Water flows into a large tank with flat bottom at the rate of 10^-4 m^3s^-1. Water is also leaking o

Q: Why do we use Torricelli's law here instead of the full Bernoulli's equation?

Torricelli's law is actually a direct derivation from Bernoulli's equation! By assuming the top surface of the large tank moves very slowly (velocity approaches zero) and both the top surface and the hole are exposed to atmospheric pressure, Bernoulli's equation simplifies exactly to $v = \sqrt{2gh}$.

Q: Why did we have to convert the area from $\text{cm}^2$ to $\text{m}^2$?

Consistency is key in physics! The inflow rate was given in $\text{m}^3/\text{s}$, and the acceleration due to gravity $g$ is in $\text{m/s}^2$. If we kept the area in $\text{cm}^2$, our units would clash, leading to a completely incorrect height.

Q: What happens if the hole is made larger?

If the hole area $a$ increases, the outflow rate for a given height increases. To maintain the same steady inflow rate, the velocity of efflux must decrease, which means the steady-state water height $h$ will be lower.

Q: Is the velocity of efflux constant?

In this specific problem, yes! Because the water level $h$ is steady, the pressure head driving the water out is constant, making the velocity of efflux $v = \sqrt{2gh}$ constant as well. If the tank were draining without any inflow, $v$ would decrease over time.

Q: Why did we use $g = 9.8$ instead of $10$?

Usually, if a problem doesn't specify, $9.8 \text{ m/s}^2$ is the standard value. Here, using $9.8$ leads to exactly $19.6$ in the denominator, which divides perfectly into $100$ to give $5.1$. Using $10$ would give $5.0$, which is not among the options.

Question

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

Options:

(A) 4 cm

(B) 2.9 cm

(C) 5.1 cm

(D) 1.7 cm

Sigma Percentile · Accepted Answer

The problem of a leaking tank with a continuous inflow is a classic example of dynamic equilibrium in fluid mechanics. It beautifully marries the principle of continuity with Torricelli's Law. Let's dive deep into the physics of what's actually happening inside that tank.

### Visualizing Dynamic Equilibrium

Imagine you are standing in front of a massive water tank. A pipe at the top is relentlessly pouring water into it at a constant rate. At the exact same time, there is a small puncture at the bottom, and water is shooting out of it.

Now, if the inflow is greater than the outflow, the water level will inevitably rise. Conversely, if the hole is too large and the outflow exceeds the inflow, the tank will eventually drain. But the problem presents a very specific, delicate situation: **the height of the water in the tank remains perfectly steady**.

This steady state is what we call **dynamic equilibrium**. The system is active—water is constantly moving—but the macroscopic observable, the height $h$, does not change. For this to happen, the universe demands a strict balance:

$$Q_{	ext{in}} = Q_{	ext{out}}$$

### The Mathematics of Outflow

We are given the inflow rate directly: $Q_{	ext{in}} = 10^{-4} 	ext{ m}^3	ext{s}^{-1}$. But what about the outflow rate?

The volume of water exiting the tank every second depends on two factors: how big the hole is, and how fast the water is moving through it. Mathematically, the volume flow rate is the product of the cross-sectional area $a$ and the velocity of efflux $v$:

$$Q_{	ext{out}} = a \cdot v$$

We know the area of the hole is $1 	ext{ cm}^2$. But to find the velocity $v$, we must invoke **Torricelli's Law**.

Evangelista Torricelli, a student of Galileo, discovered that water leaking from a tank behaves exactly like a particle in free fall. If a drop of water falls from the surface to the hole (a distance $h$), it converts its gravitational potential energy into kinetic energy. By equating $mgh = \frac{1}{2}mv^2$, we find the velocity of efflux:

$$v = \sqrt{2gh}$$

Substituting this back into our outflow equation, we get the complete expression for the rate at which water leaves the tank:

$$Q_{	ext{out}} = a \sqrt{2gh}$$

### The Master Equation and Unit Consistency

Now, we bring our two rates together. By equating the inflow and outflow, we establish our master equation:

$$10^{-4} = a \sqrt{2gh}$$

Before we rush into calculating, we must heed a critical warning: **Unit Consistency**. The inflow is in cubic meters, and gravity is in meters per second squared. We cannot simply plug in $1 	ext{ cm}^2$ for the area. We must convert it to the standard SI unit of square meters.

Since $1 	ext{ cm} = 10^{-2} 	ext{ m}$, squaring both sides gives $1 	ext{ cm}^2 = 10^{-4} 	ext{ m}^2$. Let's substitute this into our equation:

$$10^{-4} = 10^{-4} \sqrt{2gh}$$

### The Final Calculation

The beauty of physics problems often lies in how elegantly the math simplifies. Notice how the $10^{-4}$ term appears on both sides of the equation. We can divide both sides by $10^{-4}$, completely eliminating the messy decimals:

$$1 = \sqrt{2gh}$$

To isolate $h$, we square both sides of the equation:

$$1 = 2gh$$

Now, we substitute the standard acceleration due to gravity, $g = 9.8 	ext{ m/s}^2$:

$$1 = 2(9.8)h$$

$$1 = 19.6h$$

Solving for $h$, we get:

$$h = \frac{1}{19.6} 	ext{ m}$$

Our answer is in meters, but the multiple-choice options are presented in centimeters. To convert, we multiply by $100$:

$$h = \frac{100}{19.6} 	ext{ cm}$$

Performing the division, we find:

$$h \approx 5.1 	ext{ cm}$$

And there we have it! The water level will perfectly stabilize at a height of $5.1 	ext{ cm}$. This problem is a fantastic reminder of how fundamental conservation laws—like the conservation of mass leading to the continuity equation—govern the everyday phenomena around us.

Animated Solution for Physics - Properties of Solids and Liquids: 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

Select Answer:

Visualized Solution

Visualizing the Steady State

Principle of Continuity

Rate of Outflow

Torricelli's Law

Equating Inflow and Outflow

Substituting Values

Simplifying the Equation

Solving for Height

Final Calculation

The Way Forward

The Sigma Insight: Flow of Fluid

Visualizing Dynamic Equilibrium

The Mathematics of Outflow

The Master Equation and Unit Consistency

The Final Calculation

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