I have a conceptual doubt on the relation about the acceleration in a non inertial system. To explain the doubt without misunderstanding, I'm going to write the text of an exercise:
A platform rotates with $\omega=10$ rad/s around $z$-axes. A ball is connected, with a yarn to $z$. Its distance to the axes is 15 cm and it rotates with $\omega=10$ rad/s. There isn't friction between platform and ball. Suddenly, the angular velocity of the paltform is reduced to $\omega'=2$ rad/s. Find velocity and acceleration of the ball in the system of the platform.
I know that the ball acceleration calculated in an inertial reference frame is:
$\vec{a_0}=\vec{a'}+\vec{a_{cc}}+\vec{a_c}$
where $\vec{a'}$=acceleration calculted in a non inertial reference frame, $\vec{a_{cc}}=2\vec{\omega} \times \vec{v'} $, $\vec{a_c}=-\omega ^2r \vec{u_r}$.
So $\vec{a'}=\vec{a_0}-\vec{a_{cc}}-\vec{a_c}$
I have written $a'=\omega^2 r-2 w_r v'+w_r^2r$, where $\omega_r=\omega-\omega'$
But the correct formula is $a'=\omega^2r-2\omega'v'-\omega'^2r$
I don't understand
why $a_c$ has to have opposite sign
why in $a_c$ and $a_{cc}$ the angular velocity is the angular velocity of the platform insted of relative angular velocity