1. Chapter 5: Energy Energy
Energy is present in a variety of forms:
mechanical, chemical, electromagnetic, nuclear, mass, etc.
Energy can be transformed from one from to another.
The total amount of energy in the Universe never changes.
If a collection of objects can exchange energy with
each other but not with the rest of the Universe (an
isolated system), the total energy of the system is
constant.
If one form of energy in an isolated system decreases,
another form of energy must increase.
In this chapter, we focus on mechanical energy: kinetic
energy and potential energy.

2. Work when the object is displaced by Dx by the force:
The work W done on an object by a constant force F
when the object is displaced by Dx by the force:
SI unit: joule (J) = newton-meter (N m) = kg m2/s2
Work is a scalar quantity.
If the force exerted on an
object is not in the same
direction as the displacement:
component of
the force along
the direction of
the displacement
dot product or
inner product

3. Work a force exerted, then no work is done.
If an object is displaced vertical to the direction of
a force exerted, then no work is done.
If an object is displaced in opposite
direction to that of an exerted force,
the work done by the
force is negative (if F<Fg).

4. Work The friction force between two objects in contact and in relative
Work and dissipative forces
The friction force between two objects in contact and in relative
to each other always dissipate energy in complex ways.
Friction is a complex process caused by numerous microscopic
interactions over the entire area of the surfaces in contact.
The dissipated energy above is converted to heat and other forms
of energy.
Frictional work is extremely important: without it Eskimos can’t
pull sled, cars can’t move, etc.

5. Work Example 5.1: Sledding through the Yukon
(a) How much work is done if q=0?
m=50.0 kg
F= 1.20x102 N
Dx=5.00 m
(b) How much work is done if q=30o?

6. Work Example 5.2: Sledding through the Yukon (with friction)
(a) How much work is done if q=0?
m=50.0 kg
F= 1.20x102 N
Dx=5.00 m
fk=0.200
(b) How much work is done if q=30o?

7. Kinetic Energy
Kinetic energy (energy associated with motion)
Consider an object of mass m moving to the right under action of
a constant net force Fnet directed to the right.
(constant acceleration)
Define the kinetic energy KE as:
SI unit: J
work-energy theorem

8. Kinetic Energy Example 5.3: Collision analysis m=1.00x103 kg
An example
Example 5.3: Collision analysis
m=1.00x103 kg
vi = 35.0 m/s -> 0
=8.00x103 N
(a) The minimum necessary stopping distance?
(b) If Dx=30.0 m what is the speed at impact?

9. Kinetic Energy
Conservative and non-conservative forces
Two kinds of forces: conservative and non-conservative forces
Conservative forces : gravity, electric force, spring force, etc.
A force is conservative if the work it does moving an object
between two points is the same no matter what path is taken.
It can be derived from “potential energy”.
Non-conservative forces : friction, air drag, propulsive force, etc.
In general dissipative – it tends to randomly disperse the energy
of bodies on which it acts.
The dispersal of energy often takes the form of heat or sound.
The work done by a non-conservative force depends on what
path of an object that it acts on is taken.
It cannot be derived from “potential energy”.
Work-energy theorem in terms of works by conservative and non-
conservative force

10. Gravitational Potential Energy
Gravitational work and potential energy
Gravity is a conservative force and can be derived from a potential
energy.
Work done by gravity on the book:

11. Gravitational Potential Energy
Gravitational work and potential energy
Gravity is a conservative force and can be derived from a potential
energy.
Let’s define the gravitational potential energy of a system consisting
of an object of mass m located near the surface of Earth and Earth
as:
y : the vertical position of the mass to
a reference point ( often at y=0 )
g : the acceleration of gravity
SI unit: J
where

12. Gravitational Potential Energy
Reference levels for gravitational potential energy
As far as the gravitational potential is concerned, the important
quantity is not y (vertical coordinate) but the difference Dy between
two positions.
You are free to choose a reference point at any level (but usually
at y=0). yi yf

13. Gravitational Potential Energy
Gravity and the conservation of mechanical energy
When a physical quantity is conserved the numeric value of
the quantity remains the same throughout the physical process.
When there is no non-conservative force involved,
Define the total mechanical energy as:
The total mechanical energy is conserved.
In general, in any isolated system of objects interacting only
conservative forces, the total mechanical energy of the system
remains the same at all times.

14. Gravitational Potential Energy
Examples
Example 5.5: Platform diver
(a) Find the diver’s speed at y=5.00 m.
(b) Find the diver’s speed at y=0.0 m.

