Balancing a Semicircle Frame

A semicircle with a constant mass per unit length is placed on a horizontal surface such that it doesn't topple in any direction. The touch point \(P\) is assumed not to slip from the surface. Also, the imaginary line connecting the two end points of the semicircle forms an angle of \(\theta\) with the surface.

Find the measure of \(\theta\) (in degrees) to 2 decimal places.

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