As long as there is no friction or air resistance, the change in kinetic energy of the football equals the change in gravitational potential energy of the football. This can be generalized to any potential energy:
Let’s look at a specific example, choosing zero potential energy for gravitational potential energy at convenient points.
Basic properties of potential energy
A particle moves along the
x -axis under the action of a force given by
, where
. (a) What is the difference in its potential energy as it moves from
to
? (b) What is the particle’s potential energy at
with respect to a given 0.5 J of potential energy at
?
Strategy
(a) The difference in potential energy is the negative of the work done, as defined by
[link] . The work is defined in the previous chapter as the dot product of the force with the distance. Since the particle is moving forward in the
x -direction, the dot product simplifies to a multiplication (
). To find the total work done, we need to integrate the function between the given limits. After integration, we can state the work or the potential energy. (b) The potential energy function, with respect to zero at
, is the indefinite integral encountered in part (a), with the constant of integration determined from
[link] . Then, we substitute the
x -value into the function of potential energy to calculate the potential energy at
Solution
The work done by the given force as the particle moves from coordinate
x to
in one dimension is
Substituting this expression into
[link] , we obtain
The indefinite integral for the potential energy function in part (a) is
and we want the constant to be determined by
Thus, the potential energy with respect to zero at
is just
Therefore, the potential energy at
is
Significance
In this one-dimensional example, any function we can integrate, independent of path, is conservative. Notice how we applied the definition of potential energy difference to determine the potential energy function with respect to zero at a chosen point. Also notice that the potential energy, as determined in part (b), at
is
and at
is
; their difference is the result in part (a).
Check Your Understanding In
[link] , what are the potential energies of the particle at
and
with respect to zero at
? Verify that the difference of potential energy is still 7 J.
In general, a system of interest could consist of several particles. The difference in the potential energy of the system is the negative of the work done by gravitational or elastic forces, which, as we will see in the next section, are conservative forces. The potential energy difference depends only on the initial and final positions of the particles, and on some parameters that characterize the interaction (like mass for gravity or the spring constant for a Hooke’s law force).
Questions & Answers
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