# Using W = $\int\vec {F} . \vec {ds}$ in frictional force

We all know that $$W = \int_a^b \vec {F} . \vec {ds}$$

We all know that Frictional force = $$μN$$. Now, finding work done by frictional force(when a body with normal reaction N is moving on surface : $$W = -μN\int_a^b \vec {ds}$$ Now, $$\int_a^b \vec {ds} =$$ displacement from a to b, which implies that for closed path it is zero and Work done by frictional force is zero and this also implies that it is conservative force.

How is this possible since it is non - conservative force?

Imagine that the motion is around a circle of radius $$r$$.
You have to evaluate $$\displaystyle \int _{\theta_{\rm initial}}^{\theta_{\rm final}} \vec F\cdot d\vec s$$.
$$\vec F = -\mu\, N \,\hat \theta$$ and $$d\vec s = r\,d\theta\, \hat \theta \Rightarrow \displaystyle \int _{\theta_{\rm initial}}^{\theta_{\rm final}} \vec F\cdot d\vec s = \int _{\theta_{\rm initial}}^{\theta_{\rm final}}(-\mu\, N \,\hat \theta)\cdot( r\,d\theta\, \hat \theta) = -\mu\, N \,r\int _{\theta_{\rm initial}}^{\theta_{\rm final}}d\theta$$.
For one completer revolution the intergal is equal to $$2\pi$$ and the work done is $$-\mu\,N\,2\pi r$$ ie minus the magnitude of the force times the circumference of the circle.