91Ó°ÊÓ

Density is a fundamental property of matter that describes the mass of an object per unit of volume. It's expressed in units such as kilograms per cubic meter (kg/m³).
For the exercise at hand, we are dealing with two types of density:

• *Density of Ice*: 917 kg/m³
• *Density of Seawater*: 1025 kg/m³

Density plays a critical role in determining whether objects will float or sink when placed in a fluid. Substances with lower density than the fluid they are in will typically float, such as ice floating on seawater. The difference in density between the ice and seawater is key to calculating buoyancy, which allows us to understand how much weight ice can support without sinking.

Archimedes' Principle is a fundamental concept in fluid mechanics, stating that any object, completely or partially submerged in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. This is crucial for understanding how objects float.
In our exercise, this principle helps us calculate the buoyant force that acts on the ice. When the ice supports the polar bear, it displaces seawater equal to the combined weight of the ice and the bear. This principle allows us to determine the maximum weight the ice can bear by ensuring the buoyant force is equal to or greater than the total weight of the ice and the bear.

The buoyant force is the upward force exerted by a fluid on a submerged or partially submerged object. This force counteracts the gravitational pull on the object, making it float or sink slower.
In the exercise, the buoyant force acting on the ice is calculated using the volume of the ice and the density of the seawater:

• Formula: \[F_{\text{buoyant}} = \text{density of seawater} \times \text{volume of ice} \times g \]
• Calculation: \[F_{\text{buoyant}} = 1025 \, \text{kg/m}^3 \times 5.2 \, \text{m}^3 \times 9.81 \, \text{m/s}^2 = 52,377.3 \, \text{N} \]

This buoyant force indicates the maximum weight that can be supported, ensuring the ice does not sink. The force results from the weight of the displaced seawater, which is critical in maintaining the ice's buoyancy when supporting additional weight, like that of the bear.

Gravitational force is the natural phenomenon by which all things with mass or energy are brought toward one another. On Earth, it gives weight to physical objects and is an essential factor in buoyancy.
In this exercise, gravitational force ( \(g = 9.81 \, \text{m/s}^2 \) ) is a critical part of calculations for both the weight of the ice and the bear. The gravitational force is involved in determining:

• The weight of the ice: \[W_{\text{ice}} = \text{density of ice} \times \text{volume} \times g \]
• The weight that contributes to the buoyant force calculation.

Understanding gravitational force helps connect the weight of objects to the buoyant force counteracting them, which permits objects, like the polar bear in this exercise, to temporarily overcome gravity and not sink when enough buoyant force is provided by a fluid, such as seawater.