WORK
The work dW done on a particle
displaced along differential path
dr, by an object exerting force F is
defined as
 
dW  F  dr
F
A
B
dr
The SI unit of work is
1J = 1N·1m
We can define work in an integral form:
 
W   F  dr
path
For the work done on an object, one must specify on which point
associated with the object the work is done (the center of mass,
the point of the force application ... ).
kinetic energy
A particle with mass m, moving with speed v has
kinetic energy K of
mv 2
K
2
Note. Kinetic energy is used to describe the motion of an
object. When an object is approximated by a particle, the
kinetic energy defined above is called the translational kinetic
energy of the object.
Work-Energy Theorem I

 mv 2 
dv 
 

  



m
v

d
v
dK  d

m

v
dt
 dWnet

m
a

d
r

F

d
r
net

dt
 2 
In an inertial reference frame, the work dW done by all
the forces exerted on the particle (the net force) is equal
to the change in the kinetic energy dK of the particle
dW = dK
In an integral form: W = K
work & energy
Power
The power of a force is defined as the rate at which work is
done by that force.
dW
P( t ) 
dt
The SI unit of work is 1W = 1J/1s
t2
inverse relation:
W   Pt dt
t1
relation to force:
 
P  Fv
Conservative Interaction
If the work done by a “force” on an object moving
between two positions is independent of the path of
the motion, the force is called a conservative force.
B
All other forces are nonconservative.
A
(Theorem)
The work done by a conservative force around a loop
(the object returns to its initial position) is zero.
conservative interactions : gravitational, elastic, electrostatic.
Potential Energy
If a force exerted on a particle is conservative, the
change in potential energy dU from one position
to another is defined by the work dW performed
by that force
dU  - dW
(or U = -W )
This definition assigns potential energy
only with accuracy to a constant.
Gravitational Potential Energy
h
m
dr
h
W
Ug

r
y o h



U g r   U g, ref  Wro r    mgdr     mgdy  mgh

ro
y0
The gravitational potential energy Ug of a particle with
mass m, placed at a position with a vertical component
different by h from the reference location is
Ug = mgh
Elastic Potential Energy
The elastic potential energy that an
ideal spring has by virtue of being
stretched or compressed is
Us
x
2
kx
Us 
2
F = -kx
x
x
0
Us 
x
kx 2
 Ws     kx ' dx ' 
2
0
x
bungee
Gravitational Potential Energy
F
M
dr
m
Wherechooser the Mm
Mm
U G     G 2 rˆ reference?
 dr     G 2 dr 
r
r
r0
path
r
 1 11  1  Mm
 GMm     G
 rR h R  r
1


 GMm   0  2 h  o(h 2 )  


R
The gravitational potential energy of a particle
with mass m, placed at distance
GM r from
What another
about the
 m  2  h  mgh
particle with mass M is R
reference at the
Mm

U G r   G
r
surface?
Mechanical Energy
The sum of the kinetic and the potential energy
of a particle is called the total mechanical
energy of the particle.
EK+U
motion
related
energy
position
related
energy
Energy Conservation
• All concepts of energy are defined in such a way that
energy can neither be created nor destroyed, but can
be converted from one form to another.
• The total energy of an isolated system is always
constant.
Work-Energy Theorem II
If some forces exerted on a particle are
conservative, the work Wnc, done by all forces not
included in the potential energy, is equal to the
change in the mechanical energy E of the particle.
Wnc  E
E  K  U  Wnet  Wc  Wnc
bungee
Potential Energy and the Force
The conservative force is opposite to
the gradient of the potential energy
caused by this force


 U U U 
F  U    ,
, 
 x y z 
because
and
dU  dU x  dU y  dU z 
z
F
dr
y
x
U
U
U
dx 
dy 
dz
x
y
z
 
dU  dW  F  dr  Fx dx  Fy dy  Fz dz
Example. Gravitational potential energy at the surface
z
z
m
W = - mg
Ux , y, z   mgz
y
x
 U U U 
W   ,
,   0,0, mg 
 x y z 
Example. Gravitational potential energy
GMm
GMm

Ux, y, z   
r ( x, y, z)
x 2  y2  z2
U
U


Fi  
x i
x i
 GMm
 GMm
x y z
2
2
2x i

2 x 2  y2 

3
2 2
z
2
F

 G
z
r
y
Mm x i

2
r
r
x

Mm
Mm x, y, z
 G 2  r̂
Fx, y, z   G 2 
r
r
r
Example. What should be the initial speed of an
object which is supposed to escape the
gravitational field of the Earth?
vprobe
vEarth
M
m
in the reference frame of the Earth:
2
mv esc
  Mm  
  G
  0  0
2
  R 
vesc 
of the mechanical
energy
in theconservation
reference frame
of the object:
2GM
km
 11.2
R
s
2
Mv esc
  Mm  
  G
  0  0
2
  R 
vesc 
What is wrong
2Gm
m
 1.45  10 7
R
s