# 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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