Potential energy and conservation of energy
THIS LECTURE IS ONE OF THE MOST IMPORTANT OF THE
SEMESTER!
Potential Energy
We have seen that when a force does work on a system the system
may acquire motion energy, i.e. kinetic energy. However, another
possibility is the work simply stores the energy in the system
without any change in kinetic energy. This stored energy is called
potential energy.
The logic is:
F → r→ W → energy (kinetic or potential or both)
Potential energy is the energy stored in a system due to the
position of the parts of the system. In short, potential energy is
position energy. Mathematically, we stipulate that the work serves
to increase the potential (stored) energy and write
x2
U = U(x2) – U(x1)  - W = -  F(x) dx
x1
by Fund. Thm. Calc.
F(x) = - dU/dx .
The reason for the minus sign will become evident later.
Comment: Only the change in potential matters because the work
always refers to a change in position and not just one position.
Two special cases:
● constant gravitational force, mg.
We will prove that a mass m at a vertical height above the ground
h has for the mass-earth system a gravitational potential energy
Ug = mgh .
Proof
In this we have assigned the value zero to the potential energy
when the mass is at h=0. This location is called the datum or
reference location for gravitational potential energy.
● elastic potential energy
We will prove that for a mass attached to a spring of spring
constant k that has been stretched (or compressed) a distance x
from equilibrium the elastic potential energy of the system is
Uel = ½ k x2 .
Proof
In this we have assigned the value zero to the potential energy
when the spring has zero displacement x = 0. This location is
called the datum or reference location for elastic potential energy.
Conservative Force
A force is said to be conservative if the work it does depends on
only the starting and ending points of the displacement and not on
the path connecting the starting and ending points. Gravitational
and electrical forces are conservative. Friction is non-conservative
because the amount of work done by friction depends on the path.
One can associate a potential energy with a conservative force but
not with a non-conservative force.
Conservation of Energy
Recall the work-kinetic energy theorem for a mass m,
Wnet = K = Kf – Ki = ½ m vf2 – ½ m vi2 .
Everything outside the system we call the environment. Inside the
system there is this mass m. The forces acting on m are either from
objects internal to the system or from objects external to the
system. The work on m is then
Wnet = Winternal + Wexternal .
The internal forces will be either conservative or non-conservative
and we write,
Winternal = Wc + Wnc .
But for conservative forces we may define a potential energy
function
Wc = -U.

Wnet = -U + Wnc + Wexternal = K.
Rearranging this gives the fundamental result
E  K + U = Wnc + Wexternal
Suppose there are no external or non-conservative forces, then
the conservation of energy follows,
Conservation of Energy

E = 0
 Ef = Ei
or Kf + Uf = Ki + Ui
(isolated, conservative system)
Using conservation laws opens up another way of solving
mechanics problems that can be very useful.
CONSERVATION LAWS RECIPE
Step One
Define the system.
Step Two
Decide whether the relevant quantity (e.g. energy)
is conserved. If so, compute the initial value of the conserved
quantity and then the final value of the conserved quantity.
Step Three Equate the initial and final values and solve for the
unknowns in terms of the knowns.
As in all physics problems, make sure the units check out and as
the final step, make sure your final answer seems reasonable.
EXAMPLES[IN CLASS]