This paper investigates contact problems in mechanical couplings that allow free movement, focusing on the dynamic interactions between contact surfaces under varying operational conditions. Freely moving couplings, commonly used in mechanical systems to transmit torque while accommodating misalignment or axial motion, often experience complex contact behaviors that can lead to wear, vibration, or failure. Through analytical modeling and simulation, this research work analysis the effects of load, relative motion, and material properties on contact performance. The results provide insights into optimizing coupling design to minimize mechanical stress and improve system reliability.