Potential Energy of Rising Water Container

[copria]

A container with a hole in the bottom is filled with water. As the container is raised, water flows through the hole; thus the mass of the water in the container decreases. What is total potential energy of the water and water container over a specific time period as water escapes the container and the container is raised? As height increases, volume of water decreases.

Known:
Height of container
Initial volume of water
Final volume of water
Initial height of water column
Final height of water column
Mass of water container
Density of water
Gravitational acceleration
Time

Unknown:
Fluid Velocity
Total potential energy

Relevant Equations


Potential energy
PE=mgh

Flow rate
Q=Av$$_{}f$$

• A= cross sectional area (in this case, πr2)
• v$$_{}f$$= fluid velocity

Fluid velocity (as according to Bernoulli’s equation)
v$$_{}f$$= √(2gh$$_{}w$$)

• g= gravitational acceleration
• h$$_{}w$$= height of water column (measured from base of container)

Mass of water being lost
m$$_{}l$$= Qtρ

• t= time
• ρ= density of water

Velocity of container
v$$_{}c$$= h$$_{}c$$/t

• h$$_{}c$$= height of container (measured from ground)

Other Variables:
m$$_{}c$$= mass of container
m$$_{}i$$= initial mass of water
t= time
h$$_{}wi$$= initial height of water column
h$$_{}wf$$= final height of water column (zero)


Is this correct? Am I leaving any information out?

∫zero to t of {m$$_{}c$$+ m$$_{}i$$-[Atρ(∫h$$_{}wi$$ to h$$_{}wf$$ of √(2gh$$_{}w$$ )) ] }g h$$_{}c$$

 

[jbsteele]

dtThis looks correct. You have included all the relevant equations and variables needed to calculate the total potential energy of the water and water container over a specific time period.