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Chapter 8
Potential Energy and
Conservation of Energy
Conservative forces
• The net work done by a conservative force on a
particle moving around any closed path is zero
Wab,1  Wba, 2  0
Wab,1  Wba, 2
Wab, 2  Wba, 2
Wab,1  Wab, 2
• The net work done by a conservative force on a
particle moving between two points does not depend
on the path taken by the particle
Conservative forces: examples
• Gravity force
 mghup  mghdown  0
• Spring force

2
kxright
2

2
kxleft
2
0
Potential energy
• For conservative forces we introduce a definition of
potential energy U
U  W
• The change in potential energy of an object is being
defined as being equal to the negative of the work
done by conservative forces on the object
• Potential energy is associated with the arrangement
of the system subject to conservative forces
Potential energy
• For 1D case
xf
U  U f  U i  W    F ( x)dx
U ( x)    F ( x)dx  C
xi
dU ( x)
F ( x)  
dx
• A conservative force is associated with a potential
energy
• There is a freedom in defining a potential energy:
adding or subtracting a constant does not change the
force
• In 3D F ( x, y, z )   U ( x, y, z ) iˆ  U ( x, y, z ) ˆj  U ( x, y, z ) kˆ
x
y
z
Gravitational potential energy
• For an upward direction the y axis
yf
U ( y )    (mg )dy  mgy f  mgyi  mgy
yi
U g ( y )  mgy
Chapter 8
Problem 8
Chapter 8
Problem 19
Elastic potential energy
• For a spring obeying the Hooke’s law
U ( x)   
xf
xi
kx2f
kxi2
(kx)dx 

2
2
kx
U s ( x) 
2
2
Conservation of mechanical energy
• Mechanical energy of an object is
Emec  K  U
• When a conservative force does work on the object
K  W U  W K  U
K f  U f  Ki  U i
K f  K i  (U f  U i )
Emec, f  Emec,i
• In an isolated system, where only conservative
forces cause energy changes, the kinetic and
potential energies can change, but the mechanical
energy cannot change
Work done by an external force
• Work is transferred to or from the system by means
of an external force acting on that system
W  K  U  Eint
• The total energy of a system can change only by
amounts of energy that are transferred to or from the
system
• Power of energy transfer, average and intantaneous
Pavg
E

t
dE
P
dt
Conservation of
mechanical
energy:
pendulum
Pole vault
• Muscle energy becomes kinetic energy (x-direction)
Pole vault
• Kinetic energy becomes elastic potential energy
Pole vault
• Elastic energy becomes kinetic energy (y-direction)
Pole vault
• Kinetic energy becomes gravitational potential
energy
Pole vault
• Gravitational potential energy becomes kinetic
energy (y-direction)
Pole vault
• Kinetic energy energy becomes part elastic
potential energy and part internal energy
Pole vault
• The ‘pole vault’ phenomenon is ubiquitous…
Potential energy curve
dU ( x)
F ( x)  
dx
Potential energy curve: equilibrium
points
Neutral equilibrium
Unstable equilibrium
Stable equilibrium
Chapter 8
Problem 77
Answers to the even-numbered problems
Chapter 8:
Problem 18
10 cm
Answers to the even-numbered problems
Chapter 8:
Problem 68
(a) −3.8 kJ;
(b) 31 kN