Foolish Fun with COM of an IIT JEE alien.

A hypothetical alien lives on a hypothetical Planet and has a 2-dimensional face.

If he is made out of pieces of a homogeneous wire of uniform line thickness, and let mass of his two eyes (inner circles) is \(m\), and mass of the his nose and his lips is also \(m\) each ( Vertical and horizontal lines ) also mass of boundary of his face is \(6m\).

The coordinates of the centres of the different parts are :

  • Outer circle ( Face Boundary ) is (0,0)
  • Left circle ( left eye ) is (-a,a)
  • Left circle ( Right eye ) is (a,a)
  • Vertical Line ( Nose ) is (0,0)
  • Horizontal Line ( Lips ) is (0,-a)

In this situation the \(y\)-coordinate of the centre of mass of the alien is \({ y }_{ o }\).

Then an friend of this alien give an slap to him due to their Personal matters. So This alien becomes sad and curls his lips about it's centre into a semi-circle of radius \(r\) .

Then find the radius \(r\) such that the new \(y\)-coordinate of the centre of mass of this alien becomes \(-{ y }_{ o }\).

Details and Assumptions

  • \(a=10\)
  • All body parts have uniform same mass density.


This question is inspired from an question of IIT JEE 2009: Q-41
This is Part of set : Foolish things

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