Potential Energy Practice Test

A particle moves in a one-dimensional potential given by $$U(x) = x^4 - 8x^2 + 12$$. If the particle has a total mechanical energy of $$E = 4$$ J, which of the following is a turning point of its motion?

A particle of mass $$m$$ is located inside a uniform solid sphere of mass $$M$$ and radius $$R$$ at a distance $$r$$ from its center ($$r < R$$). The gravitational force on the particle is $$F_g(r) = -G\frac{Mmr}{R^3}$$. If the potential energy is defined to be zero at the center of the sphere ($$U(0)=0$$), what is the potential energy $$U(r)$$ of the particle-sphere system?