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Torricelli's theorem is a fundamental principle in fluid mechanics that describes the rate at which fluid exits an opening beneath a body of fluid under the influence of gravity. The theorem is named after Italian scientist Evangelista Torricelli. It provides a way to calculate the velocity of a fluid flowing out of a hole in a tank filled with liquid. According to the theorem, the velocity of efflux from the hole is equal to the velocity that a drop of the same fluid would acquire if it were allowed to free-fall under gravity from the same height as the fluid above the hole.

Mathematically, this is expressed as:\[ v = \sqrt{2gh} \]where:

• \(v\) is the velocity of efflux,
• \(g\) is the acceleration due to gravity, and
• \(h\) is the vertical distance between the fluid surface and the hole where the fluid exits.

This theorem is crucial for understanding how fluids behave when passing through openings, allowing us to predict the speed at which they exit. Overall, it's a powerful application in various engineering and fluid dynamics scenarios.

The velocity of efflux refers to the speed at which a fluid exits through an orifice. By using Torricelli's theorem, it's possible to determine this velocity based on the height of the fluid column above the orifice. The height dictates the potential energy available to convert into kinetic energy, which is why the formula for efflux velocity involves the square root of twice the gravitational acceleration multiplied by this height.

For instance, in the exercise, we have two holes, A and B, located at different heights. By applying the theorem:

• Hole \(A\) has a height of \(\frac{2H}{3}\) from the bottom, resulting in a velocity \(v_A = \sqrt{\frac{4gH}{3}}\).
• Hole \(B\)'s height is \(\frac{H}{3}\), giving it a velocity \(v_B = \sqrt{\frac{2gH}{3}}\).

Comparing these, it’s evident that different heights produce different efflux velocities, explaining why fluids exit with different speeds from orifices at varying depths in a container.

Flow rate, also known as discharge, refers to the volume of fluid passing through a hole in a given time. It's calculated using the formula:\[ Q = A \cdot v \]where:

• \(Q\) is the flow rate,
• \(A\) is the area of the hole, and
• \(v\) is the velocity of the fluid flowing out.

In cases with multiple holes or varying areas, like holes A and B in the exercise, the area significantly affects the flow rate. Since hole A has an area twice as large as hole B \( (2a) \) and a greater efflux velocity \( v_A \), it also exhibits a much larger discharge compared to hole B.
To understand this relationship, we compare flow rates:

• The discharge through hole \(A\) is \(Q_A = 2a \cdot v_A = 2a \cdot \sqrt{\frac{4gH}{3}}\),
• while that through hole \(B\) is \(Q_B = a \cdot v_B = a \cdot \sqrt{\frac{2gH}{3}} \).

Upon calculating, it becomes clear that the discharge through hole \(A\) is markedly higher than through hole \(B\), precisely at \(\sqrt{8}\) times, reinforcing the connection between area, velocity, and resulting flow rate.