5 CONSERVATION OF ENERGY 5.5 Potential energy
·∆K =
Of·dr. (5.34)5.5 Potential energy
Consider a body moving in a conservative force-field f(r). Let us arbitrarily pick
some point O in this field. We can define a function U(r) which possesses a
unique value at every point in the field. The value of this function associated
with some general point R is simply
U(R) = −R
f dr. (5.33)
OIn other words, U(R) is just the energy transferred to the field (i.e., minus the
work done by the field) when the body moves from point O to point R. Of
course, the value of U at point O is zero: i.e., U(O) = 0. Note that the above
definition uniquely specifies U(R), since the work done when a body moves be-
tween two points in a conservative force-field is independent of the path taken
between these points. Furthermore, the above definition would make no sense
in a non-conservative field, since the work done when a body moves between
two points in such a field is dependent on the chosen path: hence, U(R) would
have an infinite number of different values corresponding to the infinite number
of different paths the body could take between points O and R.
According to the work-energy theorem,
∫RIn other words, the net change in the kinetic energy of the body, as it moves from
point O to point R, is equal to the work done on the body by the force-field during
this process. However, comparing with Eq. (5.33), we can see that
∆K = U(O) − U(R) = −∆U. (5.35)In other words, the increase in the kinetic energy of the body, as it moves from
point O to point R, is equal to the decrease in the function U evaluated between
these same two points. Another way of putting this is
E = K + U = constant : (5.36)∫