Question

2. (15 points) A bead of mass m slides under the influence of gravity along a straight wire. The wire can pivot around the support point at the bottom so that the angle a between the wire and the vertical can change. In addition, the wire rotates around vertical axis with a constant angular velocity w

Find the Lagrangian.

Answer 1

Let o be the origin x = r sin alpha y = r cos alpha x = r sin alpha + r cos alpha alpha y = r cos alpha + r (-sin alpha)alpha x^2 + y^2 = r^2 + r^2 alpha^2 kinetic energy = 1/2 m (x^2 + y^2) potential energy pE = mgy L = kE - pE L = 1/2 m(r^2 + r^2 alpah^2) - mg r cos Alpha as Alpha = wt Alpha = w L = 1/2 m (r + r^2w^2) - mg r cos Alpha