in heights of liquid in two arms is

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a

$\frac{ω^{2}}{2 g} (r_{1}^{2} - r_{2}^{2})$

b

$\frac{ω^{2}}{g} (r_{1}^{2} - r_{2}^{2})$

c

$\frac{2 ω^{2}}{g} (r_{1}^{2} + r_{2}^{2})$

d

$\frac{ω^{2}}{2 g} (r_{1}^{2} + r_{2}^{2})$

answer is A.

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Detailed Solution

Let us take a small element of length dr from axis of rotation
$\begin{aligned} (P + dp) - P = {ρω}^{2} rdr \\ \Rightarrow P_{1} - P_{2} = {ρω}^{2} \int_{r_{1}}^{r_{2}} rdr \\ \Rightarrow P_{1} - P_{2} = \frac{{ρω}^{2}}{2} (r_{1}^{2} - r_{2}^{2}) \\ \Rightarrow h_{1} - h_{2} = \frac{ω^{2}}{2 g} (r_{1}^{2} - r_{2}^{2}) \end{aligned}$

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