Short Version : 12. Static Equilibrium

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

Example 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 F h at hinge not known.  Choose pivot point at hinge.  Another choice of pivot: Ex 15 y x 30  Tension T Gravity mg Hinge force F h 15  22 11

GOT IT? 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.

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

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

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.

Example 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. F net x : F net y : Choose pivot point at bottom of ladder.  z :    0    90  y x   mgmg n1n1 f S =  n 1 i n2n2

Example Arm Holding Pumpkin Find the magnitudes of the biceps tension & the contact force at the elbow joint. F net x : F net y : Pivot point at elbow.  z :  ~ 10 M g y x MgMg mgmg T FcFc 80 

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

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

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

Example Semiconductor Engineering A new semiconductor device has electron in a potential U(x) = a x 2 – b x 4, where x is in nm, U in aJ (10  18 J), a = 8 aJ / nm 2, b = 1 aJ / nm 4. Find the equilibrium positions for the electron and describe their stability. Equilibrium criterion :  or x = 0 is (meta) stable x =  (a/2b) are unstable equilibria Metastable

Saddle Point Equilibrium condition Saddle point stable unstable stable unstable