Water flows into a large tank with flat bottom at the rate of $10^{-4} \mathrm{~m}^{3} \mathrm{~s}^{-1}$. Water is also leaking out of a hole of area $1 \mathrm{cm}^{2}$ at its bottom. If the height of the water in the tank remains steady, then this height is:

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Fluid Mechanics

Water flows into a large tank with flat bottom at the rate of

1 0^{- 4} m^{3} s^{- 1}

. Water is also leaking out of a hole of area

1 cm^{2}

at its bottom. If the height of the water in the tank remains steady, then this height is:

(A)5.1 cm✓ Correct
(B)1.7 cm
(C)4 cm
(D)2.9 cm

Answer: (A)

Solution

In the steady state the tank neither fills nor drains, so the inflow matches the Torricelli outflow through the hole,

Q_{in} = A 2 g h

. With

A = 1 cm^{2} = 1 0^{- 4}

^{2}

and

Q_{in} = 1 0^{- 4}

^{3}

/s, the areas cancel and

2 g h = 1

m/s. Solving,

h = \frac{1}{2 g} = \frac{1}{2 \times 9.8} \approx 0.051

m, about

5.1

cm. The level self-adjusts until the exit speed set by

2 g h

carries water out exactly as fast as it arrives.

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Single correct option

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