1. The problem statement, all variables and given/known data A point mass m is fixed inside a hollow cylinder of radius R, mass M and moment of inertia I = MR^2. The cylinder rolls without slipping i) express the position (x2, y2) of the point mass in terms of the cylinders centre x. Choose x = 0 to be when the point mass is at the bottom. Show the velocity of the point mass is: x2' = x'(1-cos(x/R)) y2' = x'(sin(x/R)) ii) find the lagrangian for the generalised co-ordinate x and write down the Euler lagrange equation to obtain the equation of motion for x (DO NOT SOLVE) iii) find the frequency of small oscillations about the stable equilibrium state 2. Relevant equations L = T-V 3. The attempt at a solution i) can find this just by using the geometry of the situation and then differentiating ii) I *think* the lagrangian is: L = (x'^2)(M+m)(1-cos(x/R)) + mgRcos(x/R) using the Euler lagrange equation I think the equation of motion is: 2x''(M+m)[1-cos(x/R)] - (x')^2((M+m)/R)(sin(x/R) + mgsin(x/R) = 0 iii) this is where I am stuck.... stable equilibrium is at x = 0 if you use sin x = x then the e.o.m reduces to: 2x''(M+m)(1-cos(x/R)) + mgx/R = 0 I have neglected the middle term since x/R^2 is negligible... I was expecting to get an equation of the form x'' + w^2 x = 0 were w is the frequency ... however when expanding the cos you don't get this... Have I done something really wrong?