A new method which uses carbon-13 NMR with rapidly decelerated sample spinning to study the reorientation of a nematic and a smectic liquid crystal is presented. When a macroscopically ordered liquid crystal is subjected to a new torque of sufficient magnitude, the direction of the macroscopic ordering changes in response to the new torque. In our experiments, the liquid crystal is spun rapidly at an angle slightly less than the magic angle, so that its director aligns along the spinning axis. The rapid spinning is suddenly stopped, and the liquid crystal molecules reorient so that the director would align along the magnetic field. The carbon-13 chemical shift, which is very sensitive to the angle that the molecular segment forms with respect to the magnetic field, is observed as a function of time following the stopping of the spinner. By fitting the chemical shift data as a function of time to a physically meaningful equation, a time constant can be calculated for various molecular segments of the liquid crystal molecule. This time constant provides information about the macroscopic reorientation of the liquid crystal from the alignment caused by the spinning torque to the alignment caused by the magnetic field. A room temperature nematic, 4-n-pentyl-4’-cyanobiphenyl (5CB), and a room temperature smectic, 4-n-octyl-4’-cyanobiphenyl (8CB), are studied by this technique. The-reorientation time constant of 8CB is about 300 times larger than that of 5CB, and the reorientation time constant of the aromatic core is slightly smaller than the reorientation time constant of the aliphatic chain for both compounds. These results are among the first experimental data in the field of studying liquid reorientation on the level of the molecular segment, and a possible explanation for these results is presented.