1. Fundamentals of Physics School of Physical Science and Technology
Mechanics
(Bilingual Teaching)
张昆实
School of Physical Science and Technology
Yangtze University

2. Chapter 8 Potential Energy and Conservation of Energy
8-2 Path Independence of Conservative
Forces
8-3 Determining Potential Energy Values
8-4 Conservation of Mechanical Energy
8-5 Reading a Potential Energy Curve
8-6 Work Done on a System by an
External Force
8-7 Conservation of Energy

3. 8-1 Potential Energy Potential energy is energy that can be
associated with the configuration of a
system of objects that exert forces no one
another.
Gravitational potential energy is the energy associated with the state of seperation between objects, which attract one another via the gravitational force. ( example; a book is lifted… )
Elastic potential energy is the energy associated with the state of compression or extension of an elastic (springlike) object. ( example; a spring is stretched… )

4. 8-1 Potential Energy Work and Potential Energy Discuss the relation: ~
work-kinetic energy theorem
(7-41)
Earth
Discuss the relation: ~
Exp. A tomato is thrown upward
Rising: does , leads
Energy transferred from the tomato,
Where does it go? To
Increase the gravitational potential
energy of the tomato-earth system!
(the seperation is increased ! )
Falling: does , leads Energy transferred from the gravitational potential energy of the tomato-earth system to the kinetic energy of the tomato !

5. 8-1 Potential Energy Work and Potential Energy Discuss the relation: ~
Earth
Work and Potential Energy
Discuss the relation: ~
For either rise or fall, the change
in gravitational potential energy
is defined to equal the negative
work done on the tomato by the
gravitational force.
(8-1)
This equation also applies to a block-spring system

6. 8-1 Potential Energy Conservative and Nonconservative Forces
Key elements:
A system (two or more objects);
2. A force acts between a object and the rest part
of the system;
when configuration changes, the force does
work transffering the kinetic energy of the
object into some other form of energy.
Reversing the configuration changes, the force
reverses the energy transfer, doing work IF
is always true,
The other form of energy is a potential energy
The force is a conservative force!

7. 8-1 Potential Energy Conservative and Nonconservative Forces
Nonconservative Forces: a force that is not
conservative
Exp. : (1) the kinetic frictional force (P 141);
A block is sliding on a rough surface, the kinetic frictional force does negative work
Transfer kinetic energy thermal energy
So the thermal energy is not a potential energy!
The frictional force Nonconservative Forces !
(2) the drag force

8. 8-2 Path Independence of Conservative Forces
The closed-path test : to determine whether a force is conservative or nonconservative.
The net work done by a conservative force on a particle moving around every closed path is zero.
The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle.

9. 8-2 Path Independence of Conservative Forces
The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle.
Prove Eq.(8-2),
Sample Problem 8-1 b b 1 1 2 2 a a
If only a conservative force
acts on the particle, then
(8-2)

10. 8-3 Determining Potential Energy Values
Find the relation between a conservative force and the associated potential energy :
The work done by a variable conservative force on a particle (see Eq. 7-32):
(8-5)
(8-1)
(8-6)
Eq.(8-6) is the general relation we sought.

11. 8-3 Determining Potential Energy Values
Gravitational potential energy :
A particle is moving vertically along a y axis
From
(8-6)
(8-7)
(8-8)
(8-9)

12. 8-3 Determining Potential Energy Values
Elastic potential energy :
A block-spring system is vibrating, the
spring force does work on the block.
(8-10)
(8-11)

13. 8-4 Conservation of Mechanical Energy
Mechanical energy : THe Mechanical energy
of a system is the sum of its potential energy and
the kinetic energy of the objects within it:
( Mechanical energy )
(8-12)
A conservative force does work on the object
changing the object’s kinetic energy
(8-13)
The change in potential energy
(8-14)
Combining (8-13) , (8-14):
(8-15)
(8-16)
Rewriting:

14. 8-4 Conservation of Mechanical Energy
(8-16)
Rewriting:
(8-17)
Rearranging:
( conservation of Mechanical energy )
The sum of and for any
other state of the system
The sum of and for
any state of a system =
In a isolated system where only conservative
forces cause energy changes, the kinetic energy
and potential energy can change, but their sum,
the mechanical energy of the system, cannot
change The principle of conservation of
mechanical energy

15. 8-4 Conservation of Mechanical Energy
Combining (8-13) , (8-14):
(8-15)
(8-18)
When the mechanical energy of a system is
conserved we can relate the sum of kinetic
energy and potential energy at one instant
to that at another instant without considering
the intermediate motion and without finding
the work done by the forces involved.
The principle of conservation of
mechanical energy
Newton’s laws
of motion

16. 8-4 Conservation of Mechanical Energy
A pendulum bob swings back and forth.
= constant
Sample Problem 8-4,
Bungee jumping, P148

17. Conservation of Mechanical Energy Bungee jumping at
Victoria Falls

18. Victoria Falls (Zimbabwe- Zambia)

19. 8-5 Reading a Potential Energy Curve
Finding the force Analytically
Know Find
(8-6)
Know Find
From Eq.(8-1) :
Solving for and passing to the differential limit :
(8-20)
( one-dimensional motion )
Check Eq.(8-20) : 1. 2.

20. 8-5 Reading a Potential Energy Curve
is negative
the slope of the
curve
At ,
right
is a turning point

21. 8-5 Reading a Potential Energy Curve
unstable equilibrium
F=0 on both sides
K.E=0
deflecting force !
neutral equilibrium
neutral equilibrium
K.E=0
F=0
stable equilibrium
K.E=0
F=0 on both sides
restoring force !

22. 8-6 Work Done on a System by an External Force
Work is energy transferred to or from a
system by means of an external force
acting on that system.
system
Negative
Positive
In Chapter 7 :
(only )
The work-kinetic energy theorem
In Chapter 8 :
(in other forms)

23. 8-6 Work Done on a System by an External Force
No Friction Involved
Ball-Earth
system
Positive
throwing a ball upward,
your applied force
does work
Earth
kinetic
potential
(8-23)
(8-24)
mechanical energy
( Work done on system, no friction involved )

24. 8-6 Work Done on a System by an External Force
Friction Involved
Block-
Floor
system
(8-27)
(8-28)
(8-25)
(8-29)
(8-30)
(8-26)
(8-31)
（work done on a system, friction involved）

25. 8-7 Conservation of Energy
Countless experiments have proved:
The principal of conservation of energy
The total energy E of a system can change only by amounts of energy that
are transferred to or from the system.
Total energy
Mechanical energy
Thermal energy
Internal energy of any form
The work done on a system = the change in the total emergy
(8-33)

26. 8-7 Conservation of Energy
Isolated System
If a system is isolated from its environment
No energy transfers to or from it W=0
The total energy E of an isolated system cannot change. ★
(8-34)
(8-35)

27. 8-7 Conservation of Energy
Isolated System
The total energy E of an isolated system cannot change. ★
(8-35) ★
In an isolated system, we can relate
the total energy at one instant to
the total energy at another instant
without considering the energies at
intermediate times.
This is a powerful tool in solving problems for isolated system

28. 8-7 Conservation of Energy
Power
Power is the rate at which work is done
by a force.
Power is the rate at which energy is transferred by a force from one form to another. ★ ★
(8-36)
average power
(8-37)
instantaneous power