The vibration of a rotor with variable mass as a one-mass system with two degrees of freedom is investigated. An analytical procedure for solving of the system of two coupled second-order differential equations with slow time variable parameters is developed. The trajectory of the rotor center for various initial conditions is obtained. The method developed in the paper is applied for determining the vibration of the work piece during turning operation. The analytically obtained results show the influence of mass variation, cutting parameters and cutting force on the dynamic properties of the work piece. A decrease in mass of the work piece increases the amplitude of vibration. The amplitude increase is faster if the cutting velocity is higher. The obtained results are compared with experimentally obtained ones. The correlation between vibration and surface roughness is determined.