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If an object is floating, I know about how buoyancy force ($\text {density of liquid} \times \text{volume displaced} \times g$) must equal gravitational force ($ \text {density of object} \times \text {volume of object} \times g$). Can someone explain if it is possible for an object to float right below the surface of the liquid?

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It is possible because there is a pressure gradient which creates a density gradient. This is why (for example) a hot air balloon doesn't rise forever. Eventually the balloon will reach an area of low enough pressure (low air density) that it reaches equilibrium.

In the case of fluids, most fluids compress very poorly so large changes in pressure only result in small changes in density. In principle an object could float completely under the fluid but this configuration is extremely sensitive to very minor mass changes because the density gradient in fluids is so gradual.

For large objects like submarines though, it's possible to control the density of the sub precisely enough to raise and lower the sub at will.

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If weight / volume =$ 0.99 kg/dm^3$, then only one percent stays above water. It floats practically, all of its volume remains inside water.

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Yes, indeed it's possible but to achieve that we know,

Gravitational force is constant as we cannot add mass to system . In some cases

So, to achevie that condition we have to variate one parameter in buoyant force Because if both of them are constant then object would either

1) sink

2) float at surface

3) special case : for this case the possibility is to place object submerged under fluid. In this case there are 3 possibility they are . Sink, float , or object will float at the same point it is placed for this case to be true density must be equal to density of fluid.

Now, if we want to float inside a fluid or stay stationary with starting from surface of fluid like hot air balloon, submarine. As I mentioned we have change a parameter in buoyant force . As for case of submarine they change volume. By changing volume they make buoyant force equal to gravitation force. In case of hot air balloon the density of displaced fluid changes by altitude as it goes higher density decreases and forces get equal.

Only case required for floating at a surface is to make sum of all the forces on the body equal to 0.

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