Deriving Equation for Water Depth After Tank Hole Poked

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Homework Help Overview

The problem involves a vertical cylindrical tank containing water, which has a hole at the bottom. The task is to derive an equation for the water depth as a function of time after the hole is created, using given parameters such as the cross-sectional areas and initial water depth.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss deriving a formula for the volume of water flowing through the hole based on pressure and consider using principles from fluid mechanics, including the continuity equation and Bernoulli's equation.

Discussion Status

The discussion is ongoing, with various approaches being suggested. Some participants are exploring the application of fluid mechanics principles, while others are questioning how to proceed from initial equations. There is no explicit consensus yet on a specific method to derive the desired equation.

Contextual Notes

Participants are working within the constraints of deriving a formula based on the provided parameters and equations, with no additional information or assumptions clarified at this stage.

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Homework Statement


A vertical cylindrical tank of cross-sectional area A_1 is open to the air at the top and contains water to a depth h_0. A worker accidentally pokes a hole of area A_2 in the bottom of the tank.
Derive an equation for the depth of the water as a function of time t after the hole is poked. Use A_1,A_2 h_0 and appropriate constants


Homework Equations


P=h_0\rhog
V=h_0A_1


The Attempt at a Solution



I have no idea where to go after deriving the above equations.
 
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Can you derive a formula for the volume of water that flows through the hole per unit of time based on the pressure?
 
Could you use P=FA and Ft=impulse?
 
This is a fluid mechanics questions. i believe you need the continuity equation and bernoulli's equation to solve it.

Continuity Equation: A1*V1= A2*V2

Bernoulli's Equation: P_1+h_1pg+(1/2)pv_1^2 = P_2+h_2pg+(1/2)pv_2^2
Capital P= pressure
Small p = rho (density)
 

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