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Center of Mass, Momemtum, Collision
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Center of Mass of semi circular wire

B) Center of Mass of semi circular wire:


         

Derivation:


Total length of semi circular wire = R

Elemental length = Rdq

So

          

                       

 

 

 

C) Center of Mass of a uniform semi circular plate:



Derivation:


Here the element chosen is a thin wire (semi circular) of radius r.

As derived earlier, the for this is at .


Dumb Question:

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) Center of Mass of hemispherical shell:



Derivation:





Dumb Question:

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) Center of Mass of a hemisphere:



Derivation:




 


Dumb Question:

1)      Why is volume of elemental disc = Rdq (cosq) (pR2cos2q) and not

Rdq (pR2cos2q)?

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!

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  Electricity and Effect of Current      Centre of Mass      Work Power & Energy      Magnetic Effects of Current      Introduction to Laws of Motion      AIEEE 2009 Solutions and Analysis      Chemical Bonding      Quadratic Equations      Progression And Series      dasda      Chemical Bonding      General Principles of Extraction of Metals      Ether and epoxides      Atomic Structure      Parabola      IIT-JEE Chemistry by M.K.Tiwri      Properties of Matter      Heat & Thermodynamics      Aldehydes and ketones      Disha - Path to Success Series     
 



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