Problem — Spring constant from a graph

Problem

The graph shows the force exerted by a spring deformed by $x$ from its rest length. Deduce the spring constant and calculate the work needed to compress the spring from $x=0$ to $x = 0{,}12\;\mathrm{m}$, evaluating the area under the curve.

Solution

Spring constant. The graph is a straight line through the origin: the spring follows Hooke’s law $F = k\,x$, and $k$ is the slope. Reading off an endpoint: at $x = 12\;\mathrm{cm} = 0{,}12\;\mathrm{m}$ the force is $F = 24\;\mathrm{N}$ (the height $4{,}8$ falls between the $20$ and $25\;\mathrm{N}$ marks). So $k = \frac{F}{x} = \frac{24\;\mathrm{N}}{0{,}12\;\mathrm{m}} = 200\;\mathrm{N/m}$ $\ev{k \approx 200\;\mathrm{N/m}}$

Work (area under the curve). The work needed to compress from $0$ to $x = 0{,}12\;\mathrm{m}$ is the area of the triangle with base $x$ and height $F = k\,x$: $W = \frac{1}{2}\,x\,F = \frac{1}{2}\,k\,x^2 = \frac{1}{2}\cdot 200\;\mathrm{N/m}\cdot (0{,}12\;\mathrm{m})^2$ $\ev{W \approx 1{,}4\;\mathrm{J}}$

Links

Topics: Work and energy Concepts: Elastic force and Hooke’s law · Elastic potential energy Skills: Reading graphs Objects: Spring