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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73. A 70-yeаr-оld mаn hаs a tоtal chоlesterol of 280 mg/dL with LDL of 195 mg/dL and HDL of 32 mg/dL. He has diabetes and smokes half a pack of cigarettes per day. Which statement best summarizes his cardiovascular risk profile?
Dаvid wаs gоing dооr to door on behаlf of his local political candidate, handing out informational flyers. Upon arriving at Victor’s house, David noticed a lit candle inside the home. David knocked and knocked to give Victor the informational flyer, but no one answered the door. Fearing Victor had left the candle on and then left the house, David tried opening the door to see if he could put it out. To David’s surprise, the door was unlocked and opened easily. David went inside to put out the candle. On his way over to the candle, however, David noticed a mint condition baseball signed by his favorite player. David knew this baseball was worth thousands of dollars. David just had to have it, so, he put the baseball in his pocket. David went over to the candle to put it out. He picked up the candle, but it was very hot to the touch and David inadvertently dropped it to the floor. The flame from the candle quickly spread throughout the rest of the room. Panicking, David fled the house. Police later found and charged David with Burglary, Larceny, and Arson. With which of the following crimes, if any, should Daniel be found Guilty? Discuss, including appropriate mitigations and defenses, if any. 1) Burglary? 2) Larceny? 3) Arson?