Prove that moment of inertia of uniform ring of mass M and radius R about its geometric axis is MR^(2)

Prove that moment of inertia of uniform ring of mass M and radius R ab

1.	Prove that moment of inertia of uniform ring of mass M and radius R about its geometric axis is MR^(2)
Answer» Solution :ASSUME the DX is the element of length dx on the ring as shown in fig. The mass per unit length of the ring. `LAMBDA=(M)/(2piR)` `therefore` Mass of length dx is dx, `m=lambdadx` `therefore m=(M)/(2piR).dx....(1)` `therefore` The moment of inertia about the axis ZZ. `dI=mR^(2)` `=(M)/(2piR)R^(2)dx` `therefore dI=(MR)/(2PI)dx` `therefore` For the moment of inertia along the whole ring take the integral from `x=0` to `x=2piR` `therefore I=intdI` `=underset(0)overset(2piR)int(M)/(2pi)Rdx=(MR)/(2pi)underset(0)overset(2piR)intdx` `=(MR)/(2pi)[x]_(0)^(2piR)=(MR)/(2pi)[2piR-0]` `therefore I=MR^(2)`