浮还是沉？

Archimedes's Principle gives us a very easy way to determine whether an object will float or sink.

Let the average density of the object be $D$, and let the density of the fluid be $\rho$. Let $V$ be the volume of the object, and let $g$ be the gravitational acceleration.

The object will float if the buoyant force, denoted $F_{buoy}$ is greater than the gravitational force, $F_{gr}$.

Calculating the gravitational force, $$F_{gr}=mg$$ $$F_{gr}=VDg$$ Using the Archimedes's Principle to find the buoyant force, $$F_{buoy}=W_{fluid}$$ $$F_{buoy}=M_{fluid}g$$ $$F_{buoy}=V\rho g$$ Now, finding out if an object will float is simply a matter of inequalities: $$F_{buoy}\gt F_{gr}$$ $$V\rho g\gt VDg$$ $$D\lt\rho$$ It's as simple as that! An object will float in a fluid if its average density is less than that of the fluid in which it is placed.

Similary, to see when an object will sink: $$F_{buoy}\lt F_{gr}$$ $$V\rho g\lt VDg$$ $$D\gt\rho$$ Thus, an object will sink if its density is greater than that of the fluid.

Finally, the last case: if $D=\rho$, then the buoyant force and the gravitational force cancel out perfectly, so what the object does depends on how you place the object into the fluid. For example, if you make it stationary, it will stay stationary. On the other hand, if you set it moving under the surface of the fluid, it will continue moving at a constant velocity---of course, in real life it would eventually stop due to friction with the fluid.