15. Gravitational Potential Energy
Examples
Example 5.8: Hit the ski slopes
(a) Find the skier’s speed at the bottom (B).
(b) Find the distance traveled on
the horizontal rough surface.

16. Spring Potential Energy
Spring and Hooke’s law
Force exerted by a spring Fs
Hooke’s law
If x > 0, Fs <0
If x < 0, Fs >0
Fs to the left
x>0
Fs to the right
k : a constant of proportionality
called spring constant.
SI unit : N/m Fs
The spring always exerts its
force in a direction opposite
the displacement of its end
and tries to restore the attached
object to its original position.
Restoring force

17. Spring Potential Energy
Potential due to a spring
The spring Fs is associated with elastic potential energy.
Between xi -1/2Dx and xi+1/2Dx the work
exerted by the spring is approximately:
Between x=0 and x, the total work exerted
by the spring is approximately:
-Fs
width = Dx
xi-1/2Dx
xi+1/2Dx
-Fi
In general when the spring is stretched
from xi to xf, the work done by the spring
is: x
-Ws,i= areai xi
xi+1

18. Spring Potential Energy
Potential due to a spring (cont’d)
The energy-work theorem including a spring and gravity
elastic potential energy
Extended form of conservation of mechanical energy

19. Spring Potential Energy
Examples
Example 5.9: A horizontal spring
(a) Find the speed at x=0 without friction.
m=5.00 kg
k=4.00x102 N/m
xi= m
mk=0
(b) Find the speed at x=xi/2.

20. Spring Potential Energy
Examples
Example 5.9: A horizontal spring (cont’d)
(c) Find the speed at x=0 with friction
m=5.00 kg
k=4.00x102 N/m
xi= m
mk= 0.150

21. Spring Potential Energy
Examples
Example 5.10 : Circus acrobat
What is the max. compression
of the spring d?
m=50.0 kg
h =2.00 m
k = 8.00 x 103 N/m

22. Spring Potential Energy
Examples
Example 5.11 : A block projected up a frictionless incline
m=0.500 kg
xi=10.0 cm
k=625 N/m
q=30.0o
Find the max. distance
d the block travels up the
incline.
(b) Find the velocity at hafl height h/2.

23. Spring Potential Energy
Systems and energy conservation
Work-energy theorem
Consider changes in potential
The work done on a system by all non-conservative forces is
equal to the change in mechanical energy of the system.
If the mechanical energy is changing, it has to be going somewhere.
The energy either leaves the system and goes into the surrounding
environment, or stays in the system and is converted into non-
mechanical form(s).

24. Systems and Energy Conservation
Forms of energy
Forms of energy stored
kinetic, potential, internal energy
Forms of energy transfer between a non-isolated system and its
environment
Mechanical work :
transfers energy to a system by displacing it with a force.
Heat : transfers energy through microscopic collisions between
atoms or molecules.
Mechanical waves :
transfers energy by creating a disturbance that propagates
through a medium (air etc.).
Electrical transmission :
transfers energy through electric currents.
Electromagnetic radiation :
transfers energy in the form of electromagnetic waves such
as light, microwaves, and radio waves.

25. Systems and Energy Conservation
Principle of energy conservation:
Energy cannot be created or destroyed, only transferred from
one form to another.
The principle of conservation of energy is not only true in physics
but also in other fields such as biology, chemistry, etc.

26. Power
Power
The rate at which energy is transferred is important in the design
and use of practical devices such as electrical appliances and engines.
If an external force is applied to an object and if the work done by this
force is W in time interval Dt, then the average power delivered to the
object during this interval is the work done divided by the time interval:
SI unit : watt (W) = J/s = 1 kg m2/s3
W=FDt
More general definition
U.S. customary unit : 1 hp = 550 ft lb/s
= 746 W
1 kWh = (103 W)(3600 s) = 3.60 x 105 J

27. Power Example 5.12 : Power delivered by an elevator
Examples
Example 5.12 : Power delivered by an elevator
What is the min. power to lift the
elevator with the max. load?
M=1.00x103 kg
m=8.00x102 kg
f =4.00x103 N
v = 3.00 m/s

28. Power Example 5.14 : Speedboat power
How much power would a 1.00x103 kg speed boat need to go from
rest to 20.0 m/s in 5.00 s, assuming the water exerts a constant drag
force of magnitude fd=5.00x102 N and the acceleration is constant?

29. Power Center of mass (CM)
Energy and power in a vertical jump
Center of mass (CM)
The point in an object at which all the may be considered to be
concentrated.
h=0.40 m depth of crouch
Dt=0.25 s time for extension
m=68 kg
Stationary jump
Two phases:
(1) Extension, (2) free flight