Conceptual and laboratory exercise to apply Newton’s second law to a system of manyforces

Abstract

A pair of objects on an inclined plane are connected together by a string. The upper objectis then connected to a fixed post via a spring. The situation is first analysed as a classroomexercise in using free-body diagrams to solve Newton’s second law for a system of objectsupon 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 rollingcarts with attached force sensors (to measure the string and spring forces) and a motiondetector (to measure the position, velocity and acceleration of the objects). Aftercharacterizing the rolling friction, cart masses, incline angle and spring constant, thekinematics and dynamics of the system can be accurately modelled with no freeparameters. Representing the data in different ways, notably plotting quantitiesas a function of the displacement of the carts instead of elapsed time, greatlyassists in their interpretation. For example, the acceleration of the carts lies alongtwo straight lines when plotted in that way, the mechanical energy has a zigzagshape and the velocity of the carts traces out a set of joining half-ellipses in phasespace.