Free fall of a symmetrical gyroscope in vacuum

Abstract

This paper describes the behavior of a symmetrical gyroscope when it falls down from a certain height over the surface of the earth. The assumptions of the model are that (i) the gravitational acceleration has a constant value and (ii) no aerodynamic or other external forces are exerted on it, thus the expression ‘in vacuum’ is in the title. Using body- and space-fixed reference systems, several formulations are developed to derive the equations of motion. Among them, particular attention was paid to the conservation of the angular momentum in the space-fixed reference system. Closed-form but lengthy analytical expressions were derived for all the three Euler angles, under arbitrary initial conditions. Approximate sinusoidal expressions were derived for an initial lean angle equal to ten degrees. Through computer simulation it is shown that the initial lean angle and the spin of the gyroscope highly influence the variation of its orientation during the free fall. Based on these findings, it was concluded that low spins could cause a considerable change such as one degree in the lean angle at landing, thus may mislead the experimenter who will be cheated measuring a slightly smaller gravitational acceleration.