Nоnlineаr Spring Oscillаtоr PHY 2048C Cumulаtive Final Examinatiоn Points: 10 Suggested Time: 25–30 minutes Instructions: Show all work. Begin each derivation with an appropriate fundamental physics principle. Clearly define any additional symbols you introduce. Unsupported answers may not receive full credit. A cart of mass m moves without friction along a horizontal track. It is attached to a nonlinear spring whose restoring force is Fs(x) = −k0x(1 + αx) where x is measured from equilibrium, k0 > 0, and α is a constant with units of inverse length. The cart is released from rest at x = A, where 1 + αx remains positive over the entire motion. Tasks Part A. Derive the spring potential-energy function U(x), choosing U(0) = 0. Part B. Derive the speed of the cart as a function of position, v(x), while it moves from x = A toward equilibrium. Part C. Determine the cart's speed as it passes through x = 0. Part D. Write, but do not evaluate, a definite integral that gives the time required for the cart to move from x = A to x = 0. Part E. Linearize the equation of motion for sufficiently small displacement and determine the corresponding small-amplitude angular frequency ω0. Part F. Determine the magnitude of the cart's acceleration at the instant it is released. State how this compares with the acceleration predicted by the small-amplitude model.
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