To analyze the situation with the cylindrical tank and the water being poured in, we need to consider the dynamics of fluid flow and the principles of hydrostatics. Let's break down the options provided and see which one holds true based on the physics involved.
Understanding the Flow Dynamics
When water is poured into the tank through a tube with a cross-sectional area equal to the hole at the bottom, we can apply Bernoulli's principle and the concept of continuity. The key factors here are the speed of the water entering the tank and the area of the hole at the bottom.
Analyzing Each Option
- Option (a): Water level will keep on rising in the tank. This option suggests that as long as water is being poured in, the level will continue to rise. However, this is not entirely accurate because the water is also flowing out through the hole at the bottom. The rate at which water enters the tank must be compared to the rate at which it exits.
- Option (b): No water can be stored in the tank. This statement implies that the tank will never retain any water. While it is true that water is flowing out, if the inflow rate equals the outflow rate, the water level can stabilize at a certain height. Thus, this option is misleading.
- Option (c): Water level will rise to a height v²/2g & then stop. This option is based on the principle of energy conservation. When water is ejected from the hole, it has kinetic energy. The height to which the water can rise in the tank can be derived from the equation of motion and energy conservation, leading to the conclusion that the water level will indeed stabilize at a height of v²/2g, where g is the acceleration due to gravity. This is the correct answer.
- Option (d): Water level will oscillate. This option suggests that the water level will fluctuate over time. While there may be minor fluctuations due to the dynamics of inflow and outflow, in a steady state where the inflow equals the outflow, the water level will stabilize rather than oscillate significantly.
Conclusion on the Correct Answer
Based on the analysis, option (c) is the most accurate. The water level will rise to a height of v²/2g and then stop, as the inflow and outflow rates reach equilibrium. This height is derived from the conversion of kinetic energy of the water exiting the hole into potential energy as it rises in the tank.
Mathematical Derivation
To derive the height, we can use the following equations:
- The kinetic energy of the water exiting the hole can be expressed as: KE = ½ mv²
- The potential energy at height h is given by: PE = mgh
Setting these equal for conservation of energy gives us:
½ mv² = mgh
By canceling m from both sides (assuming mass is not zero), we get:
½ v² = gh
Rearranging this leads to:
h = v²/2g
This confirms that the water level will rise to a height of v²/2g before stabilizing. Understanding these principles helps us grasp the behavior of fluids in various scenarios, which is fundamental in fields like engineering and physics.
