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Zero Potential Energy

It is clear from above analysis that only the difference of potential energy between two points has a physical meaning as the work done in moving a particle from one point to other. The value of U (x, y, z) itself is arbitrary and carries no physical meaning. Therefore, we can assign any value to U (r₀) at a given point r₀ in space. Subsequently, value of U (r) at another point r becomes determined:

We can take r₀ as some reference point and put U (r₀) = 0. Then the value of U (r) is fixed; it is equal to negative of work done in the moving a particle from r₀ to r.

The choice of r₀ is really arbitrary, however sometimes physical arguments lead to choice of r₀. We shall explain all this by taking a few specific examples.

(i) Gravity force near earth’s surface: Considering earth as an inertial frame, take Z-axis vertically upwards and its surface as X-Y plane. Suppose a particle of mass m moves under force of gravity F = m g from position r₁ to r₂; m g is constant, g = – g k. The work done depends on the end points, the vertical co-ordinates or the particular path of motion. The zero of potential energy is implicitly set at z = 0, i.e. on the surface of earth.

The force mg is therefore conservative and can be expressed as gradient of potential function:

(ii) Inverse-square force: The inverse square force is given as,

It is a particular example of a general class of forces, called central forces, which are purely radial. That is, their magnitudes depend only on the radial distance r from the center of force and they are directed along the radial direction e_r. The well known examples of central forces are the inverse square forces given by the Newton’s law of gravitation and Coulomb’s law in electrostatics.

Suppose a particle moves from a position r₁ to r₂ under the action of given central force F along some arbitrary path.

The work done during infinitesimal displacement dl is given by,

dW = F . dl = F (r) e_r . dl

= F (r) dr