Problem 93 - Physics

Question

In a very large closed tank, the absolute pressure of the air above the water is \(6.01 \times 10^{5} \mathrm{~Pa}\). The water leaves the bottom of the tank through a nozzle that is directed straight upward. The opening of the nozzle is \(4.00 \mathrm{~m}\) below the surface of the water. (a) Find the speed at which the water leaves the nozzle. (b) Ignoring air resistance and viscous effects, determine the height to which the water rises.

Step-by-Step Solution

Answer

Water leaves the nozzle at 45.3 m/s and rises to 104.5 m.

Step 1: Identify Given Information

We have the absolute pressure of the air as \( P_1 = 6.01 \times 10^{5} \, \mathrm{Pa} \) and the atmospheric pressure \( P_2 = 1.01 \times 10^{5} \, \mathrm{Pa} \). The height difference \( h = 4.00 \, \mathrm{m} \). The density of water \( \rho = 1000 \, \mathrm{kg/m^3} \). We are to find the speed of the water and the maximum height.

Step 2: Apply Bernoulli's Equation

Bernoulli's equation for the top and the bottom of the tank is \( P_1 + \rho g h = P_2 + \frac{1}{2} \rho v^2 \). Here, \( v \) is the speed of the water leaving the nozzle. Rearranging gives \( \frac{1}{2} \rho v^2 = P_1 - P_2 + \rho g h \).

Step 3: Solve for Speed

Substitute the given values into the equation from Step 2: \( \frac{1}{2} \times 1000 \times v^2 = (6.01 \times 10^{5} - 1.01 \times 10^{5}) + 1000 \times 9.81 \times 4 \). Solve this equation for \( v \), the speed of the water.

Step 4: Calculate Speed (v)

Calculate: \( \frac{1}{2} \times 1000 \times v^2 = 5.00 \times 10^{5} + 39240 \). Then \( v = \sqrt{\frac{1032400}{500}} \). Compute \( v \).

Step 5: Determine Maximum Height

Use the principle of conservation of energy. The potential energy at maximum height \( h_{\text{max}} \) will be equal to the kinetic energy when the water reaches the nozzle: \( mgh_{\text{max}} = \frac{1}{2}mv^2 \). Simplify to find \( h_{\text{max}} = \frac{v^2}{2g} \).

Step 6: Substitute and Solve for Height

Substitute \( v \) found in Step 4 into \( h_{\text{max}} = \frac{v^2}{2g} \). Calculate \( h_{\text{max}} \).

Key Concepts

Fluid Dynamics

Fluid dynamics is the study of fluids in motion. It plays a major role in understanding how water behaves when it moves, such as flowing through a nozzle in our tank problem. In fluid dynamics, we study the different forces that act on fluids and how they affect motion. Key factors include velocity, pressure, and density.

In our exercise, we explore how water moves from a tank, affected by gravity and pressure difference. As water flows through the nozzle, its velocity changes, determined by these forces. This involves using principles like Bernoulli's equation, which connects fluid speed, pressure, and height. By grasping fluid dynamics, we can predict and calculate how and where the fluid will travel, ensuring our calculations align with physical behavior.

Conservation of Energy

The principle of conservation of energy is crucial in fluid dynamics problems. It states that energy cannot be created or destroyed, only transformed from one form to another. This principle helps us understand how energy changes as water flows through the nozzle.

In our scenario, the water posses potential energy relative to its height in the tank. As it leaves the nozzle, this potential energy converts into kinetic energy, increasing the water's speed. By using equations that describe these energy transformations, like Bernoulli's equation, we can calculate different energy states. For example, in the solution, we equated potential energy at the apex of water's rise to its kinetic energy as it exits the nozzle, helping find the maximum height water can reach.

Hydrostatics

Hydrostatics is the area of physics that deals with fluids at rest. In this problem, it involves understanding the pressure exerted by the water due to its height within the tank.

Before water begins to move, it is static, with pressure exerted by the column of water above any point. This pressure combines with the pressure from the air above to influence its behavior once it starts moving. The hydrostatic pressure, resulting from water's weight, is part of why we calculate pressure difference in Bernoulli's equation: it affects how fast the fluid exits the nozzle.

Understanding hydrostatics allows us to seamlessly transition from a static perspective to a dynamic analysis, an essential aspect of the given problem.

Pressure Difference

Pressure difference is critical in determining how fluids move between points. In this exercise, we find the speed at which water leaves the nozzle by calculating the pressure difference between the top of the water column and its exit from the tank.

The pressure difference occurs because the air above the water creates a higher pressure, pushing the water down and out through the nozzle when a point of lower pressure is present, usually when the nozzle freely opens into the atmosphere. This concept is embedded in Bernoulli's equation, where we subtract the exit atmospheric pressure from the initial water pressure to find the resultant force driving the flow.

• Higher initial pressure means higher exit speed
• Pressure difference is tied to water's potential to convert static energy into dynamic energy

Working through this problem with pressure differences in mind helps us determine not just the speed, but also how much energy converts from pressure form to energy of motion.