work and line integrals

[Rick16]

TL;DR

Taylor, example 4.1

I wonder what the work actually represents in this example. Normally, I would say that $W=\int{\vec{F} \cdot d \vec{r}}$ is the work done by the force F as it moves a particle along a path, where the path is determined by this very force. In this case, however, the force points in a direction that has nothing to do with the path of the particle. The particle obviously moves under the influence of some other force that constrains it to stay on its path. The force F does not push the particle along its path, it rather tries to push the particle away from its path. In doing so, it reduces the particle's kinetic energy, and by the work-KE-theorem this means that the force is doing work on the particle. But there is something strange about this work.

 

[wrobel]

There is nothing strange about this. For example, gravity also tries to pull a simple pendulum off its circular path, but that doesn't mean we can't calculate its work.