Work and Potential Energy

The techniques for calculating work can also be applied to the calculation of potential energy. If a certain force depends only on the distance between the two participating objects, then the energy released by changing the distance between them is defined as the potential energy, and the amount of potential energy lost equals minus the work done by the force,

ΔPE = -W .

The minus sign occurs because positive work indicates that the potential energy is being expended and converted to some other form.

It is sometimes convenient to pick some arbitrary position as a reference position, and derive an equation for once and for all that gives the potential energy relative to this position

PE_x = -W_ref→x . [potential energy at a point x]

To find the energy transferred into or out of potential energy, one then subtracts two different values of this equation.

These equations might almost make it look as though work and energy were the same thing, but they are not. First, potential energy measures the energy that a system has stored in it, while work measures how much energy is transferred in or out. Second, the techniques for calculating work can be used to find the amount of energy transferred in many situations where there is no potential energy involved, as when we calculate the amount of kinetic energy transformed into heat by a car's brake shoes.

A toy gun.

Gravitational potential energy.

Although the equations derived in the previous two examples may seem arcane and not particularly useful except for toy designers and rocket scientists, their usefulness is actually greater than it appears. The equation for the potential energy of a spring can be adapted to any other case in which an object is compressed, stretched, twisted, or bent. While you are not likely to use the equation for gravitational potential energy for anything practical, it is directly analogous to an equation that is extremely useful in chemistry, which is the equation for the potential energy of an electron at a distance r from the nucleus of its atom. As discussed in more detail later in the course, the electrical force between the electron and the nucleus is proportional to 1/r², just like the gravitational force between two masses. Since the equation for the force is of the same form, so is the equation for the potential energy.

The twin Voyager space probes were perhaps the greatest scientific successes of the space program. Over a period of decades, they flew by all the planets of the outer solar system, probably accomplishing more of scientific interest than the entire space shuttle program at a tiny fraction of the cost. Both Voyager probes completed their final planetary flybys with speeds greater than the escape velocity at that distance from the sun, and so headed on out of the solar system on hyperbolic orbits, never to return. Radio contact has been lost, and they are now likely to travel interstellar space for billions of years without colliding with anything or being detected by any intelligent species.

Discussion Questions

A	What does the graph of PE = (1/2)k (x - xo)2 look like as a function of x? Discuss the physical significance of its features.
B	What does the graph of PE = -GMm/r look like as a function of r? Discuss the physical significance of its features. How would the equation and graph change if some other reference point was chosen rather than r = ∞?
C	Starting at a distance r from a planet of mass M, how fast must an object be moving in order to have a hyperbolic orbit, i.e., one that never comes back to the planet? This velocity is called the escape velocity. Interpreting the result, does it matter in what direction the velocity is? Does it matter what mass the object has? Does the object escape because it is moving too fast for gravity to act on it?
D	Does a spring have an escape velocity?
E	Calculus-based question: If the form of energy being transferred is potential energy, then the equations F = dW/dx and W = ∫ Fdx become F = -dPE/dx and PE = -∫ Fdx. How would you then apply the following calculus concepts: zero derivative at minima and maxima, and the second derivative test for concavity up or down.