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| A cart of mass \(M\) is placed on rails and attached to a wall with the help of a massless spring with constant \(k\) (as shown in the Figure below); the spring is in its equilibrium state when the cart is at a distance \(x_0\) from the wall. A pendulum of mass \(m\) and length is attached to the cart (as shown). (a) Write the Lagrangian \(L(x, \dot{x}̇, \theta, {\dot\theta}\))for the cart-pendulum system, where \(x\) denotes the position of the cart (as measured from a suitable origin) and \(\theta\) denotes the angular position of the pendulum. (b) From your Lagrangian, write the Euler-Lagrange equations for the generalized coordinates \(x\) and \(\theta\). |  |
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