Floating & Sinking

• Pressure-Depth Relationship: The pressure $P$ at a depth $h$ in a fluid of density $\rho$ is given by $P = \rho g h$. Because the bottom of a submerged object is at a greater depth than the top, the upward pressure on the bottom is greater than the downward pressure on the top.

• Origin of Upthrust: The difference in pressure between the top and bottom surfaces of an object creates a net upward force. This resultant force is exactly what we define as upthrust.

• Archimedes' Principle: This principle states that the upthrust exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaces. Mathematically, $U = \rho_{fluid} V_{displaced} g$.

• Predicting Flotation via Density: To determine if an object will float, compare its average density ($\rho_{obj}$) to the fluid's density ($\rho_{fluid}$). If $\rho_{obj} < \rho_{fluid}$, the object floats; if $\rho_{obj} > \rho_{fluid}$, it sinks.

• Force Analysis Method: An object is in equilibrium when the upward and downward forces are balanced. For a floating object, the weight of the object must equal the upthrust ($W = U$).

• Calculating Submerged Volume: For a floating object, the fraction of the object submerged is equal to the ratio of the object's density to the fluid's density: $\frac{V_{submerged}}{V_{total}} = \frac{\rho_{obj}}{\rho_{fluid}}$.

Comparison of Buoyancy States

• Apparent Weight: When an object is submerged, it feels lighter. This 'apparent weight' is the actual weight minus the upthrust ($W_{apparent} = W_{actual} - U$).

• Unit Consistency: Always ensure density is in $kg/m^3$ and volume is in $m^3$ when using $g = 9.81 \, N/kg$. If using $g/cm^3$, ensure volume is in $cm^3$.

• Identify the 'V': In the upthrust formula $U = \rho V g$, the volume $V$ is the volume of the displaced fluid, which is only equal to the object's total volume if the object is fully submerged.

• Floating Objects: Remember that for any object currently floating, the upthrust is already equal to the weight. You do not need to calculate the full volume of the object to find the upthrust if you already know its mass.

• Sanity Check: If an object is less dense than the fluid, its apparent weight in that fluid must be zero because it is floating.