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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