5 CONSERVATION OF ENERGY 5.6 Hooke’s law
i.e., the sum of the kinetic energy and the function U remains constant as the body
moves around in the force-field. It should be clear, by now, that the function U
represents some form of potential energy.
The above discussion leads to the following important conclusions. Firstly, it
should be possible to associate a potential energy (i.e., an energy a body pos-
sesses by virtue of its position) with any conservative force-field. Secondly, any
force-field for which we can define a potential energy must necessarily be con-
servative. For instance, the existence of gravitational potential energy is proof
that gravitational fields are conservative. Thirdly, the concept of potential en-
ergy is meaningless in a non-conservative force-field (since the potential energy
at a given point cannot be uniquely defined). Fourthly, potential energy is only
defined to within an arbitrary additive constant. In other words, the point in
space at which we set the potential energy to zero can be chosen at will. This im-
plies that only differences in potential energies between different points in space
have any physical significance. For instance, we have seen that the definition of
gravitational potential energy is U = m g z, where z represents height above the
ground. However, we could just as well write U = m g (z − z 0 ), where z 0 is the
height of some arbitrarily chosen reference point (e.g., the top of Mount Ever-
est, or the bottom of the Dead Sea). Fifthly, the difference in potential energy
between two points represents the net energy transferred to the associated force-
field when a body moves between these two points. In other words, potential
energy is not, strictly speaking, a property of the body—instead, it is a property
of the force-field within which the body moves.
5.6 Hooke’s law
Consider a mass m which slides over a horizontal frictionless surface. Suppose
that the mass is attached to a light horizontal spring whose other end is anchored
to an immovable object. See Fig. 42. Let x be the extension of the spring: i.e.,
the difference between the spring’s actual length and its unstretched length. Ob-
viously, x can also be used as a coordinate to determine the horizontal displace-
ment of the mass. According to Hooke’s law, the force f that the spring exerts on