In Figure 10-20, the magnitude of the force necessary to...

In Figure $10-20$, the magnitude of the force necessary to stretch a spring is plotted against the distance the spring is stretched. a. Calculate the slope of the graph, $k,$ and show that $F=k d$, where $k=25 \mathrm{N} / \mathrm{m}$ b. Find the amount of work done in stretching the spring from 0.00 $\mathrm{m}$ to $0.20 \mathrm{m}$ by calculating the area under the graph from 0.00 m to 0.20 m. c. Show that the answer to part b can be calculated using the formula $W=\frac{1}{2} k d^{2},$ where $W$ is the work, $k=25 \mathrm{N} / \mathrm{m}$ (the slope of the graph), and $d$ is the distance the spring is stretched $(0.20 \mathrm{m})$

In Figure $10-20$, the magnitude of the force necessary to stretch a spring is plotted against the distance the spring is stretched.
a. Calculate the slope of the graph, $k,$ and show that $F=k d$, where $k=25 \mathrm{N} / \mathrm{m}$
b. Find the amount of work done in stretching the spring from 0.00 $\mathrm{m}$ to $0.20 \mathrm{m}$ by calculating the area under the graph from 0.00 m to 0.20 m.
c. Show that the answer to part b can be calculated using the formula $W=\frac{1}{2} k d^{2},$ where $W$ is the work, $k=25 \mathrm{N} / \mathrm{m}$ (the slope of the graph), and
$d$ is the distance the spring is stretched $(0.20 \mathrm{m})$

Key Concepts

Hooke's Law

Hooke's Law states that the force needed to extend or compress a spring by some distance is directly proportional to that displacement, expressed mathematically as F = kd. This principle underpins the linear relationship between force and displacement in elastic systems and provides the basis for analyzing spring behavior.

Spring Constant

The spring constant, denoted as k, is a measure of a spring's stiffness and is defined as the ratio of the force applied to the displacement produced (k = F/d). It is graphically represented as the slope of the force versus displacement graph and is crucial for determining how much force is required for a given deformation.

Graphical Analysis of Force versus Displacement

Plotting force against displacement for a spring yields a straight line whose slope is the spring constant. This linear graph confirms that the spring obeys Hooke's Law, and the area under the curve can be interpreted as the work done during stretching or compressing the spring.

Calculation of Work via Integration

The work done in stretching a spring is given by the area under the force versus displacement graph. Since the force increases linearly with displacement, the work can be calculated using the integral of F with respect to d, resulting in the formula W = (1/2)kd². This formula is a direct consequence of integrating the linear force function defined by Hooke's Law.

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