A cylinder of radius 4 cm and height 10 cm | vedclass

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Question

(A) The cylinder has a radius $r = 4 \ cm = 0.04 \ m$ and height $h = 10 \ cm = 0.1 \ m$. The cross-sectional area $A = \pi r^2 = \pi (0.04)^2 = 0.001

Vedclass · Accepted Answer

Force exerted by oil on the cylinder is zero.

Vedclass · Answer

(A) The cylinder has a radius $r = 4 \ cm = 0.04 \ m$ and height $h = 10 \ cm = 0.1 \ m$. The cross-sectional area $A = \pi r^2 = \pi (0.04)^2 = 0.0016 \pi \ m^2$.$1$. Force exerted by oil: The oil exerts pressure on the side walls of the cylinder. The net horizontal force is zero due to symmetry. However,the vertical force exerted by the oil is the buoyant force,which is $F_{oil} = ho_{oil} V_{sub, oil} g$. Here,$ho_{oil} = 0.5 	imes 1000 = 500 \ kg/m^3$ and $V_{sub, oil} = A 	imes h_{oil} = 0.0016 \pi 	imes 0.04 = 0.000064 \pi \ m^3$. Thus,$F_{oil} = 500 	imes 0.000064 \pi 	imes 10 = 0.32 \pi \ N$. Therefore,statement $A$ is incorrect.$2$. Force exerted by water: The water exerts a buoyant force $F_{water} = ho_{water} V_{sub, water} g$. The submerged height in water is $10 - 2 - 4 = 4 \ cm = 0.04 \ m$. So,$V_{sub, water} = A 	imes 0.04 = 0.0016 \pi 	imes 0.04 = 0.000064 \pi \ m^3$. Thus,$F_{water} = 1000 	imes 0.000064 \pi 	imes 10 = 0.64 \pi \ N$. Note: The pressure at the bottom of the cylinder is $P = P_{oil} + P_{water} = (ho_{oil} g h_{oil} + ho_{water} g h_{water})$. The total upward force is $F_{total} = P 	imes A = (500 	imes 10 	imes 0.04 + 1000 	imes 10 	imes 0.04) 	imes 0.0016 \pi = (200 + 400) 	imes 0.0016 \pi = 0.96 \pi \ N$. This is the net force exerted by the liquids on the bottom surface. Statement $B$ is correct.$3$. Total force: $F_{total} = F_{oil} + F_{water} = 0.32 \pi + 0.64 \pi = 0.96 \pi \ N$ (buoyant forces). Statement $C$ is incorrect as well,but $A$ is the most fundamentally incorrect statement regarding the net force on the cylinder.

$A$ cylinder of radius $4 \ cm$ and height $10 \ cm$ is immersed in two liquids as shown. Specific gravity of oil is $0.5$. $2 \ cm$ of the cylinder is in the air. Select the $INCORRECT$ statement. Neglect atmospheric pressure.

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