How to calculate differential work done by a force in circular motion?

[zenterix]

Homework Statement

When dealing with differential vectors, I still struggle sometimes.

I'd like to calculate the differential of work done by a force acting on a particle mass undergoing circular motion

Relevant Equations

Let $\vec{r}$ be the vector from the center of the circular motion to the particle.

I believe I can write the equation below, but I am not sure if it is an equality or an approximation.

$$d\vec{r}=\vec{r} \times d\vec{\theta}$$

$$dW=\vec{F} \cdot d\vec{r}$$
$$=\vec{F} \cdot (\vec{r} \times d\vec{\theta})$$

$$=(\vec{F} \times \vec{r})\cdot d\vec{\theta}$$

$$=-(\vec{r} \times \vec{F})\cdot d\vec{\theta}$$

$$\implies dW=-\vec{\tau} \cdot d\vec{\theta}$$

My question is, is this correct, and if so, why the minus sign?

 

[ergospherical]

The only mistake is in the first line, which should be $d\boldsymbol{r} = d\boldsymbol{\theta} \times \boldsymbol{r}$. The vector $d\boldsymbol{\theta} = \hat{\boldsymbol{n}}d\theta$ points along the rotation axis, and $\theta$ increases in the direction given by the right hand rule.