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Today I read a book in mechanics and encountered a funny proposition about rigid body with fixed point. Perhaps somebody will be interested to propose it to students as a task. This proposition is almost correct:)

Consider a rigid body with a fixed point ##O##. Let ##Oxyz## be a coordinate frame connected with this rigid body. Consider a unit sphere with center at ##O## as well. Now let us move the body from the initial position such that the axis $Oz$ describes a closed curve (without self-crossings) on the sphere and the projection of body's angular velocity on ##Oz## is equal to zero identically. It turns out that when the axis ##Oz## comes to the initial position other two axes will be rotated relative their initial position. The angle of rotation equals (up to the sign) the area of a figure drawn by the axis ##Oz## on the sphere.

Consider a rigid body with a fixed point ##O##. Let ##Oxyz## be a coordinate frame connected with this rigid body. Consider a unit sphere with center at ##O## as well. Now let us move the body from the initial position such that the axis $Oz$ describes a closed curve (without self-crossings) on the sphere and the projection of body's angular velocity on ##Oz## is equal to zero identically. It turns out that when the axis ##Oz## comes to the initial position other two axes will be rotated relative their initial position. The angle of rotation equals (up to the sign) the area of a figure drawn by the axis ##Oz## on the sphere.

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