How Does Total Energy Relate to Potential Energy in a Closed System?

[Mathematicsresear]

Homework Statement

Why is it that when the total energy of a system less than the max potential energy, it must equal to some potential energy in a potential energy diagram.

Homework Equations

The Attempt at a Solution

Total energy intersects multiple points in U(x) diagram. However, since potential energy is increasing before max U(x), and E< maxU(x), couldn't it be equal to some kinetic energy as well? However how I do know that that is the case? Why can't kinetic energy be greater at that point?

 

[kuruman]

E = K + U
E = constant which means K + U = constant
When E = U at the point of intersection, then U = K + U which means K = 0.
In short, if K + U = constant, then it follows that at the point of intersection K = 0. BTW, point x_T such that U(x_T) = E is the classical turning point where the particle reaches zero speed and reverses direction.

 

[CWatters]

Homework Statement

Why is it that when the total energy of a system less than the max potential energy, it must equal to some potential energy in a potential energy diagram.

If the system is "closed" (no energy gets in or out) then the total energy is constant. So the maximum amount of KE depends on the PE. The KE can't be arbitrarily high.