- #1

rtareen

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**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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