Sinking and Floating

An object placed in a fluid either floats or sinks. We can predict which by comparing densities.

Force Analysis

Consider an object of volume $V$ and density $\rho_{\text{obj}}$ fully submerged in a fluid of density $\rho_{\text{fluid}}$. Two forces act vertically: weight $W$ downward and buoyant force $F_B$ upward.

From Archimedes' principle:

The weight of the object is:

Applying Newton's second law in the vertical direction (taking upward as positive):

The Density Criterion

The net force direction depends entirely on which density is larger.

When $\rho_{\text{obj}} < \rho_{\text{fluid}}$, the net force is positive (upward). The object rises and floats.

When $\rho_{\text{obj}} > \rho_{\text{fluid}}$, the net force is negative (downward). The object sinks.

When $\rho_{\text{obj}} = \rho_{\text{fluid}}$, the net force is zero. The object has neutral buoyancy and remains at any depth.