Physics/Fluid Mechanics Problem - badly

  • Thread starter Kalookakoo
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In summary: A[sub2]/A[sub1])²*v²[sub2]Then substitute for v²[sub1] in the equation above. You then have an equation that relates v2 to v2 and p1 and p2.In summary, the conversation discusses using the principle of dimensional consistency to show that Bernoulli's equation has the dimension of pressure and that each term in the equation has the dimension of length. The conversation also includes a problem related to Bernoulli's equation, where the goal is to solve for v2 using equations for P1, v1 and P2, v2.
  • #1
Kalookakoo
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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?
 
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  • #2
Kalookakoo said:
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?
 
Last edited by a moderator:
  • #3
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,
 
  • #4
Kalookakoo said:
I don't see how the middle term can be pressure,
Hint: F=ma
 
  • #5
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.
 
  • #6
I got 1b. :)

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

Part 2 is troublesome...
 
  • #7
Kalookakoo said:
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.
 
  • #8
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?
 
  • #9
Use this equation for P1,v1 then for P2,v2
P/(ro) + (v²)/2 + gh = constant
 
  • #10
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?
 
  • #11
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?
 
  • #12
This relates v1 to v2.
A*v = A*v
Left side both have subscript 1, right side have subscript 2.
 
  • #13
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
 
  • #14
Kalookakoo said:
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.
 

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