1. Section 3.7 – Potential Energy
Gravitational Potential Energy
Let us consider the motion of a particle of mass m in proximity to another body of mass M
Newton’s Law of Universal Gravitation gives: M m êr
where êr is a unit vector in the direction of m from M

2. Recall The curl of a gradient is equal to zero, i.e.:
Thus if the curl of a vector field is equal to zero, i.e. if:
then we can define a potential function such that:
- A field whose curl is equal to zero is said to be a conservative field.

3. Gravitational Potential Energy
Let us define the Gravitational Potential Energy, Vg as:
where
and
It is customary to take the initial position r1 =  as our reference for gravitational potential energy

4. Special Case: Near Earth Problems

5. The Gravitational Potential Energy is simply the negative of the work done by the gravitational force on m.
The Gravitational Potential Energy depends only on the location of the particle (r or h).
It does NOT depend on how the particle got there.
i.e. it is path independent!

6. Elastic Potential Energy
Let us also define a potential energy for the deformation of an object such as a spring.
The Elastic Potential Energy, Ve, is defined as:
where
and

7. If we choose our reference position at x1 = 0 then:
The elastic potential energy is merely the negative of the work done by the spring on the particle.
The potential energy depends only on the endpoints of the motion.
It is path independent
Our “System” is modified to include energy stored in the spring.

8. Work Energy Equation We previously expressed the Work-Energy Equation:
Let us now include the work done by gravitational and elastic forces separately, we thus have:
Work done by all forces except for those of a gravitational or elastic nature.

9. This alternate form of the Work-Energy Equation is often far more convenient to use since the work done by gravitational and elastic forces is accounted for by using the endpoints of the motion, and therefore the path integral does not have to be evaluated for these terms.
We may rewrite the modified Work-Energy Equation as:

10. Another alternative form of the modified Work-Energy Equation is:
= Total Mechanical Energy of the Particle
where if
then we say that the total Energy of the System is conserved
Note: the “System” now includes the particle, the springs, and the gravitational field.

11. Conservative Force Fields
We have seen that then work done by a gravitational force, or an elastic force depends only on the net change in position, and not on the path taken.
Forces for which this is true are called Conservative Forces.
A Potential Energy function can be defined ONLY for Conservative Forces.
Frictional forces for example are not conservative.