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

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