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Does the buoyant force always point towards the free surface of the liquid?

For example let us take this figure enter image description here

Here let's say we have a wooden block present in between the two different liquid. Now I am sure that the liquid beneath would exert a buoyant force upward, but what should be the direction of buoyant force from the liquid present above the block.

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Let $\rho_G$ be the density of the green liquid and let $\rho_B$ be the density of the blue liquid. Let $d_T$ be the depth of the top of the block, $d_B$ be the depth of the bottom of the block, and $d_I$ be the depth of the interface. Then the pressure at the top of the block is:$$p_{T}=\rho_G g d_T$$and the pressure at the bottom of the block is $$p_B=\rho_G g d_I+\rho_B g (d_B-d_I)$$ So, the buoyant force B is:$$B=[\rho_G g d_I+\rho_B g (d_B-d_I)]A-[\rho_G g d_T]A$$But, algebraically, this is the same as:$$B=\rho_Gg(d_I-d_T)A+\rho_Bg(d_B-d_I)A$$But this can be recognized as nothing more than the sum of the weights of the displaced volumes of the two fluids.

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I think that we first need to make the terminology clear. Buoyant forces are the result of summing up all the hydrostatic stresses acting on a submerged (or partially submerged) body. Hydrostatic stresses are pushing on the wooden block in the diagram from all sides. However, the hydrostatic stresses are not uniform since there is gravity. Because of gravity, hydrostatic stresses (or pressure) increase with increasing depth going away from the free surface, so that the hydrostatic stresses acting on the lower portions of the wooden block are greater than the hydrostatic stresses acting on the upper portions of the wooden block. This results in a net force (the "buoyant force") which acts upward toward the free surface of the liquid.

In the diagram you drew, the hydrostatic stresses acting on the block from the green liquid should give a net force vector (for the green liquid) pointing downward, and the hydrostatic stresses acting on the block from the blue liquid should give a net force vector (for the blue liquid) pointing upward. The buoyant force would be the sum of these two force vectors and will point upward (again, because pressure increases with depth).

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  • $\begingroup$ Why do we still sum up the weights of the displaced blue liquid and green liquid and put it equal to the weight of the object, when we write the equation for the block? $\endgroup$ – user118752 Oct 13 '16 at 7:21
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Buoyancy is the net force of all the particles hitting the surface of the block. Regardless of the liquid type, there are forces pushing up, down, left, right, etc.

Diagram of buoyancy forces.

In general, the pressure is dependent on depth. So the highest pressure is in the lowest part. If the wood is completely submerged in a single liquid, that means the liquid on the bottom is pushing up harder than the liquid on top is pushing down. The net result is upwards buoyancy.

In this case, there isn't any of the blue liquid above the block, so the only force is pushing up. Obviously, this is a net force pushing up.

Likewise, there's none of the green liquid below the block, so the only force is pushing down. Again, there's an obvious net force, this time down.

The net buoyancy will be the combination of the two. The green liquid is necessarily less dense (or it would fall below the blue liquid), plus it's higher, so its total pressure will be lower. Ergo, the net buoyancy will be upward.

This is the same as what happens when the block is floating on top of the water. The air above the water is your second fluid in that case.

If the net buoyancy is pushing up harder than gravity pulls down, the block will move upward. If buoyancy is lower, the block will move downward. When the buoyancy exactly equals gravity, the block will sit in one spot.

Generally, if the block is more dense than both liquids, it will fall to the bottom. If it's less dense than either liquid, it will float to the top. If it's between the two densities, it will float between them like the diagram shows.

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  • $\begingroup$ Since there is no green liquid below the block, the buoyant force must be downward only, but still, when we write the equation we write weight of the displaced blue +weight of displaced green liquid =weight of the object. Why is is so then that the buoyant forces are still adding up? $\endgroup$ – user118752 Oct 13 '16 at 7:20
  • $\begingroup$ @HarshSharma: From Wikipedia, "Another common point of confusion regarding Archimedes' principle is that it only applies to submerged objects that are buoyant, not sunk objects. In the case of a sunk object the mass of displaced fluid is less than the mass of the object and the difference is associated with the object's potential energy." I think in this case, the block has sunk "to the bottom" of the green liquid, so the equation doesn't hold precisely. $\endgroup$ – MichaelS Oct 13 '16 at 8:19

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