In a smooth fixed hemisphere, a stick of mass m freely lies so that its angle with the horizon is equal to \alpha, and its end extends beyond the edge of the hemisphere. With what forces does the wand act on the hemisphere at the points of contact between A and B?

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    <h3> Statement </h3>
    <p>
        $2.8.11.$ In a smooth fixed hemisphere, a stick of mass $m$ freely lies so that its angle with the horizon is equal to $\alpha$, and its end extends beyond the edge of the hemisphere. With what forces does the wand act on the hemisphere at the points of contact between $A$ and $B$?

    <h3>Solution</h3>
    <p>

Equilibrium condition for vertical and horizontal axes respectively $$mg = N_A \sin2\alpha +N_B\cos\alpha\quad(1)$$ $$N_A \cos2\alpha = N_B\sin\alpha\quad(2)$$ Moment of forces with respect to point $A$ $$mg\cos\alpha \cdot \frac{l}{2} = N_B\cdot l_{AB}\quad(3)$$ Alternatively, moment of forces with respect to point $B$ $$mg\cos\alpha \cdot \left(l_{AB} - \frac{l}{2}\right) = N_A\sin\alpha\cdot l_{AB}\quad(4)$$ Substituting in $(2)$, $(3)$ and $(4)$ $$\frac{l}{l_{AB}} = \frac{2\cos2\alpha}{\cos^2\alpha}$$ $$\boxed{N_A = mg\tan\alpha}$$ $$\boxed{N_B = mg\frac{\cos2\alpha}{\cos\alpha}}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$F_A=mg\operatorname{tg}\alpha; F_B=mg\cos2\alpha/\cos\alpha$$
    </p>
    <p style="text-align: right; font-style: italic; font-size: 14;">
      Almaskhan Arsen<br>
    </p>


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