Conservative Forces - Yousef's Notes

For certain forces, the work is the same for all intermediate paths, and it can be computed in terms of the initial and final states only. These are called conservative forces. As a result, when the starting and ending points are the same, the total work is zero.

Gravitational work	Elastic work
![[Pasted image 20240306135755.png]]	![[Pasted image 20240306135804.png]]

$$ W_{grav}=\int_{\vec r_1}^{\vec r_2} \vec F\cdot d\vec r=\int_{y_1}^{y_2} m g dy=mg(y_1-y_2) $$ | $$ W_{elas}=\int_{\vec r_1}^{\vec r_2} \vec F\cdot d\vec r=\int_{x_1}^{x_2} kx dx=k(x^2_1-y^2_2) $$

(Notice that $\vec{w} \cdot \Delta \vec{r}=-m g \hat{\jmath} \cdot(\Delta x \hat{\imath}+\Delta y \hat{\jmath})=-m g \Delta y$)