Problem — Spring work from a graph

Problem

The graph shows the force exerted by a spring as a function of extension. Graphically calculate the work done by the force to stretch the spring from $0$ to $0{,}10\;\mathrm{m}$.

Solution

Reading the graph. The line passes through the origin (Hooke’s law, $F = k\,x$). The plotted endpoint is at $x = 0{,}04\;\mathrm{m}$, $F = 30\;\mathrm{N}$. The slope is the spring constant: $k = \frac{F}{x} = \frac{30\;\mathrm{N}}{0{,}04\;\mathrm{m}} = 750\;\mathrm{N/m}$ Extending the line to $x = 0{,}10\;\mathrm{m}$, the force is $F = k\,x = 750\cdot 0{,}10 = 75\;\mathrm{N}$.

Work = area under the curve. The work to stretch the spring from $0$ to $x = 0{,}10\;\mathrm{m}$ is the area of the triangle with base $x$ and height $F$: $W = \frac{1}{2}\,x\,F = \frac{1}{2}\cdot 0{,}10\;\mathrm{m}\cdot 75\;\mathrm{N}$ $\ev{W \approx 3{,}8\;\mathrm{J}}$ Equivalently $W = \tfrac{1}{2}k\,x^2 = \tfrac{1}{2}\cdot 750\cdot (0{,}10)^2 = 3{,}75\;\mathrm{J}$.

Collegamenti

Argomenti: Lavoro ed energia Concetti: Forza elastica e legge di Hooke · Lavoro di una forza Competenze: Lettura dei grafici Oggetti: Molla