PHY183-Lecture31

# PHY183-Lecture31 - Stability For a skyscraper or a bridge...

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1 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 2 Potential Energy and Stability E , K , and U for a roller coaster March 21, 2012 Physics for Scientists&Engineers 1 3 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 March 21, 2012 Physics for Scientists&Engineers 1 4 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 March 21, 2012 Physics for Scientists&Engineers 1 5 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? March 21, 2012 Physics for Scientists&Engineers 1 6

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2 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 7 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 March 21, 2012 Physics for Scientists&Engineers 1 8 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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## This note was uploaded on 04/02/2012 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.

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PHY183-Lecture31 - Stability For a skyscraper or a bridge...

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