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March 21, 2012
Physics for Scientists&Engineers 1
1
Physics for Scientists &
Engineers 1
Spring Semester 2012
Lecture 31
Stability of Structures and Examples
Stability
For a skyscraper or a bridge,
designers and builders need to
worry about the ability of the
the structure to remain
standing under the influence of
external forces
For example, consider the
bridge carrying Interstate 35W
across the Mississippi River in
Minneapolis, probably from
design related flaws
March 21, 2012
Physics for Scientists&Engineers 1
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Potential Energy and Stability
E
,
K
, and
U
for a roller coaster
March 21, 2012
Physics for Scientists&Engineers 1
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Potential Energy and Force
Special points on these plots
are marked by vertical gray
lines
•
x
1
and
3
represent minima in
•Ze
ro
in
F
and
a
•
2
represents maximum in
and
•
1
,
2
and
3
represent
equilibrium points
•
1
and
3
represent stable
equilibrium points
•
2
represents an unstable
equilibrium point
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Stable and Unstable Equilibrium
Definition
Mathematically, what makes an equilibrium point
stable or unstable is the second derivative of
(
)
•
Positive second derivative (curvature) means stable
•
Negative second derivative (curvature) means unstable
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Physics for Scientists&Engineers 1
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At stable equilibrium points, small perturbations
result in small oscillations around the equilibrium point.
At unstable equilibrium points, small perturbations
result in an accelerating movement away from the
equilibrium point.
Clicker Quiz
Which of the four drawings
represents a stable
equilibrium point for the ball
on its supporting surface?
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Physics for Scientists&Engineers 1
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Quantitative Condition for Stability
To quantify the stability of an equilibrium situation, we
start with the relationship between potential energy and
force from Chapter 6
Zero net force is one of our equilibrium conditions, which we
can write as
March 21, 2012
Physics for Scientists&Engineers 1
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F
r
U
r
F
x
x
dU x
dx
(one dimension)
U
r
U
r
x
ˆ
x
U
r
y
ˆ
y
U
r
z
ˆ
z
0
dU x
dx
0 (one dimension)
Case 1 Stable Equilibrium
If the second derivative of the potential energy with
respect to the coordinate is positive, then the potential
energy has a local minimum at that point
The system is in
stable equilibrium
A small deviations from the
equilibrium point creates a
restoring force that drives
the system back to the
equilibrium point
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Physics for Scientists&Engineers 1
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Stable equilibrium:
d
2
Ux
dx
2
0
x
x
0
Case 2 Unstable Equilibrium
If the second derivative of the potential energy function
with respect to the coordinate is negative at a point, then
the potential energy has a local maximum at that point
The system is in
unstable equilibrium
A small deviation from the
equilibrium position creates
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