Water in a cylindrical container is hanging by a rod from the ceiling at point O shown in the figure.
The container is initially full, and initially displaced 20 degree . There is a hole in the bottom where water
pours out and the top of the container is open to the atmosphere. As the container swings it empties
and the water level h decreases.
The cylindrical tank has a diameter D and length b of 4 cm and 0.5 m, respectively. The distance
from the ceiling to the bottom of the tank L is 5 m. The hole diameter d in the bottom of the tank is
1 mm. The combined rod and empty container's center of gravity is identified by cg and its mass is
0.0063 kg.
You are to develop a model. Use a control volume analysis of mass, linear momentum, and
angular momentum to predict the change in its period as a function of time, the number of swings
it makes before it empties, and the time it takes to empty. You are to make the following
assumptions:
i. Water is incompressible and has a density of 998 kg/m3.
ii. The linear momentum in the container's axis direction is negligible such that the
hydrostatic pressure in the tank and the centrifugal force are balanced by the fluid
momentum exiting the container.
iii. No friction between the fluid and container walls.
iv. Water's free surface remains perpendicular to the container's axis.
v. No pressure drop through hole.
In addition to the results to be determined listed above, you should address the following issues:
a. Does the period increase or decrease as the water exits the container? Explain.
b. Does it take longer to drain if the container is swinging or stationary. Explain.
c. How does the e ffect of the initial angle ø have on the time to empty. Explain.
To analyze the given problem and answer the questions, we need to apply control volume analysis of mass, linear momentum, and angular momentum. Let's break down the steps and assumptions needed to solve the problem.
Step 1: Mass Analysis
We need to analyze the mass of the water in the tank as it drains out. For this, we'll consider a control volume around the tank and apply the principle of conservation of mass. The flow of water out of the hole at the bottom of the tank is governed by Bernoulli's equation. Since no pressure drop is mentioned, we assume that the pressure outside the hole is atmospheric pressure. Therefore, the water flows out due to the difference in the pressure head and the effect of the centrifugal force.
Step 2: Linear Momentum Analysis
According to assumption ii, the linear momentum in the container's axis direction is negligible. This means the hydrostatic pressure in the tank and the centrifugal force are balanced by the fluid momentum exiting the container. By applying the principle of conservation of linear momentum, we can derive an equation relating the change in angular momentum and the change in liquid height over time.
Step 3: Angular Momentum Analysis
To analyze the angular momentum, we consider the combined rod and empty container's center of gravity, cg. The mass of the combined system is given as 0.0063 kg. Using the principle of conservation of angular momentum, we can derive an equation relating the change in angular momentum and the change in liquid height over time.
Now, let's address the specific questions:
a. Does the period increase or decrease as the water exits the container? Explain.
The period refers to the time taken for one complete swing of the container. As the water exits the container, the liquid height decreases, and the center of mass of the combined system (rod and empty container) shifts upwards. This shift in the center of mass affects the period of the swinging motion. To determine whether the period increases or decreases, we need to solve the derived equations for angular momentum and liquid height over time.
b. Does it take longer to drain if the container is swinging or stationary? Explain.
When the container is swinging, the centrifugal force acting on the water creates an additional force that aids in the drainage. This acceleration due to centrifugal force helps to increase the rate of draining. Therefore, it takes less time to drain if the container is swinging compared to when it is stationary.
c. How does the effect of the initial angle ø have on the time to empty? Explain.
The initial angle ø, at which the container is displaced from the vertical position, does not affect the time to empty. The time to empty is determined by the combined effects of the flow rate through the hole and the variation of angular momentum and liquid height over time.
By applying the control volume analysis of mass, linear momentum, and angular momentum, we can solve the derived equations to determine the change in period as a function of time, the number of swings before the container empties, and the time it takes to empty.