12. Static Equilibrium

1. 12. Static Equilibrium Conditions for Equilibrium Center of Gravity Examples of Static Equilibrium Stability

2. The Alamillo Bridge in Seville, is the work of architect Santiago Calatrava. What conditions must be met to ensure the stability of this dramatic?

3. 12.1. Conditions for Equilibrium (Mechanical) equilibrium = zero net external force & torque. Static equilibrium = equilibrium + at rest. For all pivot points Pivot point = origin of ri . Prob 55:  is the same for all choices of pivot points

4. Example 12.1. Drawbridge The raised span has a mass of 11,000 kg uniformly distributed over a length of 14 m. Find the tension in the supporting cable. Force Fh at hinge not known.  Choose pivot point at hinge.  y 2 Tension T 15 1 30 x Another choice of pivot: Ex 15 Hinge force Fh Gravity mg

5. GOT IT? 12.1. Which pair, acting as the only forces on the object, results in static equilibrium? Explain why the others don’t. (C) (A): F 0. (B):   0.

6. 12.2. Center of Gravity Total torque on mass M in uniform gravitational field : Center of gravity = point at which gravity seems to act CG does not exist if net is not  Fnet. for uniform gravitational field

7. Conceptual Example 12.1. Finding the Center of Gravity Explain how you can find an object’s center of gravity by suspending it from a string. 2nd pivot 1st pivot

8. GOT IT? 12.2. The dancer in the figure is balanced; that is, she’s in static equilibrium. Which of the three lettered points could be her center of gravity?

9. 12.3. Examples of Static Equilibrium All forces co-planar:  2 eqs in x-y plane  1 eq along z-axis Tips: choose pivot point wisely.

10. Example 12.2. Ladder Safety A ladder of mass m & length L leans against a frictionless wall. The coefficient of static friction between ladder & floor is . Find the minimum angle  at which the ladder can lean without slipping. Fnet x : n2  y Fnet y :  Choose pivot point at bottom of ladder. z : mg n1  x fS = n1i   0    90

11. Example 12.3. Arm Holding Pumpkin Find the magnitudes of the biceps tension & the contact force at the elbow joint. Fnet x : Fnet y : Pivot point at elbow. z : y T  Fc 80 x mg Mg ~ 10 M g

12. GOT IT? 12.3. • A person is in static equilibrium leaning against a wall. • Which of the following must be true: • There must be a frictional force at the wall but not necessarily at the floor. • There must be a frictional force at the floor but not necessarily at the wall. • There must be frictional forces at both floor and wall. Need frictional force to balance normal force from wall.

13. Application: Statue of Liberty Sculptor Bartholdi : lasting as long as the pyramids. Deviation from Eiffel’s plan resulted in excessive torque. Major renovation was required after only 100 yrs.

14. 12.4. Stability Stable equilibrium: Original configuration regained after small disturbance. Unstable equilibrium: Original configuration lost after small disturbance. Stable equilibrium unstable equilibrium

15. Equilibrium: Fnet = 0. Stable V at global min Unstable V at local max Neutrally stable V = const Metastable V at local min

16. Metastable equilibrium : PE at local min Stable equilibrium : PE at global min

17. Example 12.4. Semiconductor Engineering A new semiconductor device has electron in a potential U(x) = a x2 – b x4 , where x is in nm, U in aJ (1018 J), a = 8 aJ / nm2, b = 1 aJ / nm4. Find the equilibrium positions for the electron and describe their stability. equilibria Equilibrium criterion :  or Metastable x = 0 is (meta) stable x = (a/2b) are unstable

18. Saddle Point Equilibrium condition stable Saddle point stable unstable unstable

19. GOT IT? 12.4 Which of the labeled points are stable, metastable, unstable, or neutrally stable equilibria? U U N M S