A thin uniform tube is bent into a circle of radius $r$ in the virtical plane. Equal volumes of two immiscible liquids, whose densities are ${\rho _1}$ and ${\rho _2}\left( {{\rho _1} > {\rho _2}} \right)$ fill half the circle. The angle $\theta$ between the radius vector passing through the common interface and the vertical is
JEE MAIN 2018, Diffcult
Download our app for free and get started
Pressure at interface $A$ must be same from both the sides to be in equilibrium.
$\therefore \left( {R\cos \theta + R\sin \theta } \right){\rho _2}g$
$ = \left( {R\cos \theta - R\sin \theta } \right){\rho _1}g$
$ \Rightarrow \frac{{{\rho _1}}}{{{\rho _2}}} = \frac{{\cos \theta + \sin \theta }}{{\cos \theta - \sin \theta }} = \frac{{1 + \tan \theta }}{{1 - \tan \theta }}$
$ \Rightarrow {\rho _1} - {\rho _1}\,\tan \,\theta = {\rho _2} + {\rho _2}\,\tan \,\theta $
$ \Rightarrow \left( {{\rho _1} + {\rho _2}} \right)\tan \theta = {\rho _1} - {\rho _2}$
$\therefore \theta = {\tan ^{ - 1}}\left( {\frac{{{\rho _1} - {\rho _2}}}{{{\rho _1} + {\rho _2}}}} \right),so\,closest\,answer\,is\left( a \right)$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1The terminal velocity of a copper ball of radius $2.0 \;mm$ falling through a tank of oll at $20\,^{\circ} C$ is $6.5 \;cm s ^{-1} .$ Compute the viscosity of the oil at $20\,^{\circ} C .$ Density of oil is $1.5 \times 10^{3} \;kg m ^{-3},$ density of copper is $8.9 \times 10^{3} \;kg m ^{-3}$View Solution
- 2The flow speeds of air on the lower and upper surfaces of the wing of an aeroplane are $v$ and $2v$ respectively. The density of air is $\rho $ and surface area of wing is $A$ . The dynamic lift on the wing isView Solution
- 3Water is moving with a speed of $5.18 ms^{-1}$ through a pipe with a cross-sectional area of $4.20 cm^2$. The water gradually descends $ 9.66 m $ as the pipe increase in area to $7.60 cm^2$. The speed of flow at the lower level is ....... $ms^{-1}$View Solution
- 4If two liquids of same volume but different densities ${\rho _1}$ and ${\rho _2}$ are mixed, then density of mixture is given byView Solution
- 5A liquid flows through a horizontal tube. The velocities of the liquid in the two sections, which have areas of cross-section ${A_1}$ and ${A_2}$, are ${v_1}$ and ${v_2}$ respectively. The difference in the levels of the liquid in the two vertical tubes is $ h$View Solution
- 6A cylinder containing water up to a height of $25 cm$ has a hole of cross-section $\frac{1}{4}c{m^2}$ in its bottom. It is counterpoised in a balance. What is the initial change in the balancing weight when water begins to flow outView Solution
- 7The velocity of upper layer of water in a river is $36 kmh ^{-1}$. Shearing stress between horizontal layers of water is $10^{-3} Nm ^{-2}$. Depth of the river is $m$. (Co-efficiency of viscosity of water is $10^{-2} \,Pa . s$ )View Solution
- 8A spherical solid ball of volume $V$ is made of a material of density $\rho_1$ . It is falling through a liquid of density $\rho_2 (\rho_2 < \rho_1 )$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v$, i.e., $F_{viscous}= -kv^2 (k >0 )$,The terminal speed of the ball isView Solution
- 9A small wooden ball of density $ \rho$ is immersed in water of density $\sigma $ to depth $h $ and then released. The height $H$ above the surface of water up to which the ball will jump out of water isView Solution
- 10A barometer kept in an elevator reads $76 \,cm$ when it is at rest. If the elevator goes up with some acceleration, the reading will be .......... $cm$View Solution