**4. Bernoulli I.**At the output of a water treatment plant, water of density \( \rho \) flows at speed \( v \) through a pipe of radius \( 2r \). Further along the pipe, it goes down a hill of height \( 10r \), and then constricts to a radius \( r \) before emptying into a holding tank. What is the difference in pressure between the water near the plant and the water near the tank?---**Explanation:**This problem is an application of Bernoulli’s principle, which relates the speed, pressure, and potential energy (height) of flowing fluid. Key information includes:1. **Initial Conditions:** - Density of water, \( \rho \). - Flow speed, \( v \). - Pipe radius, \( 2r \).2. **Change in Conditions:** - Descent down a hill of height \( 10r \). - Change in pipe radius to \( r \).**Objective:**Calculate the pressure difference between water near the plant and the tank.This involves balancing the energy forms (kinetic, potential due to gravity, and pressure) at two points along the pipe, applying Bernoulli's equation:\[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]Where:- \( P \) is the pressure at a point.- \( \frac{1}{2} \rho v^2 \) is the dynamic pressure due to fluid speed.- \( \rho gh \) is the hydrostatic pressure due to height.By solving this equation, one can find the pressure difference as water flows from a point near the plant to a point near the tank.

Name: **4. Bernoulli I.**At the output of a water treatment plant, water of
Uploaded: 2024-03-20T06:56:55.000Z
Channel: Bartleby
Description: **4. Bernoulli I.**At the output of a water treatment plant, water of density \( \rho \) flows at speed \( v \) through a pipe of radius \( 2r \). Further along the pipe, it goes down a hill of height \( 10r \), and then constricts to a radius \( r