Work of a Constant Force

Theorem: Work of a Constant Force

If a force $\mathbf{F}$ is constant, then the work it does depends only on the displacement $\Delta \mathbf{r}$ of the point mass that it is exerted on:

$$W=\mathbf{F}\cdot \Delta \mathbf{r}$$

PROOF

We want to determine the work that $\mathbf{F}$ as the point mass moves in the time interval from $t_1$ to $t_2$.

$$W={\int }_{t_1}^{t_2}\mathbf{F}\cdot \mathbf{v}\mathrm{dt}$$

Since $\mathbf{F}=\text{const}$, we have

$$\begin{array}{rl}{\int }_{t_1}^{t_2}\mathbf{F}\cdot \mathbf{v}\mathrm{dt}& =\mathbf{F}\cdot {\int }_{t_1}^{t_2}\mathbf{v}\mathrm{dt}\\ & =\mathbf{F}\cdot (\mathbf{r}(t_2)-\mathbf{r}(t_1))\\ & =\mathbf{F}\cdot \Delta \mathbf{r}\end{array}$$