Today we begin with a very useful concept – Energy.
We will encounter many familiar terms that now have very specific definitions in physics.
 Conservation of energy
 Work
 Potential
 Power
In some cases, it can be argued that these terms have a physics definition that is similar to its
everyday usage.
The Law of Conservation of Energy
The total energy in the universe is unchanged by any physical process:
total energy before = total energy after
“In ordinary language, conserving energy means trying not to waste useful energy resources. In
the scientific meaning of conservation, energy is always conserved no matter what happens.”
Conservation of energy is one of the few universal principles of physics. No exception has ever
been found. It applies to physical, chemical, and biological systems.
“Some problems can be solved using either energy conservation or Newton’s second law.
Usually the energy method is easier.” Using Newton’s second law involves vector methods
since forces are vector quantities. Much of the time, energy involves scalar quantities, which are
much easier to deal with (and more familiar). “When deciding which of these two approaches to
use to solve a problem, try using energy conservation first.”
Kangaroos are mentioned at the beginning of this chapter.
http://www.youtube.com/watch?v=hijYSR2MFiY
Forms of Energy
At the most fundamental energy there are three kinds of energy
1. Kinetic energy – energy due to motion
2. Potential energy – energy due to interaction
a. Gravitational potential energy – interaction between the Earth and a mass
b. Elastic potential energy – interaction between a spring and a mass
3. Rest energy – internal energy to a body
Energy is measured in Joules (J)
Work


Suppose a force F causes an object to move a distance x parallel to F

F

F
x

The work done by a constant force F is defined as
W  Fx
DO NOT MEMORIZE!



Suppose a constant force F causes an object to move along r not parallel to F

F

F



r
W  Fr cos


where  is the angle between F and r . MEMORIZE.
Work is a scalar quantity and can be positive, negative, or zero.
 Positive:  between 0o and 90o
 Negative:  between 90o and 180o
 Zero:  = 90o
o Tension and normal force don’t do work.
The work done by several forces can be found from the net force
Wtotal  W1  W2    WN
 Fnet r cos
Work and Kinetic Energy
Choosing the x axis along Fnet, (using x = r cos )
Wtotal  Fnet x
 ma x x
We had an equation from chapter 2
v fx  vix  2ax x
2
2
ax x  12 (v fx  vix )
2
2
Substituting into Wtotal
Wtotal  12 m(v fx  vix )
2
2
Since the net force is in the x-direction, ay and az are both zero. Only the x-component of the
velocity changes
v f  vi  (v fx  v fy  v fz )  (vix  viy  viz )  v fx  vix
2
2
2
2
2
2
2
2
2
2
and
Wtotal  12 m(v f  vi )
2
2
The translational kinetic energy is defined as
K  12 mv 2
The work-kinetic energy theorem is
Wtotal  K
While this expression is foundational to this chapter, do not memorize. We shall derive a more
useful form.
Clicker question 1
Gravitational Potential Energy
The force of gravity can do work. Toss a ball up and it slows down. In our new language, its
kinetic energy decreases. The kinetic energy is converted into another form of energy we call
gravitational potential energy.
The change in gravitational potential energy
U grav  Wgrav
In terms of position
U grav  mgy
The gravitational potential energy is
U grav  mgy
Clicker question 2
The final form of our relation is
Wnc  K  U
This is it. You need to know it. We have another entry into our cause and effect table.
Wnc is the work done by nonconservative forces. Nonconservative forces do not have a potential
energy. A good example is friction.
The mechanical energy is
E  K U
Conservation of Mechanical Energy
When nonconservative forces do no work, mechanical energy is conserved:
Ei  E f
The zero of potential energy is arbitrary. Choose whatever is convenient.
The work done by a conservative force is independent of the path taken.
Problem: A 0.1-kg ball is thrown at 5 m/s from a 10 m tower. What is its speed when it is 5 m
above the ground? What is its speed when it hits the ground?
Solution: Use the conservation of mechanical energy.
E1  E2
U1  K1  U 2  K 2
mgy1  12 mv1  mgy2  12 mv2
2
gy1  12 v1  gy2  12 v2
2
2
2
v2  v1  2 g ( y1  y2 )
2
 (5 m/s) 2  2(9.8 m/s 2 (10 m  5 m)
 11.1 m/s
At the bottom, y2 = 0, v2 = 14.9 m/s.
Notice the direction of the throw is not mentioned. No matter which way the ball is thrown, it
has the same speed at the same height! This is very hard to prove using Newton’s second law.
Clicker question 3
For objects far from the Earth,
U 
GmM E
r
While this looks very different from mgy, the text (p. 203) shows they are equivalent.
Example 6.8 What is the escape velocity for the Earth?
Solution: Use conservation of energy.
Ei  E f
U i  Ki  U f  K f

GmmE 1
GmmE 1
2
2
 2 mvi  
 2 mv f
ri
rf
When an object escapes from the Earth, Earth’s gravity is not acting on it and rf → ∞. It the
object barely escapes the, vf = 0.

GmmE 1
2
 2 mvi  0
ri
vi 
2GmE
ri
The starting position is on the Earth’s surface and ri = RE = 6.36×106 m. The mass of the Earth is
mE = 5.97×1024 kg. The escape velocity is
vi 
2GmE
2(6.67  1011 Nm 2 / kg 2 )(5.97  1024 kg )

 11,200 m/s
ri
(6.37  106 kg )
This is
vi  11,200
m
1 mi
mi

 7.0
s 1609m
s