study materialphysicsmechanicscentre of mass
Center Of Mass 2
Center of mass, Momemtum, Collision 
Introduction
B) _{ } Derivation: Total length of semi circular wire = R Elemental length = Rdq So _{} _{ }
C) _{ } Derivation: Here the element chosen is a thin wire (semi circular) of radius r. As derived earlier, the _{ }for this is at_{ }. _{ } Dumb Question: <![if !supportLists]> 1) Why is _{ }here? Ans: The mass dm is a semi circular thin wire whose position is variable (y is not unique), so we concentrate dm mass on the COM of this wire that is at_{ }.
D) _{ } Derivation: _{ } _{ } _{ } Dumb Question: <![if !supportLists]> 1) Why is y = Rsinq here? Ans: The Center of Mass of elemental ring is at its center (by symmetry) which is at a height of Rsinq from origin; hence y = Rsinq.
E) _{ } Derivation: _{ } _{ } _{ } Dumb Question: <![if !supportLists]> 1) Why is volume of elemental disc = Rdq (cosq) (pR^{2}cos^{2}q) and not Rdq (pR^{2}cos^{2}q)? Ans: This is a solid hemisphere. If we consider the curvature at the ends of the disc as negligible (as we do in a hollow shell as in the previous case) then the integration starts yielding wrong results. So as a tip we can take this that whenever the integration is being done over solid objects, the curvature effects cannot be neglected!
