# Formula for work done for both conservative and non-conservative force are different?

We know that the formula for Work Done by an constant force is

W.D = Force x displacement x (cosine of angle between force and displacement).

Situation: A mass m travels 10 meters towards  +ve axis and 6 meters back towards -ve axis. Given that coefficient of kinetic friction is 0.1.find W.D by kinetic friction?

But while calculating W.D for friction we use distance covered instead of displacement is it because friction force is  a non-conservative force.

If yes does that means For every constant conservative force we use W.D = F.ds.cos and for every constant non conservative force we use W.D = f.distance.cos

Work has only one definition, and that is force over displacement

$$W = \int \vec{F} \cdot d\vec{r}$$

The key to answer your question is that both force and displacement are vectors, and only force component parallel (tangential) to the displacement does work. The dot in the above definition denotes scalar (dot) product

$$\vec{F} \cdot d\vec{r} = F \cos\phi dr$$

where $$\phi$$ is angle between the two vectors.

Friction force by definition acts in the direction that opposes motion. This means that scalar product $$\vec{F} \cdot d\vec{r}$$ will always be negative.

One of the properties of conservative forces is that work done in moving a particle between two points is independent of path taken. The definition of work still holds, but in this special case only final and initial value of displacement is what it matters.

• But friction isn't conservative Commented Mar 31, 2022 at 6:05
• Then how come we use displacement in the WD formula to calculate WD by friction Commented Mar 31, 2022 at 6:06