Chapter 7 Outline
Potential Energy and Energy Conservation
• Gravitational potential energy
• Conservation of mechanical energy
• Elastic potential energy
• Springs
• Conservative and non-conservative forces
• Conservation of energy
• Force and potential energy
• Energy diagrams
Potential Energy
• Kinetic energy depends on motion.
• Potential energy depends on
position.
• Energy can be converted between
these forms.
• Total energy will remain constant.
• Gravitational potential energy, 𝑈g ,
relative to some reference is equal
to the work done against gravity to
lift an object from the reference
point to some height.
∆𝑈g = 𝑚𝑔∆𝑦
Gravitational Potential Energy
• Can we discuss an exact value for
gravitational potential energy?
• We calculated the change in 𝑈g
from the gravitational force times
the distance.
• We cannot define the gravitational
potential energy without first defining
the zero point.
• Only the change in gravitational
potential energy is relevant. We
are free to set 𝑈g = 0 at any point
we want.
• Only the change in height matters;
the path is irrelevant.
Conservation of Mechanical Energy
• If we only have gravitational force, the sum of gravitational
potential energy and kinetic energy must be constant.
• This is conservation of mechanical energy.
𝐸tot = 𝐾1 + 𝑈g1 = 𝐾2 + 𝑈g2 = constant
Potential Energy Example
Elastic Potential Energy
• Last chapter, we discussed the work
needed to compress (or stretch) a spring.
• Since we were doing work on the spring,
where was the energy going?
• It was stored as elastic potential energy.
• The term elastic implies that the energy
stored in the spring can be converted
completely into kinetic energy.
𝑈el = 12𝑘𝑥 2
• The work done on the spring is equal in
magnitude but opposite in sign to the
change in the energy stored in the spring.
𝑊el = −∆𝑈el
Work Done by Other Forces
• What about work done by other forces, such as friction?
• Total energy is always conserved!
𝐾1 + 𝑈1 + 𝑊other = 𝐾2 + 𝑈2
• Where does the other energy go?
• Frictional forces are non-conservative.
• Energy is still conserved, but some of it is converted into heat.
Potential Energy Example
Conservative vs. Non-conservative Forces
• When we throw a ball in the air, its kinetic energy is
“stored” as gravitational potential energy as it approaches
its maximum height, and converted back to kinetic energy
as it comes down.
• Because this back and forth conversion between kinetic and
potential energy is possible, we call gravity a conservative force.
• Mechanical energy is conserved: 𝐸 = 𝐾 + 𝑈 = constant
• Consider instead a box sliding to a stop because of
friction.
• The kinetic energy is converted to heat by the frictional force.
• This cannot be reversed, so it is a non-conservative force.
Properties of Conservative Forces
• The work done by any conservative force has these four
properties.
1. Can be expressed as difference of potential energy.
2. Is reversible.
3. Is path-independent.
4. Closed loop work is zero.
Conservative or Non-conservative Example?
Law of Conservation of Energy
• While the total mechanical energy can vary due to work
done by non-conservative forces, energy is never created
or destroyed.
• Consider a car skidding to a stop.
• The kinetic energy is converted to heat, increasing the temperature
of the tires and pavement.
• This increases their internal energy.
• Since the work done by friction is negative and the change in
internal energy is positive, ∆𝑈int = −𝑊other .
• The law of conservation of energy is:
∆𝐾 + ∆𝑈 + ∆𝑈int = 0
• This is always true.
Force and Potential Energy (1D)
• We have found expressions for the potential energy
associated with gravity and springs from the forces.
• What if we know the potential energy function and want to
find the corresponding force? (Consider 1D first.)
• The work done by a conservative force equals the negative of the
change in potential energy.
𝑊 = −∆𝑈, or 𝑑𝑊 = −𝑑𝑈
• Since 𝑊 =
𝐹𝑥 𝑑𝑥, an infinitesimal bit of work, 𝑑𝑊 is 𝐹𝑥 𝑑𝑥, so,
𝐹𝑥 𝑑𝑥 = −𝑑𝑈
• Solving for the force,
𝑑𝑈
𝐹𝑥 = −
𝑑𝑥
Force and Potential Energy (3D)
• The potential energy will in general depend on all three
spatial dimensions.
• We can repeat the analysis for 𝑦 and 𝑧, but we need to introduce a
new mathematical notation.
• To find the total vector force, we look at each direction
independently.
• When we move only in the 𝑥 direction, 𝑦 and 𝑧 remain constant.
• We take the derivative of 𝑈 with respect to 𝑥 while treating 𝑦 and
𝑧 as constants. This is called the partial derivative.
𝜕𝑈
𝐹𝑥 = −
𝜕𝑥
• Repeating this for 𝑦 and 𝑧,
𝐹𝑦 = −
𝜕𝑈
𝜕𝑦
𝐹𝑧 = −
𝜕𝑈
𝜕𝑧
Force and Potential Energy (3D)
• Combing these in vector form,
𝜕𝑈
𝜕𝑈
𝜕𝑈
𝑭=−
𝒙+
𝒚+
𝒛
𝜕𝑥
𝜕𝑦
𝜕𝑧
• We can write this more succinctly using the “del” operator.
𝜕
𝜕
𝜕
𝛁= 𝒙
+𝒚
+𝒛
𝜕𝑥
𝜕𝑦
𝜕𝑧
• The force is the negative gradient of the potential.
𝑭 = −𝛁𝑈
Energy Diagram
• We can glean a lot of information by looking at graph of
the potential energy.
Energy Diagram Example
Chapter 7 Summary
Potential Energy and Energy Conservation
• Gravitational potential energy: 𝑈g = 𝑚𝑔ℎ
• Conservation of mechanical energy
𝐸tot = 𝐾1 + 𝑈g1 = 𝐾2 + 𝑈g2 = constant
• Elastic potential energy: 𝑈el = 12𝑘𝑥 2
• Conservative forces
• Potential energy, reversible, path-independent, zero closed loop
• Conservation of energy: ∆𝐾 + ∆𝑈 + ∆𝑈int = 0
• Force and potential energy: 𝑭 = −
• Energy diagrams
• Stable minima and unstable maxima
𝜕𝑈
𝒙
𝜕𝑥
+
𝜕𝑈
𝒚
𝜕𝑦
+
𝜕𝑈
𝒛
𝜕𝑧