Physics/Fluid Mechanics Problem - badly

[Kalookakoo]

Homework Statement

I have been staring at these two problems for a LONG time now and keep getting stuck. Please help me, and try to explain so I can understand.

ro = density
p = pressure
g = gravity
h = height/elevation
A = area
v = velocity

1a) Use the principal of dimensional consistency, show that bernoulli's equation written as:

P + (1/2)(ro)(v²) + (ro)gh = constant

has the dimension of pressure.

1b) When it is written as:

(ro)/[(ro)*g] + (v²)/(2g) + h = constant

show that each term has the dimension of length.




2) Using equations

a) A*v = A*v
Left side both have subscript 1, right side have subscript 2.

and the version of Bernoulli's equation:

P/(ro) + (v²)/2 + gh = constant

show that

v(subscript 2) = sqrt((2*[p(sub1)-p(sub2)])/ (ro(1-[(A(sub2)/A(sub1))²])

Sorry if it's hard to read. I can't do this on my own and my roommate isn't here to tutor me like he usually does.

The Attempt at a Solution

I don't even understand 1a and 1b. For 2, I can get as far as

v[sub2] = [ro =(ro)gh - p]/[(ro)*(A(sub2)/A(sub1))²]

and that's just by creating similar denominators for the second equation and substituting for v[sub2] with what i solved for the other v in the other equation.

Help?

 

[NascentOxygen]

ro = density
p = pressure
g = gravity
h = height/elevation
A = area
v = velocity

1a) Use the principal of dimensional consistency, show that bernoulli's equation written as:

P + (1/2)(ro)(v²) + (ro)gh = constant

has the dimension of pressure.

Hi Kalookakoo! http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

For starters, can you show that the terms on each side of the "+" signs all have units identical to the units of pressure?

 

[Kalookakoo]

Well. p is already pressure.

ro*gh I guess could be kg/m^3 * (g) = N/m^3 * m = N/m^2 which is pressure.

I don't see how the middle term can be pressure,

 

[NascentOxygen]

I don't see how the middle term can be pressure,

Hint: F=ma

 

[Kalookakoo]

Ohhhh.

I pull out a m on the v² to be (m/s²) which is acceleration times the mass of the density so it's Force/m^3 * m = F/A.

Wow, overlooked that, thanks.

1a) Down!

Can you help me with 2? I think I can get 2b down by myself, I'll ask if I get stuck again.

 

[Kalookakoo]

I got 1b. :)

It's really simple once I realized the F=ma part lol.

Part 2 is troublesome...

 

[NascentOxygen]

For 2, I can get as far as

v[sub2] = [ro =(ro)gh - p]/[(ro)*(A(sub2)/A(sub1))²]

You shouldn't have a "p" here; it will be p₁ or p₂. As for the "h" terms, assume h remains constant.

Your use of the "=" equal sign is too carefree. It is supposed to mean equals. Now would be a good time to start to use it more carefully, before you get into more complicated maths or science exercises. What is on the left of the "=" should be equal to what is on the right.

 

[Kalookakoo]

Sorry that second equal sign is supposed to be a subtraction sign.

Does that mean I have to use that second formula twice using p1 and v1 then v2 and p2?

 

[NascentOxygen]

Use this equation for P1,v1 then for P2,v2

P/(ro) + (v²)/2 + gh = constant

 

[Kalookakoo]

And then what?

Do I solve for p1 and p2 and set it up as (p1-p2) like it is in the end equation?

Because I get (-ro)[(v²[sub1]/2) + (v²[sub2]/2)]

Where does that come into play?

 

[Kalookakoo]

And then what?

Do I solve for p1 and p2 and set it up as (p1-p2) like it is in the end equation?

Because I get (-ro)[(v²[sub1]/2) + (v²[sub2]/2)]

Where does that come into play?

 

[NascentOxygen]

This relates v1 to v2.

A*v = A*v
Left side both have subscript 1, right side have subscript 2.

 

[Kalookakoo]

How does that fit in? I just can't see it for some reason.

I plugged in p1,v1 and p2,v2 in the bernoulli equation and solved for v2 and got:

v²[sub2] = 2(p[sub1]-p[sub2])/(ro) + v²[sub1]

It's almost right..but I'm stuck

 

[NascentOxygen]

v²[sub2] = 2(p[sub1]-p[sub2])/(ro) + v²[sub1]

Replace this v₁ with v₁ from this equation:

A*v = A*v
Left side both have subscript 1, right side have subscript 2.