# A problem in relativity

A thin rod of rest length $$L = 3R$$ is parallel to the $$x$$-axis of an inertial system, $$S,$$ and moves with constant speed, $$v=\frac{\sqrt{3}\, c}{2},$$ in the positive $$x$$ direction, remaining parallel to the $$x$$-axis, as shown in the figure. The center, $$D,$$ of the rod is initially located at $$(x,y,z)=(-h,h,0).$$ The plane in the figure has a circular hole of radius $$R$$ centered on the $$y$$-axis, is parallel to the $$xz$$-plane, and moves in the positive $$y$$ direction at constant speed, $$v,$$ remaining parallel to the $$xz$$-plane.

To an inertial observer moving along with the rod in an inertial system we can call $$S^{\prime }$$, the point $$B$$ reaches the $$xz$$-plane of $$S$$ before point $$A,$$ due to the relativity of simultaneity. Therefore, the plane is tilted for this observer, and the rod will pass through the hole without a collision. The plane and hole are rotated at an angle $$\theta$$ in the $$S^{\prime }$$-observer's $$z^{\prime}$$ direction, i.e., the line connecting $$A$$ and $$B$$ makes an angle $$\theta$$ with the $$x^{\prime }$$-axis.

Find the angle $$\theta$$ to the nearest tenth of a degree.

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