$4.2.26.$ A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius $r$ is tied to the side wall of the vessel on a thread of length $l$; during rotation the thread forms an angle $\alpha$ with the wall. Determine the angular velocity of rotation of the vessel.
Since the ball does not float, the vertical Archimedean force is completely compensated by the vertical component of the thread tension force $$T\cos\alpha=\frac{4\pi r^3}{3}\rho g$$ From where we could find the thread tension force $$T=\frac{4\pi r^3}{3\cos\alpha}\rho g\quad (1)$$ Alternatively, we could write an equilibrium condition for the Horizontal Archimedean force, which is compensated by the horizontal force of the thread tension $$T\sin\alpha = \rho V\omega^2x\quad (2)$$ It can be seen from the figure that the ball's center of mass is at a distance $l+r$ from the point of attachment of the thread: $$x=R-(l+r)\sin\alpha$$ After substituting into $(2)$ we obtain $$\rho \frac{4\pi r^3}{3}\omega^2\left(R-(l+r)\sin\alpha\right) = T\sin\alpha \quad (3)$$ Substituting $(1)$ into $(3)$
$$\omega=\sqrt{(g\operatorname{tg}\alpha)/[R-(l+r)\operatorname{sin}\alpha]}.$$
Almaskhan Arsen