Question 177 of 280

A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is $5cm$ and its rotational speed is $2$ rotations per second, then the difference in the heights between the center and the sides, in $cm,$ will be:

A$0.4$B$2.0$C$1.2$D$0.1$

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Answer: (B) $2.0$

When cylinder is rotated, water will rise at the edge and will dip at the centre. The difference in pressure at $A$ and $O$ is ${P}_{A}-{P}_{0}=\rho gh.$ Due to height difference.

pressure difference ${P}_{1}-{P}_{0}=\rho \cdot {a}_{cm}R$ Due to rotation.

$=\rho \cdot \left({\omega }^{2}.\dfrac{R}{2}\right)R$



$∴\rho gh=\rho \dfrac{{\omega }^{2}{R}^{2}}{2}\Rightarrow h=\dfrac{{\omega }^{2}\cdot {R}^{2}}{2g}={\left(2\pi \times 2\right)}^{2}\times \dfrac{{\left(0.05\right)}^{2}}{2\times 10}$

$=0.02m$

$=2cm$