# Mechanics: Chapter 12 – Conservative Forces

A force is conservative if its curl is a zero vector. In other words, a force $F$ is conservative if

• $\bigtriangledown \times \overrightarrow{F} = \overrightarrow{0}$

Two common conservative forces are the gravitational force and the spring force. In vector form, the gravitational force is

• $\overrightarrow{F} = \frac{GMm}{|r|^2}|\hat{r}|$
• $\Rightarrow \overrightarrow{F} = \frac{GMm}{x^2 + y^2 + z^2} \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}$
• $\Rightarrow \overrightarrow{F} = \frac{GMm}{x^2 + y^2 + z^2} \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}$
• $\Rightarrow \overrightarrow{F} = (GMm)\frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2+y^2+z^2)^{\frac{3}{2}}}$

Taking the curl of this, one gets

• $\bigtriangledown \times \overrightarrow{F} = (GMm)\frac{2yz\hat{i} -2yz\hat{i}}{(x^2+y^2+z^2)^{\frac{5}{2}}} + (GMm)\frac{2xz\hat{j} -2xz\hat{j}}{(x^2+y^2+z^2)^{\frac{5}{2}}} + (GMm)\frac{2xy\hat{k} -2xy\hat{k}}{(x^2+y^2+z^2)^{\frac{5}{2}}} = \overrightarrow{0}$

Similarly, in vector form the spring force can be written as

• $\overrightarrow{F} = -k \overrightarrow{s} = -k (x\hat{i} + y\hat{j} + z\hat{k})$
• $\bigtriangledown \times \overrightarrow{F} = -k (0\hat{i} + 0\hat{j} + 0\hat{k}) = \overrightarrow{0}$

Another (equivalent) way of thinking about a conservative force is that if a force does zero net work when a particle does a round trip (i.e its final position is the same as its initial position). Forces like friction are not conservative the net work done on a round trio by friction is not zero (and hence its curl is not a zero vector. These conditions are equivalent).