Conservation of Mechanical 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.
•
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.
W1→2 =
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
position 2
h
position 2
position 2
h
r2
∫r F ⋅ dr
path 1
1
path 2
position 1
path 1
path 2
path 3
Friction IS NOT a conservative force. If we change the path
from position 1 to position 2 the amount of work changes.
Potential Energy
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
•
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.
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The reference position is called the datum.
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The symbol of Potential Energy is U.
•
Types of forces that have Potential Energy:
undeformed
x1
W1 → 2
P1
1
1
= x12 − x22
2
2
x2
P2
•
•
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.
1
Potential Energy of Weight
For the weight the choice of the datum is arbitrary (i.e., we can put
it anywhere we want) but it must be clearly defined.
Potential Energy of a Spring
In this case the datum is defined for you. The datum of a spring is
ALWAYS the undeformed position.
x
block
W1 → 2 = mg (h1 ) ⇒ U1 = mg (h1 )
1
h1
2
datum
table
h2
U = 0
undeformed
x1
W2 → 2 = mg (0 ) ⇒ U 2 = 0
1
U1 = + kx12
2
P1
W3 → 2 = − mg (h2 ) ⇒ U 3 = − mg (h2 )
x2
3
Note
P2
W1 → 2 = mg (h1 ) = mg (h1 ) − 0 = U1 − U 2
In general :
Conservation of Mechanical Energy
We have seen:
W1 → 2 = U1 − U 2
But,
1
U spring = + kδ 2
2
1
U 2 = + kx22
2
where δ is the deformatio n
Given: A bob is released from rest at A1 (horizontal).
A1
Required: Velocity at A2; tension in string at A2
Solution:
Conservation of Energy: The
SUM of ALL Potential and
Kinetic energies remains constant.
L
A2
K1 + W1 → 2 = K 2
Implies,
K1 + U1 − U 2 = K 2
• Works only with conservative forces.
• Ignores thermodynamic energy, strain
energy, etc.
Rearrange,
K1 + U 1 = K 2 + U 2
Earthquake code: The gravity design load for components supported by chains or otherwise
suspended from the structural system above shall be three times their operating load.
2
Given: The 5.0 lb collar rides on a smooth
rod and has a speed of 5.0 ft/s at A. The
spring has an unstretched length of 4.0 ft.
Required: The speed of the collar at B.
Solution:
Given: The 200. kg roller coaster car is
given an initial velocity at B such that it
just barely makes it around the vertical
loop at C. Ignore friction and ρC = 25 m.
Required: The maximum height h that
the car will reach.
Solution:
3