A particle of mass is constrained to move on a fixed
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics
Action and Symmetry, Ch Constraints, generalised coordinates and Lagrange's equations twice over
Problem
A particle of mass m is constrained to move on a fixed, frictionless circular track of radius R lying in a vertical plane, under gravity g and a constant applied force of magnitude F0, tangent to the track and always directed so as to increase θ, the angle measured from the bottom of the track.
((a)) Starting from d'Alembert's principle, derive the equation of motion for θ.
((b)) Introduce the arc-length coordinate s=Rθ, measured from the bottom of the track. Working from scratch (writing T and finding the generalised force Qs afresh in terms of s, not by substituting into the answer to part (a)), derive the equation of motion for s, and show it agrees with what part (a) implies when θ is replaced by s/R.