A spring having length ℓand spring constant k is divided into two parts having lengths ℓ1 and ℓ2

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A spring having length ℓand spring constant k is divided...

A spring having length $\ell$ and spring constant $\mathrm{k}$ is divided into two parts having lengths $\ell_{1}$ and $\ell_{2}$. If $\ell_{1}=n \ell_{2}$, the force constant of the spring having length $1_{2}$ is $\ldots \ldots \ldots$ (A) $\mathrm{k}(1+\mathrm{n})$ (B) $\mathrm{k}\{(1+\mathrm{n}) / \mathrm{n}\}$ (C) $\mathrm{k}$ (D) $\{\mathrm{k} /(\mathrm{n}+1)\}$

A spring having length $\ell$ and spring constant $\mathrm{k}$ is divided into two parts having lengths $\ell_{1}$ and $\ell_{2}$. If $\ell_{1}=n \ell_{2}$, the force constant of the spring having length $1_{2}$ is $\ldots \ldots \ldots$
(A) $\mathrm{k}(1+\mathrm{n})$
(B) $\mathrm{k}\{(1+\mathrm{n}) / \mathrm{n}\}$
(C) $\mathrm{k}$
(D) $\{\mathrm{k} /(\mathrm{n}+1)\}$

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Key Concepts

Hooke's Law

Hooke’s Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, with the proportionality constant being the spring constant. This principle underpins many phenomena in mechanics and is essential for understanding how energy is stored in elastic systems.

Uniform Spring Scaling

For a uniform spring, the spring constant is inversely proportional to its length. This means that if the spring is divided into segments, each segment’s stiffness increases as its length decreases. This proportionality is a fundamental concept when analyzing the behavior of springs that are subdivided or segmented.

Segment Division of Springs

When a spring is divided into segments, the physical properties of each segment can be determined based on the original spring's characteristics and the relative lengths of the segments. Understanding how a spring's constant scales with its length allows one to calculate the effective spring constant for any segment, which is critical in both theoretical studies and practical engineering applications.

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