Spin-lattice relaxation of methyl protons in 2,6-dichlorotoluene and N-methyl phthalimide, each dissolved in CDCl$_3$, has been studied at 34$^{circ}$C and the contribution from spin-rotation interaction to the relaxation process has been separated from that due to dipole-dipole interactions among methyl protons. The results show that the spin-rotational contributions to the initial rate of relaxation in 2,6-dichlorotoluene and N-methyl phthalimide amount to 18 and 31%, respectively, of the total relaxation rate at 34$^{circ}$C. The method of separating the spin-rotational contribution from that of dipolar interactions adopted in this paper is based on the well known fact that in an A$_3$ spin system such as methyl protons in liquid phase dipolar relaxation mechanism gives non-exponential decay of the z-component of total magnetization of protons while the random field fluctuation such as spin-rotational mechanism causes exponential decay.