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 AA to points B,C,B, C, and D.D.

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

BAC=θDAC=2θ.\begin{aligned} \angle BAC &= \theta\\ \angle DAC &= 2\theta. \end{aligned}

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

Details and Assumptions

  • Gravity is constant at g=9.8 m/s2g = 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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