Explaining Pressure & Depth Relationships in Fluids Using Halliday & Resnick

[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.

 

[mjc123]

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.

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.

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 ρ_cyl = ρ_w. 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.

 

[FactChecker]

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 $y_2$

 

[rtareen]

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 ρ_cyl = ρ_w. 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.

 

[rtareen]

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!