∙∙The graph in the accompanying figure shows the magni- tude...

$\bullet$ $\bullet$ The graph in the accompanying figure shows the magni- tude of the force exerted by a given spring as a function of the distance $x$ the spring is stretched. How much work is needed to stretch this spring: (a) a distance of $5.0 \mathrm{cm},$ starting with it unstretched, and (b) from $x=2.0 \mathrm{cm}$ to $x=7.0 \mathrm{cm} ?$

$\bullet$ $\bullet$ The graph in the accompanying figure shows the magni-
tude of the force exerted by a given spring as a function of the
distance $x$ the spring is stretched. How much work is needed to
stretch this spring: (a) a distance of $5.0 \mathrm{cm},$ starting with it
unstretched, and (b) from $x=2.0 \mathrm{cm}$ to $x=7.0 \mathrm{cm} ?$

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Key Concepts

Work by a Variable Force

Work in physics is defined as the integral of force over displacement. When the force is not constant but changes with position, as in the case of a spring, the work is determined by integrating the force function over the displacement interval. This integral represents the total energy needed to perform the task, in this context, stretching the spring.

Integration in Mechanics

Integration is a mathematical tool used to sum up continuously varying quantities. In mechanics, it is particularly useful for calculating work done when a force varies with distance. By breaking the displacement into infinitesimal segments and summing the contributions of force over these segments, integration provides a precise measure of the work done on the system.

Graphical Interpretation of Work

In a force-displacement graph, the work done by the force is represented by the area under the curve between the initial and final positions. This visual method is applicable to any varying force and offers an intuitive way to understand how the accumulation of force over a distance results in work, which can be computed through integration.

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