A spring of spring constant 10^3 N / m is stretched initially 4 cm from the unscratched positi

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A spring of spring constant 10^3 N / m is stretched...

A spring of spring constant $10^{3} \mathrm{~N} / \mathrm{m}$ is stretched initially $4 \mathrm{~cm}$ from the unscratched position. How much the work required to stretched it further by another $5 \mathrm{~cm}$ ? (A) $6.5 \mathrm{NM}$ (B) $2.5 \mathrm{NM}$ (C) $3.25 \mathrm{NM}$ (D) $6.75 \mathrm{NM}$

A spring of spring constant $10^{3} \mathrm{~N} / \mathrm{m}$ is stretched initially $4 \mathrm{~cm}$ from the unscratched position. How much the work required to stretched it further by another $5 \mathrm{~cm}$ ?
(A) $6.5 \mathrm{NM}$
(B) $2.5 \mathrm{NM}$
(C) $3.25 \mathrm{NM}$
(D) $6.75 \mathrm{NM}$

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Key Concepts

Hooke's Law

Hooke’s Law defines the restoring force in a spring as being directly proportional to its displacement from the equilibrium position. It is expressed by F = -kx, where k is the spring constant and x is the displacement. This principle explains the elastic behavior of materials and is fundamental in analyzing spring systems.

Elastic Potential Energy and Work

The concept of elastic potential energy describes the energy stored in a deformed spring, which is given by the formula (1/2) k x². The work done in stretching or compressing the spring is equivalent to the change in this potential energy. This approach, involving the calculation of energy differences, is central to understanding energy transfer in elastic systems.

Integration in Calculating Work

When the force applied varies with displacement—as is the case with springs—the work done must be computed using integration. This involves integrating the force function over the range of displacement, which, for a spring, leads to the work expression W = (1/2) k (x_final² - x_initial²). This technique is essential for dealing with non-constant force fields in mechanics.

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