The concept of potential energy can be a little difficult to wrap your mind around. Kinetic energy is easy: Things that are moving obviously have energy. A baseball can knock you on the head (and hurt), a moving train could flatten you, and so on. But an object (stationary or moving) also has a kind of energy just by virtue of where it is located with respect to other
objects in the universe, and that's a little weird (as first).
Forces are the reality that create the concept of potential energy. Without forces, there would be no potential energy. The three examples below illustrate the concept of the force field, a sort of map of how forces between objects change as their relative position changes. The force field and the potential energy are related.
Gravity is a force that we can model mathematically very well, but we don't really understand what it is. Any object with mass exerts a gravitational force on any other object with mass. Planets and stars, of course, are very massive, thus they exert the largest forces.
The gravitational force produced by any body on another is inversely proportional to the square of the distance between. That means that if I double My distance from the center of the Earth, the gravitational attraction that Earth and I exert on each other is reduced by a factor of 1/4 (not 1/2).
In the diagram, the dashed lines represent spheres of constant force. The spacing between the rings is roughly proportional to the amount of Earth-ward force; they're closer near the surface and farther apart far away where the force is weaker.
This image is a representation of the gravitational potential energy around a planet. The planet would lie at the center of the "well".
At far distances where the gravitational force is weak, the potential well is not steep. Near the planet it becomes much steeper. The force felt by an object attracted to the planet is proportional to the steepness of this potential.
If the graph represents the potential energy function, then its steepness (its derivative in calculus) at any point is the gravitational force at that point.
Source: AllenMcC., Wikimedia Commons
The gradient of a force field or a potential energy function at any point is a vector describing the direction and magnitude of the steepest increase. We say that motion in a force field is caused by the field gradient. If there is no gradient, then there's no force and no motion.
Many devices can store elastic potential energy and release it as kinetic energy: a spring, a rubber band, an archer's bow, a catapult ...
The animation on the right shows a compressed spring at rest. In its compressed state, the spring stores elastic potential energy. The downward force it exerts on the hanging mass is proportional to the amount of compression (see Hooke's law).
Once the mass reaches the middle of its travel, and is at its equilibrium (unstretched / uncom-pressed) length, it will then begin to be stretched, storing more potential energy in that way. The force exerted on the hanging mass is greatest at either end, less in the middle, thus the mass has its greatest elastic potential energy when the spring is most compressed or most stretched.
Of course, it's also possible to over-stretch a spring, ruining its potential energy storage properties.
Charged objects, like electrons (-) and protons (+) exert forces on one-another. These are called electrostatic forces. Like gravity, these are non-contact forces (one object doesn't have to touch another in order to exert a force on it). But unlike gravity, electrostatic forces can be attractive or repulsive. (Gravity is always attractive — no one ever just gets ejected off the planet while walking down the street.)
A charge placed near another will have potential energy if it "sees" a gradient in the electrostatic force field. That gradient is steeper closer to the charge and less steep farther away.
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