(a) A particle of mass m slides down a smooth spherical bowl, as in Fig. 7.8. The particle remains in a vertical plane (the xz-plane). First, assume that the bowl does not move. Write down the Lagrangian, taking the angle I‘ with respect to the vertical direction as the generalized coordinate. Hence, derive the equation of motion for the particle. (b) Assume now that the bowl rests on a smooth horizontal table and has a mass M , the bowl can slide freely along the x-direction. (i) Write down the Lagrangian in terms of the angle T and the x - coordinate of the bowl, x. (ii) Starting from the corresponding Lagrange's equations, obtain an equation giving (x··) in terms of T, (T·) and (T··) and an equation giving (T··) in terms of (x··) and T . (iii) Hence, and assuming that M >> m, show that for small displacements about equilibrium the period of oscillation of the particle is smaller
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