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

Answer

`(a) (i) D=(5)/(4)d,(ii) p = P_(0)+(1)/(4)(6H+L)dg;(b) (ii)x=sqrt(h(3H-4h))(iii)x_("max")=(3)/(4)H`

Explanation

`W=B`
`(A)/(5)LDg=[(L)/(4)xx(A)/(5)xx2d xx g+(3L)/(4)xx(A)/(5)xxd xx g]`
`(D)/(5)=[(d)/(10)+(3d)/(20)]impliesD=(5)/(4)d`
(b) P=(Pressure due to weight of liquid and cylinder on base + atmospheric pressure)
`=([(H)/(2)xxAxxd+(H)/(2)xxAxx2d+Lxx(A)/(5)xx(5d)/(4)]xxg)/(A)+P_(0)`
`=[(3Hd)/(2)+(Ld)/(4)]g+P_(0)=((6H+L))/(4)dg+P_(0)`
(b) (i) Applying Burnouli's theorem at interface of liquid and at tiny hol at bottom
`[P_(0)+((H)/(2)dg)+((H)/(2)-h)2dg]+0=P_(0)+(1)/(2)(2d)v^(2)+0`
`P_(0) +(H )/(2) dg + Hdg-2hdg= P_( 0) +dv ^(2)`
`((3H-4h ))/(2) dg=dv ^ (2)`
`v=sqrt(((3H-4h)g)/(2))`
(ii) `x=vtimpliesx=sqrt(((3H-4h)g)/(2))xxsqrt((2h)/(g))impliesx=sqrt(h(3H-4h))`
(iii) For x to be maximum
`(dx)/(dh)=0`or`(dh(3H-4h))/(dh)=0impliesh=(3H)/(8)`
`X_(max)=sqrt((3H)/(8)(3H-4xx(3H)/(8)))`
`X_(max)=(3H)/(4)`