Conceptual and laboratory exercise to apply Newton's second law to a system of many forces

A pair of objects on an inclined plane are connected together by a string. The upper object is then connected to a fixed post via a spring. The situation is first analysed as a classroom exercise in using free-body diagrams to solve Newton’s second law for a system of objects upon which many different kinds of force are acting (string tension, spring force, gravity, normal force, and friction). Next, the setup is replicated in the laboratory using rolling carts with attached force sensors (to measure the string and spring forces) and a motion detector (to measure the position, velocity and acceleration of the objects). After characterizing the rolling friction, cart masses, incline angle and spring constant, the kinematics and dynamics of the system can be accurately modelled with no free parameters. Representing the data in different ways, notably plotting quantities as a function of the displacement of the carts instead of elapsed time, greatly assists in their interpretation. For example, the acceleration of the carts lies along two straight lines when plotted in that way, the mechanical energy has a zigzag shape and the velocity of the carts traces out a set of joining half-ellipses in phase space.