Home
Class 12
PHYSICS
A container of a large uniform cross-sec...

A container of a large uniform cross-sectional area `A` resting on a horizontal surface holds two immiscible, non viscous and incompressible liquids of densities `d` and `2d`, each of height `H//2` as shown in figure. The lower density liquid is open to atmosphere. A homogeneous solid cylinder of length `L(L lt (H)/(2))`, cross-sectional area `A//5` is immersed such that it floats with its axis vertical of the liquid-liquid interface with length `L//4` denser liquid. Determine
(a) density `D` of the solid and
(b) the total pressure at the bottom of the container. (Atmospheric pressure `= P_0`).
.

Text Solution

Verified by Experts

The correct Answer is:
(i) (a) `(5d)/(4), (b) `P_0 + (dg(6H + L))/(4) `
(i) (a) `(5 d ) /(4)`, (b) ` P_0 + (dg (6 H + L))/(4) `
(i) (a) Considering vertical equilibrium of cylinder weight of cylinder = upthrust due to upper liquid + upthrust due to lower liquid
` therefore ((A)/(5)) (L) D . G = ((A)/(5)) ((3L)/(4)) (d) g + ((A)/(5)) ((L)/(4)) (2d ) (g) `
` therefore D = ((3)/(4)) d + ((1)/(4)) (2d) rArr D = (5)/(4) d `
(b) Considering vertical equilibrium of two liquids and the cylinder.
` (p - p _ 0 ) A ` = weight of two liquids + weight of cylinder
` therefore P = P_ 0 + ("Weight of liquids + Weight of cylinder")/(A) " " ...(i) `
Now, weight of cylinder ` = ((4)/(5)) (L) (D) (g) = ((A)/(5) Lg ) ((5)/(4) d ) = (ALd g )/(4) `
weight of upper liquid ` = ((H)/(2) Ad g ) ` and weight of lower liquid ` = (H)/(2) A (2d ) g = HA gh `
` therefore ` Total weight of two liquids ` = (3)/(2) HA dg `
` therefore ` From equation (i) pressure at the bottom of the container will be
` therefore p = p _ 0 + ( ((3)/(2)) HA dg + (ALdg )/(4))/ (A) or p = p _ 0 + (dg (6H + L))/(4 ) `
Promotional Banner

Similar Questions

Explore conceptually related problems

A container of large uniform cross-sectional area A resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities d and 2d, each of height H/2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_0 . (a) A homogeneous solid cylinder of length L(LltH//2) , cross-sectional area A//5 is immersed such that it floats with its axis vertical at the liquid-liquid interface with length L//4 in the denser liquid. Determine: (i) the density D of the solid and (ii) the total pressure at the bottom of the container. (b) The cylinder is removed and the original arrangement is restored. A tiny hole of area s(slt ltA) is punched on the vertical side of the container at a height h(hltH//2) . Determine: (i) the initial speed of efflux of the liquid at the hole, (ii) the horizontal distance x travelled by the liquid initially, and (iii) the height h_m at which the hole should be punched so that the liquid travels the maximum distance x_m initially. Also calculate x_m .

Fluids at rest exert a normal force to the walls of the container or to the sruface of the body immersed in the fluid. The pressure exerted by this force at a point inside the liqid is the sum of atmospheric pressure and a factor which depends on the density of the liquid, the acceleration due to gravity and the height of the liquid, above that point. The upthrust acting on a body immersed in a stationary liquid is the net force acting on the body in the upward direction. A number of phenomenon of liquids in motion can be explain by Bernoulli's theorem which relates the pressure, flow speed and height for flow of an ideal incompressible fluid. A container of large uniform corss sectional area. A resting on a horizontal surface holds two immiscible, non viscous and incompressile liquids of densities d and 2d , each of height H//2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_(0) . Situation I: A homogeneous solid cylinder of length L(LltH//2) . cross sectional area A//5 is immersed such that it floats with its axis vertical at liquid -liquid interface with lenght L//4 in the denser liquid. The total pressure at the bottom of the container is

A container of large uniform cross-sectional area A resting on a horizontal surface, holes two immiscible, non-viscon and incompressible liquids of densities d and 2d each of height H//2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_(0) . A homogeneous solid cylinder of length L(LltH//2) and cross-sectional area A//5 is immeresed such that it floats with its axis vertical at the liquid-liquid interface with length L//4 in the denser liquid, The cylinder is then removed and the original arrangement is restroed. a tiny hole of area s(sltltA) is punched on the vertical side of the container at a height h(hltH//2) . As a result of this, liquid starts flowing out of the hole with a range x on the horizontal surface. The density D of the material of the floation cylinder is

A container of large uniform cross-sectional area A resting on a horizontal surface, holes two immiscible, non-viscon and incompressible liquids of densities d and 2d each of height H//2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_(0) . A homogeneous solid cylinder of length L(LltH//2) and cross-sectional area A//5 is immeresed such that it floats with its axis vertical at the liquid-liquid interface with length L//4 in the denser liquid, The cylinder is then removed and the original arrangement is restroed. a tiny hole of area s(sltltA) is punched on the vertical side of the container at a height h(hltH//2) . As a result of this, liquid starts flowing out of the hole with a range x on the horizontal surface. The total pressure with cylinder, at the bottom of the container is

A container of large uniform cross-sectional area A resting on a horizontal surface, holes two immiscible, non-viscon and incompressible liquids of densities d and 2d each of height H//2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_(0) . A homogeneous solid cylinder of length L(LltH//2) and cross-sectional area A//5 is immeresed such that it floats with its axis vertical at the liquid-liquid interface with length L//4 in the denser liquid, The cylinder is then removed and the original arrangement is restroed. a tiny hole of area s(sltltA) is punched on the vertical side of the container at a height h(hltH//2) . As a result of this, liquid starts flowing out of the hole with a range x on the horizontal surface. The initial speed of efflux without cylinder is

A container of large uniform cross-sectional area A resting on a horizontal surface, holes two immiscible, non-viscon and incompressible liquids of densities d and 2d each of height H//2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_(0) . A homogeneous solid cylinder of length L(LltH//2) and cross-sectional area A//5 is immeresed such that it floats with its axis vertical at the liquid-liquid interface with length L//4 in the denser liquid, The cylinder is then removed and the original arrangement is restroed. a tiny hole of area s(sltltA) is punched on the vertical side of the container at a height h(hltH//2) . As a result of this, liquid starts flowing out of the hole with a range x on the horizontal surface. The initial value of x is

Fluids at rest exert a normal force to the walls of the container or to the sruface of the Body immersed in the fluid. The pressure exerted by this force at a point inside the liqid is the sum of atmospheric pressure and a factor which depends on the density of the liquid, the acceleration due to gravity and the height of the liquid, above that point. The upthrust acting on a body immersed in a stationary liquid is the net force acting on the body in the upward direction. A number of phenomenon of liquids in motion can be explain by Bernoulli's theorem which relates the pressure, flow speed and height for flow of an ideal incompressible fluid. A container of large uniform corss sectional area. A resting on a horizontal surface holds two immiscible, non viscous and incompressile liquids of densities d and 2d , each of height H//2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_(0) . Situation I: A homogeneous solid cylinder of length L(LltH//2) . cross sectional area A//5 is immersed such that it floats with its axis vertical at liquid -liquid interface with lenght L//4 in the denser liquid. The density of the solid is

A container of cross-section area A resting on a horizontal surface, holds two immiscible and incompressible liquid of densities rho and 2 rho as shown in figure. The lower density liquid is open to the atmosphere having pressure P_(0) . The pressure at the bottom of the container is