Lecture 12
Motion with constraints

In this lecture we are going to deal with motion of particles when they move under constraints applied on their motion. Of course the motion is determined by Newton 's second law i.e., by solving the equation of motion

where is the total force – which is the sum of the externally applied and those arising from other particles as well as the constraints in the system - acting on a body of mass m and is producing an acceleration . Recall from lecture 9 that constraints are the restrictions applied on the movement of a body by various means and are brought about by constraint forces . For example, I may restrict the body to move along a straight wire (see figure 1). In that case the component of only along the wire will affect the motion of the mass (if there is no friction) and its perpendicular component will be nullified by the normal reaction of the wire, which is the constraint force in this case. As another common example of constrained motion take the motion of two masses at the end of a rope going over a frictionless pulley (Atwood's machine) also shown in figure 1.

In this case also, the motion of one mass is determined by not only by the gravitational force on it alone but also by the weight of the other mass. Thus the two masses are not fully free to move under their own weight and the motion is constrained. The constrained is brought about through tension in the rope, which is then the constraint force.

We have seen two simple examples of constrained motion. We make an observation that constraints can be caused either by restricting the motion externally, as was the case for a mass on a wire, or by the presence of other bodies that are themselves moving, as in the example of two masses over a pulley. In lecture 9 we had introduced these concepts and stopped at that. However, for obtaining the positions and velocities of particles under constraints, we wish to express these constraints mathematically and account for them while solving the equations of motion. This is what this lecture is going to be about.