What is the force acting on a rotating body in a non-inertial frame?

[Silviu]

Hello! The angular velocity in the non-inertial frame of a rotating body of mass m is $\Omega$ and I need to find the force acting on the body (in the non-inertial frame associated with the body). In the book they say (without any derivation, they just state it) that the force is: $$\bar{F}=m\bar{g}+2m\bar{v}\times\bar{\Omega}+m(\bar{r}\times\bar{\dot{\Omega}})+\frac{\partial}{\partial \bar{r}} \frac{m( \bar{\Omega}\times \bar{r} )^2}{2}$$ So we have gravity, Coriolis force and centrifugal force. But what is the second to last term and how do you derive it? Thank you!

 

[BvU]

The $\dot\omega$ shlould give you a clue

 

[Silviu]

The $\dot\omega$ shlould give you a clue

Well it looks like something associated with a torque

 

[BvU]

Ever hear of $\vec \tau = I\vec \alpha$ ?