Question

A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. (2 pts) In software, plot the potential U) Set your axis ranges: -2 SxS2 and 0s b. (5 pts) Find the equilibrium positions and determine whether they are stable or c. (8 pts) Describe the motion of the particle for total energy values E-О.0.05. 1.0, 2.0 unstable. Explain how you arrived at your answers. (all in Joules). What I am looking for here is a written description of how the particle moves in x for each different energy value. You only need to consider motion for a particle that is "inside" the potential (that is, in the region -2 Sr2) d. (5 pts) Find an expression for ), the velocity as a function of position, given that this is a conservative system, in terms of the total energy E. e. (6 pts) Use the vx expression you found to make a phase diagram of the motion showing the phase paths for E 0.5, 1.0, 2.0. Assume-1. Note that you can get a complete phase diagram by plotting +r) and- on the same graph. Set your axis ranges: -2sx S2.-2SYS2 The potential in the previous problem [U(x)--2x1+x*+1] can be approximated as U(x)- x2-1 for small values around x= 1 4. (5 pts) Find an expression for the force in this case, given that the system is conservative (5 pts) Write the equation of motion for this case, for particle of mass m (2 pts) As your phase diagram in the previous problem should show, the motion in this region is oscillatory. (Make sure you got a result that fits this!) What is the frequency of oscillation? a. b. c.

Answer 1

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2. 2 21 2 m2 o.5, 1, 2