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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