Two tanks A and B are filled with different liquids of density ρ and 2ρ respectively and the height of both the liquids is the same in both the tanks. If both the tanks have a small hole at their bottom then find the ratio of the speed of efflux of the liquid in tank A to tank B:

CONCEPT:

Speed of Efflux: Torricelli’s Law:

• The word efflux means fluid outflow. Torricelli discovered that the speed of efflux from an open tank is given by a formula identical to that of a freely falling body.
• Consider a tank containing a liquid of density ρ with a small hole in its side at a height y₁ from the bottom as shown in the figure.
• The air above the liquid, whose surface is at height y₂, is at pressure P.
• If the cross-sectional area of the tank is much larger than that of the hole, then the speed of efflux is given as,

⇒ \(v =\sqrt{2gh}\)

Where g = gravitational acceleration and h = y₂ - y₁

CALCULATION:

Given ρ_A = ρ, ρ_B = 2ρ and h_A = h_B = h

• If the cross-sectional area of the tank is much larger than that of the hole, then the speed of efflux is given as,

⇒ \(v =\sqrt{2gh}\)     -----(1)

By equation 1, the speed of efflux for the liquid in tank A is given as,

⇒ \(]v_A =\sqrt{2gh_A}\)

⇒ \( v_A =\sqrt{2gh}\)     -----(2)

By equation 1, the speed of efflux for the liquid in tank B is given as,

⇒ \(v_B =\sqrt{2gh_B}\)

⇒ \( v_B =\sqrt{2gh}\)     -----(3)

By equation 2 and equation 3,

​⇒ \(\frac{v_A}{v_B} =\frac{\sqrt{2gh}}{\sqrt{2gh}}\)

⇒ \( \frac{v_A}{v_B} =\frac{1}{1}\)

• Hence, option 3 is correct.