To tackle this problem, we need to analyze the dynamics of the water flow into and out of the cylindrical tank. The situation involves two key aspects: the inflow of water through a tube and the outflow through a hole at the bottom of the tank. Let's break down the options provided and see which one holds true based on fluid dynamics principles.
Understanding the Flow Dynamics
When water is poured into the tank through a tube with a cross-sectional area A, it enters at a speed v. Simultaneously, water is exiting the tank through a hole at the bottom, also with an area A. The relationship between the inflow and outflow will determine the behavior of the water level in the tank.
Analyzing the Options
- Option A: The water level in the tank will keep on rising. This statement is not accurate because the outflow will counterbalance the inflow. If the inflow rate equals the outflow rate, the water level will stabilize rather than continuously rise.
- Option B: No water can be stored in the tank. This is incorrect as well. Water can be stored in the tank until the inflow and outflow rates reach equilibrium.
- Option C: The water level will rise to a height v²/2g and then stop. This option is based on Torricelli's law, which states that the speed of efflux of fluid under gravity through an orifice is given by v = √(2gh). When the inflow and outflow rates balance, the water level will stabilize at a height determined by the inflow speed and gravitational acceleration.
- Option D: The water level will oscillate. While oscillation can occur in certain systems, in this case, the inflow and outflow will likely reach a steady state rather than oscillating.
Determining the Correct Statement
Given the analysis above, option C is the most accurate. When water is poured into the tank, it will rise until the gravitational force acting on the water exiting through the hole balances the kinetic energy of the water entering. The height at which this balance occurs can be derived from the equation of motion for fluids, leading to the conclusion that the water level will stabilize at a height of v²/2g.
Conclusion
In summary, the correct statement is that the water level will rise to a height of v²/2g and then stop. This reflects the balance between the inflow and outflow rates, demonstrating a fundamental principle of fluid dynamics in a controlled system.
