Q.An aluminium sphere is dipped into water. Which of the following is true?

The key idea is Buoyancy and the Anomalous Expansion of Water.

The buoyant force on a submerged object is directly proportional to the density of the fluid it displaces. Water exhibits anomalous expansion, meaning its density is maximum at $4^\circ$C. As the temperature decreases from $4^\circ$C to $0^\circ$C, the density of water decreases.

1. The buoyant force ($F_B$) is given by Archimedes' principle: $F_B = \rho_f V_s g$, where $\rho_f$ is the density of the fluid, $V_s$ is the volume of the submerged part of the sphere, and $g$ is the acceleration due to gravity.
2. For water, the density is maximum at $4^\circ$C. This means $\rho_{\text{water at } 4^\circ\text{C}} > \rho_{\text{water at } 0^\circ\text{C}}$. …

When an object is dipped into a fluid, it experiences an upward force called buoyancy. This force is a direct consequence of the pressure difference between the top and bottom surfaces of the submerged object. The deeper you go in a fluid, the greater the pressure. This means the pressure on the bottom of the object is greater than the pressure on its top, resulting in a net upward force.

The magnitude of this buoyant force is given by Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. For a fully submerged object, the volume of fluid displaced is simply the volume of the object itself.

The crucial point here is how the density of water changes with temperature. Unlike most substances, water exhibits an anomalous expansion: its density increases as it is heated from $0^\circ$C to $4^\circ$C, reaching a maximum density at $4^\circ$C. Beyond $4^\circ$C, its density decreases with increasing temperature, as is typical for most liquids.

Let's break down the problem:

1. Understand Buoyancy: The buoyant force ($F_B$) acting on a submerged object is given by Archimedes' principle:

where:
*   $\rho_f$ is the density of the fluid.
*   $V_s$ is the volume of the submerged part of the object (in this case, the entire sphere, as aluminium is denser than water and will sink).
*   $g$ is the acceleration due to gravity.

2. Identify Constant Terms:

For the aluminium sphere, its volume $V_s$ remains constant regardless of the water temperature. The acceleration due to gravity $g$ is also constant. Therefore, the buoyant force $F_B$ is directly proportional to the density of the fluid, $\rho_f$.

3. Recall Water's Anomalous Expansion: Water has a unique property: its density is maximum at $4^\circ$C. As temperature decreases from $4^\circ$C to $0^\circ$C, water expands, meaning its density decreases. …

A one-line sanity check using a fact every student already knows: ice floats on water because water is less dense as it approaches $0^\circ$C than it is at $4^\circ$C (the anomalous-expansion region). If density drops on cooling from $4^\circ$C to $0^\circ$C, then by $F_B = \rho_f V g$, the buoyant force must drop too — no separate density-vs-temperature cu …