A body of mass \( m_{1} \mathrm{~kg} \), is placed on a rough horizontal table, the coefficient of kinetic friction between them is \( \mu_{\mathrm{k}} \) the body connected by a light string passing over a smooth pulley fixed at the edge of the table and another body of mass \( m_{2} \mathrm{~kg} \). is hanging vertically at the other end of the string, if the system moves with uniform velocity, then the coefficient of kinetic friction between the body and the table \( =\ldots \ldots \ldots \ldots \ldots \). \( \begin{array}{llll}\text { (a) } \frac{m_{1}}{m_{2}} & \text { (b) } m_{2}-m_{1} & \text { (c) } \frac{m_{2}}{m_{1}} & \text { (d) } m_{1} m_{2}\end{array} \)

When the system moves with uniform velocity, it means that the forces acting on the bodies are balanced. The tension in the string pulling the mass \(m_1\) on the table is equal to the frictional force opposing its motion. By Newton's second law and the conditions of equilibrium, you find that the tension is also equal to the weight of the hanging mass \(m_2\). This gives us the equation \(\mu_k m_1 g = m_2 g\), leading to the conclusion that the coefficient of kinetic friction \(\mu_k\) can be expressed as \(\frac{m_2}{m_1}\). So when you look for an answer to the question posed, the correct choice is (c) \(\frac{m_{2}}{m_{1}}\). In practical applications, this scenario is important in various engineering fields where pulleys and friction play key roles, like in cranes, elevators, and pulling systems in machinery. Understanding how to calculate the right friction coefficients can help in designing systems that operate efficiently and safely under such conditions. Balancing forces and understanding how the friction works becomes crucial for safe operations in these systems!