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 $L(L < H/2)$, 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 $s(s \ll A)$ is punched on the vertical side of the container at a height $h(h < H/2)$. 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 $x_m$. (Neglect the air resistance in these calculations.)

Mechanical Properties of Fluids 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 $L(L < H/2)$, 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 $s(s \ll A)$ is punched on the vertical side of the container at a height $h(h < H/2)$. 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 $x_m$. (Neglect the air resistance in these calculations.) Watch this solved on the board The full walkthrough, written out a line at a time with narration explaining each step as it appears. Pause anywhere and ask why a line is there. Free to watch

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