Question

A force $F=\alpha +bx$ acts on a particle in the x-direction, where a and b are constants. Find the work done by this force during a displacement from x = 0 to x = d.

Answer 1

Given that force is a function of displacement, i.e. $\mathrm{F}=a+bx$,
where a and b are constants.
So, work done by this force during the displacement x = 0 to x = d,
$$\begin{gathered} \mathrm{W}=\underset{0}{\overset{d}{\int }}\mathrm{F}dx \\ W=\underset{0}{\overset{d}{\int }}\left(a+bx\right)dx \\ W={\left[ax+\frac{b{x}^2}{2}\right]}_0^d \\ W=ad+\frac{b{d}^2}{2} \\ \Rightarrow W=\left(a+\frac{bd}{2}\right)d \end{gathered}$$