Write an expression for the KE of a body rolling without slipping with point of contact as reference.

Explanation

Consider a body rolling without slipping with point of contact as reference we can also arrive at the same expresssion by taking the momentary rotation happening with respect to the point of contact. If we take the point of contact as O, then
`KE=1/2 I_(0)omega^(2)`
Here `I_(0)` is the moment of inertia of the object about the point of contact.
By parallel axis theorem `I_(0)=I_(CM)+MR^(2)`
Further we can write` I_(0)=MK^(2)+MR^(2)`
`v_(CM)=R omega` or `omega=(v_(CM))/R`
`KE=1/2 (MK^(2)+MR^(2))(v_(CM)^(2))/(R^(2))`
`KE=1/2 Mv_(CM)^(2)(1+(K^(2))/(R^(2)))`