# Explaining Pressure & Depth Relationships in Fluids Using Halliday & Resnick

• rtareen
In summary: Correct. The pressure will increase with depth and with density of the object within the water.Furthermore, if we let ##y_1 = 0## be the position of the liquids surface where it comes in contact with the atmosphere and let ##y_2 = -h## be any depth below the surface, we get the equation:## p = p_0 + \rho g h## where p is the pressure at -h and ##p_0## is the pressure of the atmosphere at ##y = 0##. This seems to imply the pressure is downwards as atmosphere pushed downwards on the water.This equation is describing the pressure at a certain depth, -h
rtareen
Homework Statement:: Not a homework problem. I need a conceptual explanation.
Relevant Equations:: ##p_2 = p_1 + \rho g(y_1 - y_2)## (1)
## p = p_0 + \rho gh## (2)

When deriving equation (1) we use the example of a submerged cylinder in static equilibrium with its top at position ##y_1## and bottom at poition ##y_2##. Using Newton's law for translational equilibrium we get that ##F_2 = F_1 + mg## where ##F_2## is an upward force due to the water below the cylinder and ##F_1## is downward force due to the water above. Then we substitute using ##F = pA##, ##m = \rho V##, and ## V = A (y_1 - y_2)## and we get:

##p_2A = p_1A + \rho g A (y_1 - y_2)## and dividing by A we get:
## p_2 = p_1 + \rho g (y_1 - y_2)##

Im assuming that ##p_2## is the upward pressure (since ##F_2## is upward) associated with depth ##y_2## and that ##p_1## is a downward pressure (since ##F_1## is upward) associated with depth ##y_1##. Is this correct or is pressure omnidirectional? What I don't understand is why the upwards pressure from the water below (##p_2##) depends on the downward presure from above (##p_1##). This is not explained well in the book. What is this equation actually describing?

Since ##\rho ## is the density of the object then pressure from the water does not only depend on depth but also on the density of the object within?

Furthermore, if we let ##y_1 = 0## be the position of the liquids surface where it comes in contact with the atmosphere and let ##y_2 = -h## be any depth below the surface, we get the equation:

## p = p_0 + \rho g h## where p is the pressure at -h and ##p_0## is the pressure of the atmosphere at ##y = 0##. This seems to imply the pressure is downwards as atmosphere pushed downwards on the water. But in the first equation we derived this from an upwards force. Can somebody please explain these equations to me? I am using the derivation from Halliday and Resnick.

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rtareen said:
Im assuming that p2 is the upward pressure (since F2 is upward) associated with depth y2 and that p1 is a downward pressure (since F1 is upward) associated with depth y1. Is this correct or is pressure omnidirectional?
Within the water the pressure is omnidirectional; when the water impinges on a solid surface it exerts a pressure normal to the surface.
rtareen said:
What I don't understand is why the upwards pressure from the water below (p2) depends on the downward presure from above (p1). This is not explained well in the book. What is this equation actually describing?
It describes the difference in pressure between the top and bottom of the cylinder due to the difference in water depth. The absolute pressure values will depend on how deep the cylinder is immersed, but the difference will be the same.
rtareen said:
p=p0+ρgh where p is the pressure at -h and p0 is the pressure of the atmosphere at y=0.
In this case ρ is the density of the water. Your equation 1 is only true if ρcyl = ρw. The cylinder is in equilibrium only if its density is equal to that of water. If it is not, it will either rise or sink. F2 ≠ F1 + mg.

rtareen said:
Homework Statement:: Not a homework problem. I need a conceptual explanation.
Relevant Equations:: ##p_2 = p_1 + \rho g(y_1 - y_2)## (1)
## p = p_0 + \rho gh## (2)

When deriving equation (1) we use the example of a submerged cylinder in static equilibrium with its top at position ##y_1## and bottom at poition ##y_2##. Using Newton's law for translational equilibrium we get that ##F_2 = F_1 + mg## where ##F_2## is an upward force due to the water below the cylinder and ##F_1## is downward force due to the water above. Then we substitute using ##F = pA##, ##m = \rho V##, and ## V = A (y_1 - y_2)## and we get:

##p_2A = p_1A + \rho g A (y_1 - y_2)## and dividing by A we get:
## p_2 = p_1 + \rho g (y_1 - y_2)##

Im assuming that ##p_2## is the upward pressure (since ##F_2## is upward) associated with depth ##y_2## and that ##p_1## is a downward pressure (since ##F_1## is upward) associated with depth ##y_1##.
Is this correct or is pressure omnidirectional?
The pressures are omnidirectional. They will push on any surface.
What I don't understand is why the upwards pressure from the water below (##p_2##) depends on the downward presure from above (##p_1##). This is not explained well in the book. What is this equation actually describing?
There is no upward and downward. This equation is just comparing the omnidirectional pressures at two different depths, ##y_1## and ##y_2##
Since ##\rho ## is the density of the object then pressure from the water does not only depend on depth but also on the density of the object within?
There is no "object within". This can be the pressures per unit area that the fluid presses on a hypothetical surface at exactly the heights,##y_1## and ##y_2##

mjc123 said:
Within the water the pressure is omnidirectional; when the water impinges on a solid surface it exerts a pressure normal to the surface.

It describes the difference in pressure between the top and bottom of the cylinder due to the difference in water depth. The absolute pressure values will depend on how deep the cylinder is immersed, but the difference will be the same.

In this case ρ is the density of the water. Your equation 1 is only true if ρcyl = ρw. The cylinder is in equilibrium only if its density is equal to that of water. If it is not, it will either rise or sink. F2 ≠ F1 + mg.
Thanks for clarifying. I don't understand the derivation completely, but you atleast cleared up what this equation is describing for me.

FactChecker said:
The pressures are omnidirectional. They will push on any surface.There is no upward and downward. This equation is just comparing the omnidirectional pressures at two different depths, y1 and y2There is no "object within". This can be the pressures per unit area that the fluid presses on a hypothetical surface at exactly the heights,y1 and y2
Thanks for clearing it up!

## 1. What is the relationship between pressure and depth in fluids?

The relationship between pressure and depth in fluids is known as Pascal's Law, which states that the pressure exerted by a fluid is equal in all directions and increases with depth due to the weight of the fluid above.

## 2. How does Halliday & Resnick explain pressure and depth relationships in fluids?

In their book "Fundamentals of Physics," Halliday & Resnick explain that pressure in a fluid is caused by the collisions of molecules with the walls of the container and with each other. As depth increases, the weight of the fluid above also increases, resulting in a higher pressure at greater depths.

## 3. What is the formula for calculating pressure in a fluid at a given depth?

The formula for calculating pressure in a fluid is P = ρgh, where P is pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth.

## 4. How does pressure change in a fluid when the depth increases?

As the depth in a fluid increases, the pressure also increases. This is because the weight of the fluid above increases, causing more collisions between molecules and resulting in a higher pressure.

## 5. What is the practical application of understanding pressure and depth relationships in fluids?

Understanding pressure and depth relationships is crucial in many real-life applications, such as scuba diving, submarine operations, and hydraulic systems. It allows us to predict the behavior of fluids and design structures that can withstand different pressures at varying depths.

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