Grooves through a sphere

A sphere, as shown in the image is placed on a horizontal surface with three grooves drilled into it from point \(A\) to points \(B, C,\) and \(D.\)

\(\overline{AC}\) 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.


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