A U - shaped tube contains a liquid of density rho and it is...

A U - shaped tube contains a liquid of density rho and it is rotated about the left dotted line as shown in the figure. Find the difference in the levels of liquid column.

Options

A $\frac{\omega^{2} \mathrm{ L}^{2}}{2 g}$
B $\frac{\omega^{2} L^{2}}{2 \sqrt{2} g}$
C $\frac{2 \omega^{2} \mathrm{ L}^{2}}{g}$
D $\frac{2 \sqrt{2} \omega^{2} \mathrm{ L}^{2}}{g}$

Correct answer

A. $\frac{\omega^{2} \mathrm{ L}^{2}}{2 g}$

Step-by-step solution

Weight of liquid of height

\mathrm{H}=\frac{\pi \mathrmd^{2}}{4} \times \mathrm{H} \rho \times \mathrm{g} \rightarrow) (i) Let us consider a mass dm situated at a distance

from

as shown in the figure. The centripetal force required for the mass to rotate (=(\mathrm{dm}) \omega^{2}) (\therefore) The total centripetal force required for the mass of length

to rotate (=\int_{0}^{1}\left(\rho \mathrm{x} \frac{\pi \mathrmd^{2}}{4}\right) \mathrm{xdx} \times\left(\mathrm{x} \omega^{2}\right)=\rho \times \frac{\pi \mathrmd^{2}}{4} \times \omega^{2} \times \frac{1^{2}}{2} \rightarrow) (ii) This centripetal force required is provided by the weight of liquid of height 'H' From (i) and (ii)

\begin{aligned}& \frac{\pi \mathrmd^{2}}{4} \times \mathrm{H} \times \rho \times \mathrm{g}=\rho \times \frac{\pi \mathrmd^{2}}{4} \times \omega^{2} \times \frac{1^{2}}{2}\\ & \Rightarrow \mathrm{H}=\frac{\omega^{2} \mathrml^{2}}{2 \mathrm{ g}} \end{aligned}

Practice Mechanical Properties of Fluids on Quantrex Academy →

More from Mechanical Properties of Fluids

Pressure inside two soap bubbles are 1.01 and 1.02 atmosphere, respectively. The ratio of… BITSAT 2024 A cube of ice floats partly in water and partly in kerosene oil. The radio of volume of… BITSAT 2024 Eight equal drops of water are falling through air with a steady speed of 10 cm / s . If… BITSAT 2023 If 1000 droplets of water of surface tension 0.07 N / m . having same radius 1 ~mm each… BITSAT 2023 A capillary tube is attached horizontally to a constant heat arrangement. If the radius… BITSAT 2022 A spherical drop of radius R is divided into eight equal droplets. If surface tension is… BITSAT 2021 A body of density ρ^ is dropped from rest at a height h into a lake of density ρ where ρ… BITSAT 2020 A vessel with water is placed on a weighing pan and it reads 600 g . Now a ball of mass… BITSAT 2020 Full Mechanical Properties of Fluids list All BITSAT PYQs