# Centripetal force acting on a rotating rod

Assuming a rod is set into vertical translation in the positive y direction, while being rotated around it's center of mass with uniform angular velocity: what direction will centripetal force act on? Will it depend on the orientation of the rod at that instant?

• If the translational velocity is constant, it doens't matter. It is always possible to be in its rest frame. And the problem becomes a body in pure rotation. – Claudio Saspinski May 30 '20 at 20:14
• Even for a rod in pure rotation then, I have no idea what direction centripetal force might act – Soumil May 31 '20 at 3:57
• a) Centripetal force measured where? b) the constant motion of the center of mass does not change the problem. You can always create a co-moving inertial reference frame where the center of mass isn't moving. – John Alexiou May 31 '20 at 4:33

## 1 Answer

Centripetal force always points towards the axis of rotation. It is a vector field with different values depending on the location measured. Its magnitude varies linearly with distance from the axis of rotation, and its direction points towards this axis.

$$\boldsymbol{F}_{\rm cp}(\boldsymbol{r}) = m ( \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \boldsymbol{r}) )$$

where $$\boldsymbol{\omega}$$ is the rotational velocity vector, and $$\boldsymbol{r}$$ the position vector relative to the axis of rotation.

when viewed on a plane perpendicular to the axis of rotation it is

$$\boldsymbol{F}_{\rm cp}(\boldsymbol{r}) = -m\, \omega^2\, \boldsymbol{r}$$

where $$\omega$$ is the rotational speed and $$\boldsymbol{r}$$ the position vector on the plane (as a 2D vector).