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When an object is submerged in a fluid (e.g., water), there is a pressure on the object due to weight of water by p=mgh, but why is there an upward buoyant force?

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Because there is pressure from everywhere and the effective force you find is upwards.

Suppose a cube is deep inside water. The upper face is at depth $H$ and the lower face is at depth $H+h$ where $h$ is side length of cube.

Note that pressure varies with depth as $\rho gr$ where $r$ is the depth. Clearly, all horizontal forces on cube cancel and net upwards force is : $$\text{Area}(\rho g (H+h)-\rho g H)=A\rho g h = V\rho g $$

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  • $\begingroup$ there is pressure downward due to weight of water but how the pressure exert up ward direction? $\endgroup$ – serwan May 2 '14 at 16:15
  • $\begingroup$ Pressure does not have any direction. $\endgroup$ – evil999man May 2 '14 at 16:26
  • $\begingroup$ how the pressure of water exerts on the lower face of object? $\endgroup$ – serwan May 2 '14 at 16:48
  • $\begingroup$ You press water, it presses you back. $\endgroup$ – evil999man May 2 '14 at 17:05
  • $\begingroup$ @serwan Pressure from below: youtube.com/watch?v=j8flXtndQO4 $\endgroup$ – Bernhard May 6 '14 at 19:12
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Consider a cube of height $\Delta h$ submerged in a liquid as seen in the figure. As you said, the pressure $p$ at the depth $h$ is $p = \rho g h$. The force is then $F = \rho g h A$, where $A$ is the area of the top and bottom surfaces.

If the force from the pressure at the top is $F_{top} = \rho g h A$, then the force at the bottom is $F_{bot} = \rho g A (h + \Delta h)$ and the resulting force is $F = \rho g A \Delta h$.

The reason the water exerts pressure in the first place is because of gravity. The water on the bottom pushes up because it is being pressured by the water at its sides, which is being pressured by the water from the top.

Another way to view this through Newton's third law. If you put an ice cube into a glass of water the water level will rise; the cube exerts a force on the water, and by Newton's third law, the water will exert a force on the cube.

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Because the liquid exerts the same force on the object as it would have exterted on the portion of liquid the object has displaced . We can consider the the liquid displaced too to be an object placed in the same position as the original one . In that case since the portion of liquid was stationary it meant that the rest of the liquid was exterting a force on it equal and opposite to its weight . When another object is kept in its place the rest of the liquid still exerts the same amount of force on it in the same direction(archimedes principle) which we call buoyant force.

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