# Bernoulli Equation and Velocities

• XanMan
In summary, the problem involves a cubic wine box with a tap at the bottom, releasing wine at a speed of ##v_0## when full and lying on a horizontal plane. The first question asks for the speed of the wine when the box is half empty, which is found to be ##v_1 = \sqrt{gh}## using the Bernoulli Equation. The second question involves tilting the box by 45 degrees and finding the speed of the wine, which is ##v_2 = \sqrt{gh}##, using the same equation and considering the height difference only, as the angle of the tap does not affect the result.
XanMan

## Homework Statement

A cubic wine box of dimensional length ##h## has a small tap at an angle at the bottom. When the box is full and is lying on a horizontal plane with the tap open, the wine comes out with a speed ##v_0##.

i) What is the speed of the wine if the box is half empty? (Neglect the speed of the liquid at the top of the box.)
ii) What is the speed of the wine if the box is tilted by 45 degrees? (See attached figure)

[Assume the pressure at the top and bottom is equal]

## Homework Equations

Bernoulli Equation: ##\cfrac{p}{\rho} + \cfrac{v^2}{2} + \phi = ## const.

where ##\phi## is the potential energy for a unit mass as a function of the height ##z##.

It follows from the Bernoulli Equation that is ##p_0## is constant at the top and bottom, and the initial height is ##h##, we get: $$v_0 = \sqrt{2gh}$$

## The Attempt at a Solution

[/B]
i) By Bernoulli Equation, we get:

$$\cfrac{p_0}{\rho} + \cfrac{0}{2} + \cfrac{gh}{2} = \cfrac{p_0}{\rho} + \cfrac{v_1^2}{2} + 0$$
(since at the bottom ##z = 0##, and if I understand right, the velocity of the liquid at ##h/2## is 0 (?)).

Simplifying, we get:

$$\cfrac{gh}{2} = \cfrac{v_1^2}{2}$$

Answer: $$v_1 = \sqrt{gh}$$

ii) I am not quite sure about this part of the question. I tried using Pythagoras' Theorem to find the height of the liquid in terms of ##h##, and got ##\cfrac{\sqrt{2}h}{2}##.

Following a similar procedure as in i), I got the following result, which I think is incorrect, and would like your help on it (thanks!):

Answer: $$v_2 = \sqrt{\sqrt{2}gh}$$

Note: I am unsure where to use ##v_0## in the problem, or the fact that the tap is "at an angle"!

Looks good to me! But I assume they want the answers in terms of ##v_0##, not ##h##. (You can convert between the two.)

The tap being at an angle should not matter.

XanMan
Doc Al said:
Looks good to me! But I assume they want the answers in terms of ##v_0##, not ##h##. (You can convert between the two.)

The tap being at an angle should not matter.

Ah, how silly of me...I neglected to use ##v_0 = \sqrt{2gh}##! Thanks DocAl : - )

Doc Al said:
Looks good to me! But I assume they want the answers in terms of ##v_0##, not ##h##. (You can convert between the two.)

The tap being at an angle should not matter.
sir I have a doubt why component of gravitational force doesn't matter here when tap is at an angle
considering a fluid part it has 2 forces on it 1 by the pressure or weight created by fluid part above it and gravitational force on it.
please coeerct me if I am wrong

akshay86 said:
sir I have a doubt why component of gravitational force doesn't matter here when tap is at an angle
considering a fluid part it has 2 forces on it 1 by the pressure or weight created by fluid part above it and gravitational force on it.
please coeerct me if I am wrong

Where do forces come in however?

Also, why does the angle of the tap not matter?

akshay86 said:
sir I have a doubt why component of gravitational force doesn't matter here when tap is at an angle
considering a fluid part it has 2 forces on it 1 by the pressure or weight created by fluid part above it and gravitational force on it.
please coeerct me if I am wrong
The effect of gravity is included in the potential energy term.

XanMan said:
Also, why does the angle of the tap not matter?
All that matters is the height difference (review the derivation). Of course, the subsequent motion of the fluid does depend on whether it is sent out at an angle (like any other projectile).

Doc Al said:
The effect of gravity is included in the potential energy term.All that matters is the height difference (review the derivation). Of course, the subsequent motion of the fluid does depend on whether it is sent out at an angle (like any other projectile).

Yes, that's what I was thinking as well - it's the velocity *just* as it is leaving the pipe. Sorry for all the questions lately, but this was in my first university physics exam and I'm a bit scared about it to be honest! Never actually properly had physics before - quite a challenge but I'm enjoying it! :-)

XanMan said:
Yes, that's what I was thinking as well - it's the velocity *just* as it is leaving the pipe.
Exactly.

XanMan said:
Sorry for all the questions lately, but this was in my first university physics exam and I'm a bit scared about it to be honest! Never actually properly had physics before - quite a challenge but I'm enjoying it!
Never apologize for asking questions! I'm glad you're enjoying your physics journey and I'm sure you are up to the challenge.

Doc Al said:
Exactly.Never apologize for asking questions! I'm glad you're enjoying your physics journey and I'm sure you are up to the challenge.

Doc Al, did anyone tell you how awesome you are? Thanks for the support! :D

## What is the Bernoulli equation?

The Bernoulli equation is a mathematical equation that describes the relationship between fluid velocity, pressure, and elevation along a streamline. It is based on the principle of conservation of energy and is commonly used in fluid mechanics and aerodynamics.

## What is the significance of the Bernoulli equation?

The Bernoulli equation is significant because it allows us to predict the behavior of fluids in various situations, such as in pipes, wings, and other flow systems. It also helps us understand the relationship between fluid velocity and pressure, and how changes in one affect the other.

## How are velocities related to the Bernoulli equation?

The Bernoulli equation states that as the velocity of a fluid increases, the pressure decreases. This means that if the velocity of a fluid increases at a certain point, the pressure at that point will decrease. Similarly, if the velocity decreases, the pressure will increase.

## What is the difference between stagnation and dynamic pressure in the Bernoulli equation?

In the Bernoulli equation, stagnation pressure refers to the pressure at a point where the fluid is brought to rest, such as at the front of a streamlined object. Dynamic pressure, on the other hand, refers to the pressure exerted by a fluid in motion. Both of these pressures are included in the Bernoulli equation to determine the total pressure at a certain point in a fluid flow.

## What are some applications of the Bernoulli equation?

The Bernoulli equation has many practical applications, such as in the design of aircraft wings, water turbines, and pumps. It is also used in calculating airspeed in weather forecasting and in studying the flow of blood in the human body. Other examples include airfoil design, pipe flow, and flow through nozzles and diffusers.

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