A block of mass m is attached to a wedge of mass M by a spring with spring constant k. The inclined frictionless surface of the wedge makes an angle d to the horizontal. The wedge is free to slide on a horizontal frictionless surface as shown in Fig. 7.11. (a) Show that the Lagrangian of the system is L=(M+m)/2(x·)^2+1/2m(s·)^2+m(x·)(s·)cosak/2(sl)^2mg(hs sin a), where l is the natural length of the spring, x is the coordinate of the wedge and s is the length of the spring. (b) By using the Lagrangian derived in (a), show that the equations of motion are as follows: (m + M)(x··) + m(s··) cos d = 0, m (x··) cos a + m(s··) + k(ss0) = 0, where s0 = l + (mg sin a)/k. (c) By using the equations of motion in (b), derive the frequency for a small amplitude oscillation of this system.
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