A spring of force constant K is cut into two pieces such...

A spring of force constant $K$ is cut into two pieces such that one piece is double the length of the other. Then the long piece will have a foree constant of
(a) $2 / 3 K$
(b) $3 / 2 \mathrm{~K}$
(c) $3 K$
(c) $6 K$

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Key Concepts

Uniform Spring Properties

Uniform springs are characterized by consistent material and cross-sectional properties throughout their length. This uniformity ensures that scaling relationships—such as the inverse proportionality of the spring constant to its length—are valid and can be used to determine the force constant of any segment of the spring.

Hooke’s Law

Hooke’s Law states that the restoring force exerted by a spring is directly proportional to the displacement from its equilibrium position. This principle underlies the behavior of springs and is fundamental to understanding how forces and displacements relate in elastic systems.

Dependence of Spring Constant on Length

In uniform springs, the spring constant is inversely proportional to the length of the spring. This means that when a spring is cut into shorter segments, each segment will have a spring constant that is higher than the original, owing to the reduced length in which the elastic deformation occurs.

Segmenting Springs

When a uniform spring is divided into parts, the mechanical properties, particularly the spring constant, change proportionally with the segment's length. This concept is crucial when understanding how modifications like cutting a spring affect its elastic response under applied forces.

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