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.

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    <h3> Statement </h3>
    <p>
        $4.2.26.$ 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.

    <h3>Solution</h3>
    <p>

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 $(3)$

From where we find the angular velocity $$\boxed{\omega = \sqrt{\frac{g\tan\alpha}{R-(l+r)\sin\alpha}}}$$

    <h4>Answer</h4>
    <p>
        $$\omega=\sqrt{(g\operatorname{tg}\alpha)/[R-(l+r)\operatorname{sin}\alpha]}.$$
    </p>

    <p style="text-align: right; font-style: italic; font-size: 14;">
      Almaskhan Arsen<br>
    </p>

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