Question

A vertical U-tube is rotated about the vertical axis passing through the left limb, with an angular speed $\omega$ such that the liquid columns in the limbs have height $hand3h$ where h is 205 mm, as shown in figure. The value of $\omega$(in rad/s) is(Take $g=10m{s}^{-2}$) (Answer upto two digit after the decimal point)

Answer 1

Taking an element at distance r from the axis as shown in figure.
The pressure difference across the element,
$dp=\left({\rho }_{liq}{\omega }^{2}r\right)dr$
On integrating (from $4h$ density changes, therefore integrating it separately), we get
$P_2-P_1=P_o+6\rho gh-P_o-\rho gh=5\rho gh\Rightarrow 5\rho gh=\rho {\int }_0^{4h}\left({\omega }^{2}r\right)dr+2\rho {\int }_{4h}^{7h}\left({\omega }^{2}r\right)dr$
$\Rightarrow 5g=41{\omega }^{2}h\omega =\sqrt{\frac{5g}{41h}}=2.44rad/s$