Problem — Work from a triangular force graph

Problem

The graph of the component $F_x$ of a force applied to a body, as a function of position $x$, has the shape of a triangle with base from $x = 0$ to $x = 4{,}0\;\mathrm{m}$ and maximum height $F_{\max} = 10\;\mathrm{N}$ at $x = 2{,}0\;\mathrm{m}$. Calculate the work done by the force between $x = 0$ and $x = 4{,}0\;\mathrm{m}$ and explain the method used.

Solution

When the force varies with position, the work is the area under the curve $F_x(x)$: $L = \int_0^{x_f} F_x\,\dd x = \text{area under the graph}$ Here the graph is a triangle with base $b = 4{,}0\;\mathrm{m}$ (from $x=0$ to $x=4{,}0\;\mathrm{m}$) and height $F_{\max} = 10\;\mathrm{N}$. The area of the triangle is $L = \frac{1}{2}\,b\,F_{\max} = \frac{1}{2}\cdot 4{,}0\;\mathrm{m}\cdot 10\;\mathrm{N}$ $\ev{L = 20\;\mathrm{J}}$

The area method also holds when the force increases and then decreases (as here): one simply has to compute the geometric area enclosed between the curve and the $x$-axis.

Links

Topics: Work and energy Concepts: Work done by a force Skills: Reading graphs