## Librational Periods

Libration takes place when a body on periodic motion does not have enough energy to complete a full rotation. When it reaches the end points the sign ofchanges. Librational motion is illustrated in the phase diagram below, the librations taking place betweenand

The time to travel betweenandis given by

This is equal to the timetaken to travel betweenandfor a mechanical system in which energy is conserved. The total period is then

Example: For the linear oscillator,and the turning points are

Substitutingchanges the upper and lower limits toand respectively . The integrand becomesand usinggives