Acceleration in a non-inertial reference frame

Question

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

1. why $a_c$ has to have opposite sign

2. why in $a_c$ and $a_{cc}$ the angular velocity is the angular velocity of the platform insted of relative angular velocity

Answer 1

1) why $a_c$ has to have opposite sign.

$a_c$ is the centrifugal acceleration in the rotating frame experienced by the tennis ball. $$\vec{a_c} = (\vec{\omega} \times \vec{r}) \times \vec{\omega}$$ You’re answer for $\vec{a_c}$ gives the correct sign and magnitude. If you draw the vectors for $\vec{\omega}$ and $\vec{r}$ and apply the right hand rule (RHR) it points away from the center of rotation. Whatever sign you choose for $\vec{a_0}$, the centrifugal acceleration must have the opposite sign. In this case it's negative.

2) why in $a_c$ and $a_{cc}$ the angular velocity is the angular velocity of the platform instead of relative angular velocity

You need to use $\omega’$ here because it is the rotation of the platform in the inertial system. In the last line of the exercise text, you are being asked for velocity and acceleration in the platform system. If you use the relative angular velocity, your answers will be different than what is being asked for.

Ref Taylor Classical Mechanics chapter on Mechanics in Non Inertial Frames (Chapter nine in the pre-publication Dec 2001 edition I have)