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12. Static Equilibrium Conditions for Equilibrium Center of Gravity
Examples of Static Equilibrium Stability
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The Alamillo Bridge in Seville, is the work of architect Santiago Calatrava.
What conditions must be met to ensure the stability of this dramatic?
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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
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Example 12.1. Drawbridge y 2 Tension T 1 x Hinge force Fh
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
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GOT IT? 12.1. (C) (A): F 0. (B): 0.
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.
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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
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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
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GOT IT? 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?
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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.
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Example 12.2. Ladder Safety n2 y mg n1 x fS = n1i
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
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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
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GOT IT? 12.3. Need frictional force to balance normal force from wall.
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.
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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.
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12.4. Stability Stable equilibrium: Original configuration regained after small disturbance. Unstable equilibrium: Original configuration lost after small disturbance. Stable equilibrium unstable equilibrium
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Equilibrium: Fnet = 0. Stable V at global min Unstable V at local max Neutrally stable V = const Metastable V at local min
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Metastable equilibrium :
PE at local min Stable equilibrium : PE at global min
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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 (1018 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
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Saddle Point Equilibrium condition stable Saddle point stable unstable
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GOT IT? 12.4 Which of the labeled points are stable, metastable, unstable, or neutrally stable equilibria? U U N M S
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