Two identical holes each of cross-sectional area A are opened on the opposite sides of a wide vertical vessel filled with water. The difference in vertical height of the two holes is h .The resultant

Question

Two identical holes each of cross-sectional area A are opened on the opposite sides of a wide vertical vessel filled with water. The difference in vertical height of the two holes is h .The resultant force of reaction of water flowing out of vessel is-

TestHub · Accepted Answer

To find the resultant force, we first determine the velocity of efflux from each hole using Torricelli's Law. Let $h_{1}$ and $h_{2}$ be the depths of the upper and lower holes, respectively, from the free surface.

Velocity of efflux from the upper hole: $v_{1} = 2 g h_{1}$

Velocity of efflux from the lower hole: $v_{2} = 2 g h_{2}$

The mass flow rate from each hole is $\overset{m}{˙} = ρ A v$ .

The reaction force due to efflux from each hole is $F = \overset{m}{˙} v = ρ A v^{2}$ .

Force from the upper hole: $F_{1} = ρ A (2 g h_{1}) = 2 ρ A g h_{1}$

Force from the lower hole: $F_{2} = ρ A (2 g h_{2}) = 2 ρ A g h_{2}$

Since the holes are on opposite sides, the forces are in opposite directions. The resultant force is the difference between these two forces:

$F_{res u lt an t} = ∣ F_{2} - F_{1} ∣ = ∣2 ρ A g h_{2} - 2 ρ A g h_{1} ∣$

$F_{res u lt an t} = 2 ρ A g (h_{2} - h_{1})$

The difference in vertical height of the two holes is $h = h_{2} - h_{1}$ .

Therefore, the resultant force of reaction is $F_{res u lt an t} = 2 ρ A g h$ .

The final answer is $2 h ρ g A$ .

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