Download Work Energy Theorem

WORK
Work provides a means of determining
the motion of an object when the force
applied to it is known as a function of
position.
Physical Modeling, Fall 2006
1
Work Energy Theorem
Wnet is the work done by
Fnet the net force acting on a body.
Wnet   Fnet ( x ) dx
xf
xi
Physical Modeling, Fall 2006
2
Work Energy Theorem (continued)
Wnet   Fnet dx
xf
xi
dv
  madx  m
dx
dt
v dx
v
 mv
dv  mv vdv
dt
xf
xi
xf
xi
f
i
Physical Modeling, Fall 2006
f
i
3
Work Energy Theorem (continued)
Wnet  m vdv
vf
vi
2 vf
v 
2
2
1
 m   2 m(v f  vi )
 2 v
i
Wnet  mv  mv
1
2
Physical Modeling, Fall 2006
2
f
1
2
2
i
4
Work Energy Theorem (concluded)
Define Kinetic Energy
K  mv
1
2
2
Then,
Wnet = Kf - Ki
Wnet = DK
Physical Modeling, Fall 2006
5
WORK - GENERAL DEFINITION

r  
W   Fds
r
 
Fds  (iˆFx  ˆjFy )(iˆdx  ˆjdy )
 
Fds  Fx dx  Fy dy
f
i

rf
W   (Fx dx  Fy dy )
ri
xf
yf
xi
yi
W   Fx dx   Fy dy
Physical Modeling, Fall 2006
6
CONSERVATIVE FORCES
A force is conservative if the work it
does on a particle that moves through a
closed path is zero, otherwise, the force
is nonconservative.
 
 F  dr  0
Physical Modeling, Fall 2006
7
CONSERVATIVE FORCES (cont)
A force is conservative if the work it
does on a particle that moves between
two points is the same for all paths
connecting those points, otherwise, the
force in nonconservative
Physical Modeling, Fall 2006
8
Potential Energy
If
Then,
DU = - Wc
Wnet = DK
Wnet = Wc
DK = - DU
DK + DU = 0
D(K + U) = 0
Physical Modeling, Fall 2006
9
MECHANICAL ENERGY
U+K=E
E = Mechanical Energy
DE = 0
Ei = Ef
Ki + Ui = Kf + Uf
Physical Modeling, Fall 2006
10
Mass and Spring
E  U  K  kx  mv
2
1
2
1
2
mv  E  kx
2
1
2
2
1
2
2
1
2
2
E  kA
1
2
2
2 E k 2 2 kA
2 2
2
2
2
v 
 x 
 x   (A  x )
m m
m
2
v   A  x
2
Physical Modeling, Fall 2006
2
11