$1.5.12.$ A bead can move along a ring of radius $R$, pushed by a spoke that rotates uniformly with angular velocity $\omega$ in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead.
The bead will move along the spoke with the velocity $\upsilon_{\parallel}$, and since the spoke rotates, the bead will have a velocity component perpendicular to the spoke: $\upsilon_{\perp}$
Then the absolute velocity $\vec{v}$ in Non-inertial reference frame will be equal to the vector sum of $\vec{v}_{rel}$ and $\vec{v}_{cl}$
$$ \upsilon_{\perp}=\omega r=\omega\cdot 2R\cos\alpha $$On the other hand,
Then the absolute velocity $\vec{v}$ in Non-inertial reference frame will be equal to the vector sum of $\vec{v}_{rel}$ and $\vec{v}_{cl}$
$$ \upsilon_{\perp}=\upsilon \cos\alpha $$Since the speed of a bead moving in a circle is directed perpendicular to the radius.
Then
$$ \omega\cdot 2R\cos\alpha=\upsilon \cos\alpha $$ $$ \upsilon=2R\omega=\text{const} $$Since the speed does not change in magnitude, the bead has only normal acceleration
$$ a_n=\frac{\upsilon^2}{R}=\frac{4\omega^2R^2}{R}=4\omega^2R $$$$a_n=4\omega^2R$$