In the figure shown,the heavy cylinder (radiu | vedclass

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(B) For the cylinder to be in equilibrium,the net horizontal force acting on it must be zero.The horizontal force exerted by a liquid on a curved surf

Vedclass · Accepted Answer

$R\sqrt{3/2}$

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(B) For the cylinder to be in equilibrium,the net horizontal force acting on it must be zero.The horizontal force exerted by a liquid on a curved surface is equal to the force exerted by the same liquid on the vertical projection of that surface.Let the length of the cylinder be $L$.The force exerted by the liquid of density $2ho$ on the left side is $F_1 = P_{avg} 	imes A = (2ho g \frac{h}{2}) 	imes (hL) = ho g h^2 L$.The force exerted by the liquid of density $3ho$ on the right side is $F_2 = P_{avg} 	imes A = (3ho g \frac{h}{2}) 	imes (hL) = 1.5 ho g h^2 L$.However,the problem implies the cylinder is in equilibrium under the pressure of these liquids. For the cylinder to not move horizontally,the net force must be zero. This implies the pressure forces must balance. Given the geometry,the effective force is calculated by integrating pressure over the surface. The horizontal force on a submerged vertical projection of height $h$ is $F = \int_0^h (ho g y) L dy = \frac{1}{2} ho g h^2 L$.Equating the forces: $(2ho) g \frac{h^2}{2} L = (3ho) g \frac{h^2}{2} L$ is not possible unless $h=0$. Re-evaluating: the cylinder is in equilibrium if the net torque or net force is zero. The horizontal force is $F_H = \int P dA_x$. For a cylinder of radius $R$ and height $h$ submerged,the horizontal force is $F = \int_0^h ho g y (L dy) = \frac{1}{2} ho g h^2 L$. Setting $2ho g h^2 = 3ho g (R^2 - (R-h)^2)$ is incorrect. The correct equilibrium condition for the cylinder resting on a surface is that the net horizontal force is zero. Since the liquids are on opposite sides,the forces are $F_1 = \frac{1}{2}(2ho)gh^2$ and $F_2 = \frac{1}{2}(3ho)g(R^2)$. Thus,$2h^2 = 3R^2$,which gives $h = R\sqrt{3/2}$.

In the figure shown,the heavy cylinder (radius $R$) resting on a smooth surface separates two liquids of densities $2\rho$ and $3\rho$. The height '$h$' for the equilibrium of the cylinder must be

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