The potential-energy parabola

When a body moves along one dimension under the effect of conservative forces (spring, gravity), its total potential energy $E_\text{pot}(x)$ is a function of position. The graph of this function is extremely powerful: a single curve reveals the entire behaviour of the motion without solving any equation. Just draw over it the horizontal line of the total energy $E_\text{tot}$, which stays constant as long as the forces in play are conservative, and compare the two.

The idea is that, at every position $x$, the kinetic energy is the vertical difference between the $E_\text{tot}$ line and the $E_\text{pot}(x)$ curve: $E_\text{cin}(x) = E_\text{tot} - E_\text{pot}(x)$. Since kinetic energy can never be negative, the body is confined to where the curve lies below the line. This turns the graph into a complete map of the motion.

Principle — The potential-energy parabola

• The equilibrium point is the minimum of the parabola: there $E_\text{pot}$ is minimum and the net force is zero.
• The turning points are where $E_\text{pot}(x) = E_\text{tot}$: there $E_\text{cin} = 0$ (the body stops and turns back).
• The accessible region is where $E_\text{pot}(x) \leq E_\text{tot}$.

The minimum of the curve is a point of equilibrium because the conservative force is tied to the slope of the potential energy: where the curve is flat, the force vanishes. It’s also the point where kinetic energy reaches its maximum value, since there the vertical distance from the $E_\text{tot}$ line is greatest: the body passes through equilibrium at its highest speed. At the two extremes, instead, the curve meets the line: all the energy is potential, kinetic energy is zero, the body stops for an instant and reverses its motion. Between these two turning points the body oscillates back and forth, trapped in the potential well.

Potential-energy graph: the body oscillates between the turning points $x_1$ and $x_2$ where $E_\text{pot} = E_\text{tot}$. At equilibrium the kinetic energy is at its maximum.

Further detail

With friction, the total energy decreases over time: the $E_\text{tot}$ line drops, the turning points move closer together, and the body converges towards equilibrium with damped oscillations.

Links

Topics: Lavoro ed energia Concepts: Energia potenziale elastica · Forze conservative · Conservazione dell’energia meccanica Skills: Lettura dei grafici Objects: Molla

Related exercises: Problema — Due molle e una scatola · Problema — Blocco su piano inclinato con molla (equilibrio e oscillazioni) · Esercizio svolto — mela e molla