Download Work – Energy Principle

Work – Energy Principle
Units of work and energy:
SI: N-m or Joule
USC: ft-lb
1
K = mv 2
2
Define kinetic energy:
Work-Energy Principle:
1
1
2
2
mv1 + ∑W1− 2 = mv2
2
2
⇒
K1 +
∑W1− 2 = K 2
Work-Energy Principle for a System of Particles:
K1 +
∑W1− 2 = K 2
⎛1
i =1 ⎝
n
∑ ⎜ 2 mv1
⇒
for the system
2
⎞
⎟ +
⎠i
⎛1
i =1 ⎝
n
n
∑ ( ∑W1− 2 )i = ∑ ⎜ 2 mv2
i =1
2
⎞
⎟
⎠i
Work
The total work from position 1 to position 2:
r2
s2
r1
s1
W1− 2 = ∫ F • dr = ∫ F cosθ ds
Work is the integral over the displacement of the component of the force in
the direction of the displacement.
Work of sliding friction from position 1 to position 2:
W
W1− 2 =
x
F
Rough Surface
s2
∫ f cosθ ds = 0
s1
W1− 2 = − f (s2 − s1 )
F
f
f = μk N
N
1
Work – Specific Cases
F
θ
B. Constant force in rectilinear motion
W1− 2 = F cosθ (s2 − s1 )
s2
y
W1− 2 = − mg ( y2 − y1 )
C. Work of a weight
s
s1
mg
x
z
D. Work of a linear spring
x
1
1
W1− 2 = kx12 − kx22
2
2
Frictionless Surface
Work by spring on body is negative when the body is moving away from the
undeformed position, and positive when returning to the undeformed position.
Given: Block A has a mass of 20. kg and block B has a mass of 10. kg. The
system starts from rest and μ k = 0.20 .
Required: The speed of block A after it moves 2.0 m down the incline.
Solution:
2
y
x
Conservation of Energy
•
An alternative form of Work and Energy.
•
Very useful with Conservative Forces.
•
A force is conservative if the work done by
the force in going from position 1 to
position 2 ONLY depends on the positions
1 and 2 and DOES NOT DEPEND ON
THE PATH from position 1 to 2.
•
A conservative force is path independent.
•
Therefore, a non-conservative
force is path dependent.
•
W1→2 =
position 2
r
∫r F ⋅ dr
That is, the work done by a nonconservative force in going from position 1
to position 2 DOES DEPEND ON THE
PATH from position 1 to 2.
2
1
position 1
path 1
path 2
path 3
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Example
position 1
Given: The crate
slides down the
incline from position
1 to position 2.
Assume the distance
h is held constant
for all values of θ .
μs
μk
h
position 2
Required: Exam
the work for all
forces for various
values of θ .
θ
Example Cont.
position 1
F
θ
Remember:
hold h constant.
N
θ
W
y
s
^
j
^
i
h
x
position 2
F
Normal:
Weight:
N
θ
W
Friction:
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Example Continued: Conclusions
Weight (force due to gravity) is a conservative force. The work
only depends on the vertical position at positions 1 and 2
position 1
position 1
h
position 2
position 2
h
path 1
path 2
Friction IS NOT a conservative force. If we change the path
from position 1 to position 2 the amount of work changes.
Other Forces
What about the normal force?
Who cares? It does NO work!
What about the spring force?
The spring force is conservative
since the work only depends on the
deformation at positions 1 & 2.
x
undeformed
W1 → 2 =
x1
P1
1 2
1
x1 − x22
2
2
x2
P2
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Potential Energy
•
The amount of work that a conservative force would do if
the force were to move from its current position to a
given (defined) reference position.
•
The reference position is called the datum.
•
The symbol of Potential Energy is U.
•
Types of forces that have Potential Energy:
•
•
Weight (i.e., the force due to gravity).
•
Spring.
•
ANY Constant Conservative Force.
Friction and ALL other Non-Conservative forces DO
NOT have a Potential Energy.
Conservation of Energy
We will see that:
W1 → 2 = U1 − U 2
But,
Conservation of Energy: The
SUM of ALL Potential and
Kinetic energies remains constant.
K1 + W1 → 2 = K 2
Implies,
K1 + U 1 − U 2 = K 2
• Works only with conservative forces.
• Ignores thermodynamic energy, strain
energy, etc.
Rearrange,
K1 + U 1 = K 2 + U 2
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