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As I currently understand it, Hyperconjugation occurs in carbocations and alkenes via the interaction of the electrons in the MO of a C-H bond at an adjacent carbon to bring stability to the system.The interaction requires for in phase overlap and occurs when the C-H bond is in the same plane as the vacant p orbital. (or the pi bond in case of alkenes) It is thus not seen if the beta hydrogens are in the plane perpendicular to them.

If this is true, then how does hyperconjugation work, if at all in cyclic systems where free bond rotation is impossible and thus the orbitals will never align perfectly? What conceptual bridge am i missing here?

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Cyclic systems aren't planar. The system orients itself in such a way that the vacant p orbital is in plane in order to gain the ability to engage in hyperconjugation which stabilizes it very well.

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