A spherical ball of radius r and relative density 0.5 is...

A spherical ball of radius $r$ and relative density $0.5$ is floating in equilibrium in water with half of it immersed in water. The work done in pushing the ball down so that whole of it is just immersed in water is (where $\rho$ is the density of water) (a) $\frac{5}{12} \pi r^{4} \rho g$ (b) $0.5 \rho r g$ (c) $\frac{4}{3} \pi r^{3} \rho g$ (d) $\frac{2}{3} \pi r^{4} \rho g$

A spherical ball of radius $r$ and relative density $0.5$ is floating in equilibrium in water with half of it immersed in water. The work done in pushing the ball down so that whole of it is just immersed in water is (where $\rho$ is the density of water)
(a) $\frac{5}{12} \pi r^{4} \rho g$
(b) $0.5 \rho r g$
(c) $\frac{4}{3} \pi r^{3} \rho g$
(d) $\frac{2}{3} \pi r^{4} \rho g$

Key Concepts

Archimedes' Principle

This principle states that any object wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. In problems involving floating or sinking objects, it provides the fundamental basis for relating the displaced fluid volume to the gravitational forces acting on the object.

Buoyant Force Variation

As an object is pushed deeper into a fluid, the volume of the displaced fluid increases, leading to a change in the buoyant force acting on the object. This variation must be taken into account when calculating the work done against the buoyant force, since the force is not constant during the displacement.

Work-Energy Principle

Work is defined as the integral of force over a displacement. In the context of submerging an object in a fluid, the work done is calculated by integrating the varying buoyant force over the distance the object is moved, linking physical displacement to energy considerations.

Spherical Geometry

Understanding the geometry of a sphere is crucial when determining aspects like volume or the center of mass of submerged portions (e.g., spherical segments). Such calculations are necessary to accurately model how the immersed volume changes as the sphere is pushed deeper into the water.

Center of Buoyancy

The center of buoyancy is the point through which the resultant buoyant force acts, typically at the centroid of the displaced volume. Knowing its position is important for analyzing stability and the effective point of application of the variable buoyant force during the submergence process.

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