Water is filled in a cylindrical container to a height of $3m. $ The ratio of the cross-sectional area of the orifice and the beaker is $ 0.1. $ The square of the speed of the liquid coming out from the orifice is ....... $m^2/s^2$ ($g = 10 m/s^2$)
IIT 2004, Diffcult
Download our app for free and get started
(a)Let $A = $ cross-section of tank
$a = $ cross-section hole
$V =$ velocity with which level decreases
$v = $ velocity of efflux
From equation of continuity $av = AV \Rightarrow V = \frac{{av}}{A}$
By using Bernoulli's theorem for energy per unit volume
Energy per unit volume at point A
= Energy per unit volume at point B
$P + \rho gh + \frac{1}{2}\rho {V^2} = P + 0 + \frac{1}{2}\rho {v^2}$
==> ${v^2} = \frac{{2gh}}{{1 - {{\left( {\frac{a}{A}} \right)}^2}}} = \frac{{2 \times 10 \times (3 - 0.525)}}{{1 - {{(0.1)}^2}}} = 50{(m/\sec )^2}$
$a = $ cross-section hole
$V =$ velocity with which level decreases
$v = $ velocity of efflux
From equation of continuity $av = AV \Rightarrow V = \frac{{av}}{A}$
By using Bernoulli's theorem for energy per unit volume
Energy per unit volume at point A
= Energy per unit volume at point B
$P + \rho gh + \frac{1}{2}\rho {V^2} = P + 0 + \frac{1}{2}\rho {v^2}$
==> ${v^2} = \frac{{2gh}}{{1 - {{\left( {\frac{a}{A}} \right)}^2}}} = \frac{{2 \times 10 \times (3 - 0.525)}}{{1 - {{(0.1)}^2}}} = 50{(m/\sec )^2}$
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
- 1An object with uniform density $\rho$ is attached to a spring that is known to stretch linearly with applied force as shown below.When the spring object system is immersed in a liquid of density $\rho_1$ as shown in the above figure, the spring stretches by an amount $x_1\left(\rho > \rho_1\right)$. When the experiment is repeated in a liquid of density $\left(\rho_2 < \rho_1\right)$, the spring stretches by an amount $x_2$. Neglecting any buoyant force on the spring, the density of the object isView Solution
- 2View SolutionThe term 'fluid' is used for ............
- 3A solid sphere of radius $r$ is floating at the interface of two immiscible liquids of densities $\rho_1$ and $\rho_2\,\, (\rho_2 > \rho_1),$ half of its volume lying in each. The height of the upper liquid column from the interface of the two liquids is $h.$ The force exerted on the sphere by the upper liquid is $($ atmospheric pressure $= p_0\,\,\&$ acceleration due to gravity is $g) $View Solution
- 4A log of wood of mass $120 Kg$ floats in water. The weight that can be put on the raft to make it just sink, should be ....... $Kg$ (density of wood = $600 Kg/m^3$)View Solution
- 5Sixty four spherical rain drops of equal size are falling vertically through air with terminal velocity $1.5\, m/s$. All of the drops coalesce to form a big spherical drop, then terminal velocity of big drop is ........... $m/s$View Solution
- 6Three identical vessels are filled with equal masses of three different liquids $A, B$ and $C$ $({\rho _A} > {\rho _B} > {\rho _C})$. The pressure at the base will beView Solution
- 7A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If $\rho_0$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell isView Solution
- 8A wooden cube just floats inside water with a $200 \,gm$ mass placed on it. When the mass is removed, the cube floats with its top surface $2 \,cm$ above the water level. the side of the cube is ......... $cm$View Solution
- 9A uniform rod of density $\rho $ is placed in a wide tank containing a liquid of density ${\rho _0}({\rho _0} > \rho )$. The depth of liquid in the tank is half the length of the rod. The rod is in equilibrium, with its lower end resting on the bottom of the tank. In this position the rod makes an angle $\theta $ with the horizontalView Solution
- 10View SolutionPressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. This law was first formulated by