A particle of mass m is constrained to move on the inside surface of a smooth cone
Question:
A particle of mass m is constrained to move on the inside surface of a smooth cone of half-angle α. The particle is subject to a gravitational force. Let the axis of the cone correspond to the z-axis (gravity acts in the minus z direction) and let the apex of the cone be located at the origin. Work in cylindrical polar coordinates. Note: in general, the particle is not constrained to follow a circular orbit.
a) Write down the Lagrangian in cylindrical polar coordinates. The cone imposes a constraint on the motion of the particle. Determine this constraint (not the constraint force) and use it to eliminate the z-coordinate from the Lagrangian.
b) Derive the equations of motion.
c) Identify any conserved quantities in this system.
d) What value of θ dot is required for circular motion?
e) Reduce this system to 1D by showing that it is possible to write down an effective potential Ueff(r) = mgr cotα + l^2/2mr^2. What is l? Express l in terms of the other variables in this problem.
f) On a single set of axes, draw Ueff(r) as well as the two individual terms on the right-hand-side of the equation for Ueff(r) above. Comment on the trajectories.
Vector Mechanics for Engineers Statics and Dynamics
ISBN: 978-0073212227
8th Edition
Authors: Ferdinand Beer, E. Russell Johnston, Jr., Elliot Eisenberg, William Clausen, David Mazurek, Phillip Cornwell