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copria

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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

PE=mgh

Q=Av[tex]_{}f[/tex]

v[tex]_{}f[/tex]= √(2gh[tex]_{}w[/tex])

m[tex]_{}l[/tex]= Qtρ

v[tex]_{}c[/tex]= h[tex]_{}c[/tex]/t

m[tex]_{}c[/tex]= mass of container

m[tex]_{}i[/tex]= initial mass of water

t= time

h[tex]_{}wi[/tex]= initial height of water column

h[tex]_{}wf[/tex]= final height of water column (zero)

Is this correct? Am I leaving any information out?

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

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[tex]_{}f[/tex]

- A= cross sectional area (in this case, πr2)
- v[tex]_{}f[/tex]= fluid velocity

__Fluid velocity__(as according to Bernoulli’s equation)v[tex]_{}f[/tex]= √(2gh[tex]_{}w[/tex])

- g= gravitational acceleration
- h[tex]_{}w[/tex]= height of water column (measured from base of container)

__Mass of water being lost__m[tex]_{}l[/tex]= Qtρ

- t= time
- ρ= density of water

__Velocity of container__v[tex]_{}c[/tex]= h[tex]_{}c[/tex]/t

- h[tex]_{}c[/tex]= height of container (measured from ground)

__Other Variables:__m[tex]_{}c[/tex]= mass of container

m[tex]_{}i[/tex]= initial mass of water

t= time

h[tex]_{}wi[/tex]= initial height of water column

h[tex]_{}wf[/tex]= final height of water column (zero)

Is this correct? Am I leaving any information out?

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

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