Potential energy
Gravitational potential energy
Let’s return to the case of lifting weights to think about another way to compute energy. If we were to draw a free body diagram of the weight being lifted, we would see the force required to lift it is \[\begin{equation} \vec{F}=m\vec{g} \end{equation}\] where \(m\) is mass and \(\vec{g}=\qty{-9.81}{\meter\per\second\squared}\) is the acceleration of gravity. Consider then lifting the weight a height \(h\). The work required to lift the weight is \[\begin{align} W &= \vec{F}\cdot\Delta\vec{x} \\ &= (m g) \cdot (h) \\ &= mgh \end{align}\] where \(m\) is mass, \(g=\qty{9.81}{\meter\per\second\squared}\), and \(h\) is height. The last expression is often called gravitational potential energy: \[\begin{equation} \text{gravitational potential energy}\ GPE = mgh . \end{equation}\]
Gravitational potential energy measures the work done to raise a weight to some height; it represents how much energy would be released in allowing the weight to fall back down to zero height. As we will see in a bit, the equivalence between these is due to conservation of energy.
Potential energy
A system composed of two or more objects has potential energy if the objects within that system only interact with each other through conservative forces.
Potential energy is a scalar quantity associated with the position of objects within a system. It is measured in joules (J), same as for other energy type quantities.
The definition of zero potential energy for a given system is a decision made by the observer considering the situation to simplify or otherwise assist in analysis.
Relationship between conservative forces and potential energy
The relationship between conservative forces exerted on a system and the system’s potential energy is \[\begin{equation} \Delta U = -\int_a^b \vec{F}_{cf}(r)\cdot d\vec{r} \end{equation}\]
The conservative forces exerted on a system in a single dimension can be determined using the slope of the system’s potential energy with respect to position in that dimension; these forces point in the direction of decreasing potential energy. \[\begin{equation} F_x = - \dfrac{dU(x)}{dx} \end{equation}\]
Stability of a system
Graphs of a system’s potential energy as a function of its position can be useful in determining physical properties of that system.
Stable equilibrium is a location at which a small displacement in an object’s position results in a force exerted on the object opposite to the direction of the small displacement, accelerating the object back toward the equilibrium position. In a given dimension, stable equilibrium positions exist at locations where the potential energy as a function of position in that dimension has a local minimum.
Unstable equilibrium is a location at which a small displacement in an object’s position results in a force exerted on the object in the same direction as the small displacement, accelerating the object away from the equilibrium position. In a given dimension, unstable equilibrium positions occur at locations where the potential energy as a function of position in that dimension has a local maximum.
Example forms of potential energy
The potential energy of common physical systems can be described using the physical properties of that system.
Elastic potential energy of a spring
The elastic potential energy of an ideal spring is given by the following equation, where \(\Delta x\) is the distance the spring has been stretched or compressed from its equilibrium length. \[\begin{equation} U = \dfrac{1}{2} K (\Delta x)^2 \end{equation}\]
Gravitational potential energy of a system
The general form for the gravitational potential energy (GPE) of a system consisting of two approximately spherical distributions of mass (e.g., moons, planets, or stars) is given by the equation \[\begin{equation} U_g = -G \dfrac{m_1 m_2}{r}. \end{equation}\]
Because the gravitational field near the surface of a planet is nearly constant, the change in gravitational potential energy in a system consisting of an object with mass \(m\) and a planet with gravitation field of magnitude \(g\) when the object is near the surface of the planet may be approximated by the equation \[\begin{equation} \Delta U_g = m g \Delta y = mgh. \end{equation}\]
Superposition
The total potential energy of a system containing more than two objects is the sum of the potential energy of each pair of objects within the system.
See also
- list here