Bernoulli's Equation & Fluid Dynamics LAB

[Danneskjöld]

Before I describe the problem, I would like to apologize for my lack of comprehension
of this subject. It is the first semester of Physics, but the end, and has become quite
complicated.

Homework Statement

There is a 2.0 L soda bottle used, with a hole a certain height (h_hole) from
the bottom (and a small mini-hose to assist in the motion of the exiting of the fluid from the
bottle). The bottle is aligned with the opening of a sink, and a hose is attached to the top
so as to fill the bottle to a certain height and remain at that height constantly (h₁).
Then the distance the water travels out the hole is measured and recorded as a value
of x₁.
Subsequently, this is done one more time with the water level at a new higher height (h₂) and a new value of x₂.

Okay, hopefully that makes sense.

Next, we are given two equations that are used to derive some other new equations
meant to discover the velocity (I'm uncertain if I should include the messy method of
proving these equations weren't pulled out of my ear, or just cut to the chase).

THEN... we are to solve for velocity.
Then we are asked to comment on our results, giving probable reasons for difference
in values obtained from equations 1 & 2.

Homework Equations

Now for the equations.

Equation 1:

Point 1_____Point 2_____Point 3
___P₀___= P₀ + $$\rho$$gh₁ = P₀+($$\rho$$v_H2O²)/2

OR

$$\rho$$v_H2O² = $$\rho$$gh₁ ; v_H2O = $$\sqrt{(2gh[sub1])}$$
___2

Equation 1: v_H2O = $$\sqrt{(2gh[sub1])}$$

Equation 2:

h₂ = ½ gt² OR t = $$\sqrt{[2h(sub2)]/g}$$ ; X = v_H2Ot ; OR t = x/v_H2O = $$\sqrt{(2gh[sub1])}$$

THEN

v_H2O = X*$$\sqrt{g/(2h(sub2))}$$ ; v_H2O = 2.21 * X/$$\sqrt{h(sub2)}$$

Equation 2: v_H2O = 2.21 * X/$$\sqrt{h(sub2)}$$

The Attempt at a Solution

h_hole = h₂ = 6.9 cm
h₁ = 8.4 cm
x₁ = 8.6 cm
v_1H2O=________?

h_hole = h₂ = 6.9 cm
h₁ = 11.1 cm
x₂ = 9.8 cm
v_2H2O = _______?

Using Equation 1:

v_1H2O = $$\sqrt{2gh(sub1)}$$ = $$\sqrt{(2)(9.8)(0.084m)]}$$ = 1.6464 m/s

v_2H2O = $$\sqrt{2gh(sub2)}$$ = $$\sqrt{(2)(9.8)(0.11m)]}$$ = 2.156 m/s


Using Equation 2:

v_1H2O = (2.21) (X/$$\sqrt{h(sub2)}$$ = [(2.21)(0.086m)]/$$\sqrt{0.084m}$$ = 0.655769301 m/s

v_2H2O = (2.21) (X/$$\sqrt{h(sub2)}$$ = [(2.21)(0.098m)/$$\sqrt{0.11m}$$ = 0.65302 m/s

So to discuss these differences and probably reasons, I came to the conclusion
that the 1st equation is lacking the value of X. I purport that because we normally
consider volume (not cross-sectional area) when we are dealing with pressure,
and also because the X value is not taken into consideration within the first equation,
that perhaps it is wrong for those reasons.

However, the truth is, I don't even know what I am talking about, and am simply
left in a complete state of perplexity. Any and all help is appreciated.

P.S.: I have even asked my class mate and my father for help, both of whom
are equally confused.
P.P.S.: My roommate mentioned something along the lines of the potential for
the two Bernoulli equations in question to be from two different relatives from
the Bernoulli family (the roommate has read many a Mathematician's biography,
which is how they know about the relations)... I believe this to be nonsense. ;)

 

[Andrew Mason]

What does Bernouilli's equation predict about the value of X as a function of H? That is the question you have to answer.

Since the hole is at the same height above the bottom of the bottle and bottom of the sink, the distance of fall of the exiting water is constant for all H (height of the water in the bottle above the hole). This means that the time of fall is the same. So X is proportional to the horizontal speed of the exiting water: $$X \propto v_{H_2O}$$

What does Bernouilli say the relationship is between H and v? So what is the relationship that you should find between H and X? Does your data support that relationship?

AM

 

[kerimek]

Dear student,

First of all, don't apologize for not understanding a subject. I can assure you that it is completely normal to struggle with complex concepts at first. What matters is that you are actively trying to understand and learn, and that is commendable.

Now, let's address the problem at hand. From what I understand, you are conducting an experiment to measure the velocity of water flowing out of a bottle through a small hole. You are using two different equations, both derived from Bernoulli's equation, to calculate the velocity. However, you are getting different values for the velocity from each equation and are unsure of the reasons for this difference.

First, let's discuss Bernoulli's equation. It is a fundamental equation in fluid dynamics that relates the pressure, velocity, and height of a fluid in a continuous flow. It states that the total energy of the fluid (kinetic energy + potential energy) remains constant along a streamline. This means that if the velocity of the fluid increases, the pressure decreases and vice versa.

Now, let's look at the two equations you are using. Equation 1 is derived from the Bernoulli equation and relates the pressure at two different points in the fluid. Equation 2 is also derived from the Bernoulli equation, but it relates the velocity at two different points in the fluid. Both equations are valid and can be used to calculate the velocity of the fluid. The difference in values you are getting could be due to several reasons.

One possible reason could be experimental error. It is possible that your measurements of the heights and distances were not accurate, leading to different values for the velocity. Another reason could be the assumptions made in deriving the equations. In both equations, the fluid is assumed to be incompressible and the flow is assumed to be steady and inviscid. These assumptions may not hold true in a real-life scenario, leading to differences in the calculated values.

Additionally, the equations might be applicable in different scenarios. Equation 1 is applicable when there is a change in height between the two points, while equation 2 is applicable when there is a change in velocity between the two points. Depending on the setup of your experiment, one equation may be more suitable than the other.

In conclusion, it is important to understand the assumptions and limitations of the equations you are using and to make accurate measurements in order to get reliable results. It is also important to critically analyze your results and try