A 14.5 kg mass, fastened to the end of a steel wire of...

A $14.5 \mathrm{~kg}$ mass, fastened to the end of a steel wire of unstretched length $1.0 \mathrm{~m}$, is whirled in a vertical circle with an angular velocity of $2 \mathrm{rev} / \mathrm{s}$ at the bottom of the circle. The cross-sectional area of the wire is $0.065 \mathrm{~cm}^{2} .$ Calculate the elongation of the wire when the mass is at the lowest point of its path.

Key Concepts

Circular Motion and Centripetal Force

In circular motion, an object moving in a circle at a given angular velocity experiences an inward acceleration called centripetal acceleration, which is given by the product of the square of the angular velocity and the radius of the circle. This acceleration requires a corresponding centripetal force that keeps the object moving along the circular path. Understanding this relationship is essential in problems involving rotating systems, as it establishes the connection between the motion parameters (angular velocity, radius) and the forces acting on the object.

Force Balance in Vertical Circular Motion

For an object moving in a vertical circle, the forces acting on it vary with position. At the lowest point of the path, the tension in the supporting element (such as a wire) must counteract not only the force caused by the object’s weight (gravitational force) but also provide the necessary centripetal force for circular motion. Recognizing and calculating this composite force is crucial for accurately analyzing the dynamics of the system.

Elastic Deformation and Hooke's Law

When a force is applied to an elastic material, such as a steel wire, it undergoes a deformation that is proportional to the applied force, as stated by Hooke’s Law. In the context of elongation, the extension of the wire can be determined using the relationship between the applied force, the material’s cross-sectional area, the original length, and its Young’s modulus. This concept links the mechanical properties of material with the resulting deformation under tension, which is fundamental in understanding how materials respond to applied loads.

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