A peculiar definition of work on Wikipedia

[etotheipi]

I came across this here:

The work done W by an external agent which exerts a force $\vec{F}$ (at $\vec{r}$) and torque $\vec{\tau}$ on an object along a curved path C is: $$W = \int_{C} (\vec{F}\cdot d\vec{r} + \vec{\tau} \cdot \vec{n} d\theta)$$

Is this incorrect? If we setup any coordinate system and take torques about that coordinate system, then I would have thought we say the work done in that frame is $$W = \int_{C} \vec{F} \cdot d\vec{r} = \int_{C} \vec{\tau} \cdot d\vec{\theta} \quad \left( = \int_{C} \vec{\tau} \cdot \vec{n} d\theta \right)$$ So long as the curve $C$ represents the path of the point of application of the force. The definition Wikipedia gives appears to be double counting the work since we can show that the two expressions I equated are equivalent. I would however agree that another correct expression be $$W = \int \vec{F} \cdot d\vec{r}_{CM} + \int \vec{F} \cdot d\vec{r}' = \int \vec{F} \cdot d\vec{r}_{CM} + \int \vec{\tau}_{CM} \cdot d\vec{\theta}_{CM}$$ if $\vec{r}'$ represents the position of the point of application of the force w.r.t. the centre of mass.

I wondered whether anyone could clarify, since I'm not sure if the relation they gave makes any sense. Surely work is just $\int \vec{F} \cdot d\vec{r}$? Thank you!

 

[A.T.]

I came across this here:

Is this incorrect? If we setup any coordinate system and take torques about that coordinate system,

Are you sure they take the torque around the origin, and not around the point defined by r?

 

[etotheipi]

Are you sure they take the torque around the origin, and not around the point defined by r?

It doesn't specify, but even if it were about the the point $\vec{r}$ wouldn't it still be incorrect?

The only decomposition of that form which is meaningful is when $\vec{r}$ is to the centre of mass. But from the preceding statement it doesn't specify that the curve C is the path of the centre of mass. That is to say the the force need not be applied at the centre of mass of the body.

 

[etotheipi]

I think it's a blunder; I'll ignore this definition!