Question

4. A particle of mass m is constrained to slide without friction on the surface of a smooth circular bowl of mass M with inner radius R as shown in the figure. The bottom of the bowl lies on a horizontal table and is free to slide without friction along the table. All motion is constrained to the plane of the page. Assume uniform gravitati acceleration. =T-V- State the Lagrangian for this system. Derive the differential equations of motion for the particle and the bowl. You de not have to solve these equations. a. b.

Answer 1

Co-ordinates of bowl = (x, 0) co-ordinates of mass = (x + R sin x, R cos x) kinetic energy of system T = 1/ 2 m x^2 + 1/ 2 M (x + R cos alpha alpha)^2 + 1/2 M (- R sin alpha alpha)^2 T = 1/ 2 h x^2 + 1/ 2 M (x^2 + R^2 alpha^2 + 2 R cos x alpha) potential energy U = mg R cos alpha a. Lagrangian L = T - U = 1/ 2 h x^2 + 1 /2 M (x^2 + R^2 alpha^2 + 2 R cos alpha x alpha - m g R cos alpha) b. For x d/dt (partial differential L/ partial differential x) - partial differential L/ partial differential x = 0 d/d t (m x + M x + m R alpha cos alpha) - 0 = 0