1. Short Version : 12. Static Equilibrium

2. 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

3. 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

4. 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.

5. 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

6. 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

7. 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.

8. 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

9. 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

10. 12.4. Stability Stable equilibrium: Original configuration regained after small disturbance. Unstable equilibrium: Original configuration lost after small disturbance. Stable equilibrium unstable equilibrium

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

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

13. 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

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