(Solved): 1. Suppose a system has a potential energy: U=A(x2R2)2 where A is a constant and R is some fixed ...

1. Suppose a system has a potential energy: $$U=A{\left(x^2?R^2\right)}^2$$ where $$A$$ is a constant and $$R$$ is some fixed distance from the origin (on the positive and negative side) (a) Sketch the potential energy, and identify all equilibria. Are they stable or unstable. Note: Please try to sketch it over a range in $$x$$ such that all equilibria are clearly visible. (b) Consider a particle at a position $$x=R+dr$$ (where $$dr$$ is assumed to be small). What is the forc on the particle? Be sure to include the direction. (c) Suppose a particle is released from rest at $$x=R+dr$$ at $$\mathrm{t}=0$$. Describe the subsequent motio $$x(t)$$, in terms of $$R,A$$, and $$dr$$. If (for instance) the motion is periodic, be sure to give the angular frequency.