A small block of mass m slides from rest at point A on the inner surface if a smooth hemispherical surface of radius R.

a) Write down the Lagrangian function of the system using r and theta as generalized coordinates.

b) Using the constant equation r - R = 0 and Lagrange multiplier lambda, write down the Lagrange equations of motion for r and theta.

c) Find the angular speed and magnitude of the force between the block and the hemispherical surface as a function of theta.

d) Find the frequency of small oscillations of the block