The analysis of a cantilever beam with a circular cross-section subjected to impact loading at its free end illustrates the conversion of potential energy from a dropped object into kinetic energy, which is then absorbed by the beam as strain energy. This process is crucial for understanding how materials behave under dynamic loads, which is important in fields such as construction and aerospace.

When an object is dropped onto the free end of a cantilever, its potential energy due to gravity is transformed into kinetic energy at the point of impact. This energy causes the beam to bend, creating a bending moment that varies along the length of the cantilever. The strain energy, which is the energy stored due to this bending, reaches its maximum at the fixed end of the beam. Integrating the strain energy across the beam helps assess the maximum load the beam can withstand before failure.

This maximum load is critical for determining the maximum stress experienced by the beam. Stress depends on both the maximum load and the geometric properties of the beam, specifically the moment of inertia, which involves its radius for a circular section.

Ultimately, understanding the maximum stress in terms of the material's modulus of elasticity and the developed strain energy is essential for designing structures that are capable of withstanding unexpected dynamic loads without failing.