# 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.

Note

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