That's one huge yo-yo!

In an alien planet, an alien wants to play yo-yo. The gigantic yo-yo consists of two solid disks, each of mass \(15 kg\) and radius \(5 m\), connected by a central spindle of radius \(2 m\) and negligible mass. A string is coiled around the central spindle. The yo-yo then is placed upright on a rough flat surface and the string is pulled gently with a tension of \(10 N\) at an angle \(30^\circ\) to the horizontal. The pull is gentle enough to ensure that that yo-yo does not slip nor lifts off the ground.

The acceleration of the center of mass of the yo-yo can be expressed in the form \(\dfrac{a\sqrt{b}-c}{d}\). Where \(a\) is as large as possible, \(gcd(a,d)=5\) and \(gcd(c,d)\)=1. What is the value of \(a+b+c+d\)?

  • Assume that the only difference between the planet and earth is the size of the inhabitants and the planets, nothing else.
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