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Archimedes' Principle provides the foundation for calculating buoyant force. This principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by that object. To find the buoyant force on our sphere, we utilize the formula:

• \[ F_b = \rho V g \]

where:

• \( F_b \) is the buoyant force,
• \( \rho \) represents the density of the fluid, here 1000 kg/m³ for water,
• \( V \) is the volume of the submerged part of the object, which is 0.650 m³ in this scenario, and
• \( g \) is the acceleration due to gravity, approximately 9.8 m/s².

By substituting these values into the equation, we find the buoyant force:

• \[ F_b = 1000 \times 0.650 \times 9.8 = 6370 \, \text{N} \]

This shows that the force exerted by the water on the sphere is 6370 Newtons.

In physics, the concept of equilibrium helps us understand how forces interact with an object. For an object held in place by multiple forces, like our sphere attached to the bottom of a lake, we apply the equilibrium of forces.This equilibrium means that the sum of all forces acting on an object must be zero if the object isn’t accelerating. In the sphere's case, the tension in the cord, the gravitational force (or weight) of the sphere, and the buoyant force are the key players. The tension force (T) counteracts the combination of the sphere's buoyant force (\(F_b\)) and its weight (\(mg\)). The equilibrium equation becomes:

• \[ F_b = T + mg \]

By rearranging, we find the sphere's mass (m):

• \[ m = \frac{F_b - T}{g} \]

Plugging in our known values, the sphere's mass is found to be approximately 556.12 kg.

Volume displacement is a key concept in understanding buoyancy, as it relates to the volume of fluid displaced by an object. This displacement is directly linked to the buoyant force according to Archimedes' Principle. For a floating or submerged object, the volume of the displaced liquid equals the volume of the object below the liquid's surface.
Given our exercise with the sphere having a total volume of 0.650 m³, when it is fully submerged, that entire volume of water is displaced. This forms the basis of the buoyant force calculated initially. Further, understanding volume displacement is crucial when considering changing conditions, such as when the sphere ascends to the surface of the lake. This ongoing change can affect how much of the sphere stays submerged once it reaches the surface and comes to rest.

The last part of this exercise involves determining how much of the sphere remains submerged once it ascends to the surface. When the cord breaks, the sphere is now subject only to its own weight and the buoyant force of the water.Once it is floating at the surface and not moving vertically, the sphere reaches equilibrium again. To find the submerged volume fraction, we apply the formula:

• \[ f = \frac{m}{\rho V} \]

where:

• \( f \) is the fraction of the volume submerged,
• \( m \) is the mass of the sphere,
• \( \rho \) is the density of water (1000 kg/m³), and
• \( V \) is the total volume of the sphere.

By substituting the values, we discover:

• \[ f = \frac{556.12}{1000 \times 0.650} \approx 0.855 \]

This means that approximately 85.5% of the sphere will remain underwater. Understanding this balance of forces leads to insights into why objects float or sink according to their material and shape.