Problem 57 - Fundamentals of Physics

Question

A cylindrical tank with a large diameter is filled with water to a depth \(D=0.30 \mathrm{~m} .\) A hole of cross-sectional area \(A=6.5 \mathrm{~cm}^{2}\) in the bottom of the tank allows water to drain out. (a) What is the drainage rate in cubic meters per second? (b) At what distance below the bottom of the tank is the cross-sectional area of the stream equal to one-half the area of the hole?

Step-by-Step Solution

Answer

(a) The drainage rate is \(1.58 \times 10^{-3} \ m^3/s\). (b) Distance is \(0.30 \ m\).

Step 1: Understand the Problem

We need to find two things: (a) the drainage rate of water from the tank, and (b) the distance below the tank where the stream has half the cross-sectional area of the hole.

Step 2: Determine Drainage Rate

To find the drainage rate, use Torricelli's theorem: \( v = \sqrt{2gD} \), where \( g \) is the acceleration due to gravity (\( 9.81 \ m/s^2 \)), and \( D = 0.30 \ m \). First, calculate the velocity, \( v \).\[ v = \sqrt{2 \times 9.81 \times 0.3} \approx 2.43 \ m/s \]Now, find the discharge rate using \( Q = Av \), where \( A = 6.5 \ cm^2 = 6.5 \times 10^{-4} \ m^2 \).\[ Q = 6.5 \times 10^{-4} \times 2.43 = 1.58 \times 10^{-3} \ m^3/s \]

Step 3: Determine Distance for Half Area

To find the distance below the tank where the stream area is half, first calculate the velocity change. Use the fact that \( A_2 = \frac{A}{2} \) and continuity equation \( A_1v_1 = A_2v_2 \).Rearrange for \( v_2 \):\[ v_2 = 2v_1 \approx 2 \times 2.43 = 4.86 \ m/s \]Using the speed equation: \( v_2^2 = v_1^2 + 2gh \) for height \( h \) below the hole,\[ (4.86)^2 = (2.43)^2 + 2 \times 9.81 \times h \]Solve for \( h \):\[ h = \frac{(4.86)^2 - (2.43)^2}{2 \times 9.81} = 0.30 \ m \]

Key Concepts

Torricelli's Theorem

Torricelli's theorem is a fundamental principle in fluid mechanics that helps predict the speed of fluid flowing out of an orifice. Named after the Italian scientist Evangelista Torricelli, this theorem is based on the conservation of energy.
When dealing with a tank, Torricelli's theorem tells us that the speed of efflux, or the velocity (\( v \) of a fluid flowing from a sharp-edged hole at the bottom of the tank, can be calculated using the formula: \\[ v = \sqrt{2gD} \] where

• \( g \) is the acceleration due to gravity (approximately \( 9.81 \, m/s^2 \)).
• \( D \) is the height or depth of the fluid above the hole.

This formula is derived under the assumption that the fluid is incompressible and there is no viscosity. It simplifies the study of fluid motion by providing a direct relationship between the velocity of the outflow and the height of fluid above the outlet. Understanding Torricelli's theorem is key in predicting how fast a fluid exits a container, which is critical in many practical applications, such as hydraulic systems and even designing certain sports equipment.

Continuity Equation

The Continuity Equation is another essential concept in fluid mechanics. It is based on the conservation of mass, which implies that for an incompressible fluid, the volume flow rate must remain constant from one cross-section to another.
This principle is expressed mathematically as \\[ A_1v_1 = A_2v_2 \] where

• \( A_1 \) and \( A_2 \) are the cross-sectional areas at two different points along the flow path.
• \( v_1 \) and \( v_2 \) are the fluid velocities at these points.

In the example of the cylindrical tank, this equation helps us understand how changes in cross-sectional area affect fluid velocity when no other forces are acting on the system.
The Continuity Equation ensures that even as the stream of water narrows and its cross-sectional area decreases, the velocity must increase to keep the flow rate consistent. It's indispensable for analyzing situations in which fluid streams change shape or path.

Discharge Rate

Understanding discharge rate is crucial for calculating how effectively fluids are transported or drained within systems. The discharge rate, often denoted by \( Q \), represents the volume of fluid flowing per unit time, typically in \( m^3/s \).
It can be calculated using the formula \\[ Q = Av \] where

• \( A \) is the cross-sectional area of the opening through which the fluid exits.
• \( v \) is the velocity of the fluid at this point.

For the cylindrical tank example, the known velocity (derived from Torricelli’s theorem) is combined with the known area of the hole to find the discharge rate. This calculation is significant in various engineering applications, like determining flow through pipelines or water spillways.
Mastering discharge rate calculations helps in planning effective drainage systems and ensuring systems can handle the required fluid volumes without overflow or undue pressure.

Cross-Sectional Area

The cross-sectional area, symbolized as \( A \), is a critical factor in determining how fluids flow through an opening. In fluid mechanics, knowing the cross-sectional area is imperative for calculations involving the discharge rate and velocity of fluid.
In the context of fluid flowing out of a tank, the cross-sectional area pertains to the size of the hole through which the fluid exits. When the area changes, it affects the velocity, as dictated by both Torricelli’s theorem and the Continuity Equation.
For example, when the cross-sectional area of the stream becomes half of the area of the hole, the velocity of the fluid must adapt. This adjustment is directly tied to principles discussed earlier, ensuring the flow rate remains constant.

• Increases in cross-sectional area lead to decreases in velocity, given a consistent discharge rate.
• Decreases in area necessitate increased velocity.

Understanding how these area changes impact fluid behavior is key in designing systems like pipelines, hydraulic networks, and even simple tasks such as draining water tanks effectively.