Grooves through a sphere

\(\overline{AC}\) is a diameter to the sphere, while \(\overline{AB}\) and \(\overline{AD}\) are chords, such that:

\[\begin{align} \angle BAC &= \theta\\ \angle DAC &= 2\theta. \end{align}\]

An object of mass \(M\) is dropped thrice, each time through each groove, and the time taken for it to emerge out of the groove is measured.

Now, let \(a:b:c\) (where \(a,b,\) and \(c\) are coprime integers) be the ratio of the respective times that the object takes to exit through grooves \(\overline{AB}\,\) \(\overline{AC}\) and \(\overline{AD}.\) Find \(a+b+c.\)

Details and Assumptions

• Gravity is constant at \(g = 9.8\text{ m/s}^2\) vertically downwards at all times.
• The grooves have negligible friction, and the object is considered to be point sized.

Please do note that this problem was taken from a teacher of mine.