In this paper, the global finite time tracking problem of desired trajectories for a wide group of nonlinear systems subjected to unbounded disturbances and uncertainties is addressed. The considered class for nonlinear systems, containing a chain of interacting double integrator subsystems, can describe and model variety of practical systems. To tackle the mentioned problem, the adaptive control approach is incorporated with a nonsingular terminal sliding mode control and a new control method is suggested. Finite time adaptation laws are designed to estimate unknown constant coefficients of upper bound of uncertainties and disturbances within a finite time. Analytical studies demonstrate that by applying the proposed control scheme, tracking errors will converge to zero exactly after a tunable finite time. Also, a relation is presented for calculating an upper bound for the mentioned time which depends on initial conditions of the states and some arbitrary parameters. The convergence rate of the tracking errors could be tuned by proper choice of these parameters. Finally, analytical results are simulated on two practical systems including thrust active magnetic bearing (TAMB) and two-link robot manipulator to demonstrate the effectiveness of the suggested control technique.