1. The problem statement, all variables and given/known data A point particle of mass m slides without friction within a hoop of radius R and mass M. The hoop is free to roll without slipping along a horizontal surface. What is the frequency of small oscillations of the point mass, when it is close to the bottom of the hoop? 2. Relevant equations Euler Legrange Equations and taylor approximation for cosine of phi. cosine of phi is approximated as being porportional to phi squared 3. The attempt at a solution Well, I have the solution in the above. I just don't understand why cos(phi)=1 going from line 3 to line 4. It make sense because the angle is nearly 0, but still, isn't it more proper to do second order taylor approximation? Also, in line 5, a small angle approximation was used. But why here? Why in the potential energy expression and not the kinetic? Edit: I think I figured it out. So at the bottom, or very near it, the height which is directly related to distance to the equilibrium point, is very near R. The situation is observed in both the potential and kinetic. But kinetic is tangential, but since the particle is virtually at the bottom, the tangential is in the x, and the displacement forward in the x is very small, almost 0, while the height from the bottom is very small as well but somehow does not approach 0 as fast as the displacement in the x. Is this correct?