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PHY 3221 Classical Mechanics, Homework #5, due 8:30 am, 02/27/12 1 Consider a mass moving on a frictionless surface and attached to a (horizontal) spring. a) Write down an expression for the conserved energy of the mass-spring system. b) Express the total energy in terms of the maximum amplitude of the motion for an oscillating mass. c) Express the total energy in terms of the maximum velocity of the oscillating mass. d) Write down a relation between the maximum amplitude and the maximum velocity. e) Using your answers above, solve for an expression giving the velocity of the motion in terms of the displacement from equilibrium. f) Integrate this relation to find a relation between displacement and time. 1 2 Consider an elastic collision of a moving mass m1 and a stationary mass m2 , where the masses are not equal. a) Write down equations for the conservation of momentum. b) Write down an equation expressing the conservation of energy. c) Algebraically combine the equations to find the cosine of the angle between the particles after the collision. d) Find the limit for the angle when m1 >> m2 . e) Find the limit or the angle when m1 << m2 . 2 3 An isolated system consists of three interacting particles. a) Write out expressions for the potential energies of each of the three particles in terms of their relative separations. b) Find the forces on each of the three particles. c) Write out an expression for the total energy of the system, and identify a potential energy function for the whole system. d) Show that you can take derivatives of this potential energy function which give the three forces you found in part b). 3