Show that the units for all 3 terms in bernouliis equation are the same

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In summary, the three terms in the left-hand side of the equation have the same units to the power of 3, so the constant must be in the same units.
  • #1
victoriafello
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Homework Statement



bernouliis equaton

P + 1/2 Rho v^2 + Rho g h = constant

find the units of the three terms on the left hand side & show they are the same and then the units of the constant on the right

The Attempt at a Solution



all i can think to do here is check the SI units for each term

units for pressure = Pa or NM^-2
units for density = kg/m^3
units for velocity =m/s
units for g = m/s
units for h = m

but i can't see how i combine them or how they all come out the same ?
 
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  • #2
Pressure P = NM^-2 = kg*M*s^-2*M^-2 = kg*M^-1*s^-2.
Similarly work out for the other two expression. You can see that all the tree terms have the same dimensions.
 
  • #3
victoriafello said:
but i can't see how i combine them or how they all come out the same ?
Express each factor in terms of the basic units: m, kg, s. Then work out the units of each term in the equation by multiplying the units of each factor.
units for pressure = Pa or NM^-2
Hint: Express the pressure in terms of those three basic units.
units for g = m/s
Those units should be ms^-2.
 
  • #4
Ok so pressure becomes kgm^-1 s^-2, then the other terms expressed in
M , kg, s are

½ pv^2 is Kg m^-3 m/s^-1
Pgh is kg m^-3 m/2^2

I think I am still having a problem with the powers thou, I am not combining the units correctly
 
  • #5
victoriafello said:
Ok so pressure becomes kgm^-1 s^-2,
Good.
then the other terms expressed in
M , kg, s are

½ pv^2 is Kg m^-3 m/s^-1
Pgh is kg m^-3 m/2^2
Show how you got these.

What are the units for density? For v?
 
  • #6
density is mass / volume so its units are Kg m^-3
for velocity units are m/s^-1

so combined you get Kg m^-3 m/s^-1

and for the second part
density - Kg m^-3
units for g = m/s^-2
units for h - m

combines to give Kg m^-3 m/s^-2

i must be getting this wrong but i have re read my textbook and i can't see where to correct it,
 
  • #7
The dimensions of velocity are [tex][L]^1\times [T]^{-1}[/tex]

The dimensions of velocity squared are [tex][L]^2 \times [T]^{-2}[/tex]
 
  • #8
victoriafello said:
density is mass / volume so its units are Kg m^-3
for velocity units are m/s^-1
The density units are correct, but velocity has units of m/s or m*s^-1.

so combined you get Kg m^-3 m/s^-1
Not quite. The units for ½ρv² would be:
[kg*m^-3]*[m*s^-1]² = [kg*m^-3]*[m*s^-1]*[m*s^-1]

See if you can simplify that by collecting all the powers of m and s. (For example, what's m^-3*m*m simplify to?)
 
  • #9
Ok I think I see it now,

So m^-3*m*m simplifies to m^-1
And s^-1*s^-1 simplifies to s^-2

So the units for ½ρv² would be kg m^-1s^-2

Then for the second term
Units for density are kg*m^-3
Units for g are m*s^-2
Units for h are m

This is

[kg*m^-3]*[m*s^-2]*[m]

Collecting the m terms gives m^-1 so the units are

Kg m^-1s^-2

Then all I need is the units for the constant at the end, if the three terms all have the same units then the constant must be in the same units to the power 3, like if you have a volume then m*m*m the result is m^3 ?

if so how do I do this for kg m^-1s^-2 ?
 
  • #10
victoriafello said:
Ok I think I see it now,

So m^-3*m*m simplifies to m^-1
And s^-1*s^-1 simplifies to s^-2

So the units for ½ρv² would be kg m^-1s^-2

Then for the second term
Units for density are kg*m^-3
Units for g are m*s^-2
Units for h are m

This is

[kg*m^-3]*[m*s^-2]*[m]

Collecting the m terms gives m^-1 so the units are

Kg m^-1s^-2
Good.

Then all I need is the units for the constant at the end, if the three terms all have the same units then the constant must be in the same units to the power 3, like if you have a volume then m*m*m the result is m^3 ?
Don't confuse three terms (which add) with three factors (which multiply). All the terms have the same units, including the constant. So you're done.
 
  • #11
thanks so much for your help !
 

1. What is Bernoulii's equation?

Bernoulli's equation is an equation that relates the pressure, density, and velocity of a fluid at a certain point in a pipeline or flow system. It is used to predict the behavior of fluids in motion.

2. What are the units for each term in Bernoulli's equation?

The units for each term in Bernoulli's equation are as follows: pressure is measured in Pascals (Pa), density is measured in kilograms per cubic meter (kg/m^3), and velocity is measured in meters per second (m/s).

3. How do you show that the units for all 3 terms in Bernoulli's equation are the same?

To show that the units for all 3 terms in Bernoulli's equation are the same, you can simply plug in the units for each term into the equation and simplify. This will result in the units on both sides of the equation being equal.

4. Why is it important for the units to be the same in Bernoulli's equation?

It is important for the units to be the same in Bernoulli's equation because it ensures that the equation is dimensionally consistent. This means that the equation is mathematically correct and physically meaningful.

5. Can Bernoulli's equation be used for all types of fluids?

Yes, Bernoulli's equation can be used for all types of fluids, as long as the fluid is incompressible (meaning its density remains constant) and there are no significant external forces acting on the fluid.

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