12. Static Equilibrium 靜力平衡 Conditions for Equilibrium 平衡的條件 Center of Gravity 重心 Examples of Static Equilibrium 靜力平衡的例子 Stability 穩定性

The Alamillo Bridge in Seville, is the work of architect Santiago Calatrava. 塞維亞的阿拉米略橋是建築師聖地亞哥·卡拉特拉瓦的作品。 What conditions must be met to ensure the stability of this dramatic structure? 要滿定甚麼條件才能保証這個誇張建物的穩定性？

12.1. Conditions for Equilibrium 平衡的條件 (Mechanical) equilibrium = zero net external force & torque. (機械)平衡 = 外力和力距的淨值皆為零 Static equilibrium = equilibrium + at rest. 靜力平衡 = 平衡 + 靜止 For all pivot points 所有支點都是 Pivot point = origin of ri . 支(撐)點 = ri 的原點。 Prob 55:  is the same for all choices of pivot points 對任一個支點的值都一樣

Example 12.1. Drawbridge 吊橋  y 2 Tension 張力 T 1 x Hinge force Fh The raised span has a mass of 11,000 kg uniformly distributed over a length of 14 m. 拉起的橋身的 11,000 kg 質量，均勻地分佈在 14m 的長度上。 Find the tension in the supporting cable. 求支撐纜的張力。 Force Fh at hinge not known. 在鉸鏈處的力 Fh 不詳。  Choose pivot point at hinge. 故取鉸鏈為支點。  y 2 Tension 張力 T 15 1 30 x Hinge force Fh 鉸鏈處的力 Another choice of pivot: Ex 15 選另一支點 Gravity 重力 mg

GOT IT 懂嗎? 12.1. (C) (A): F  0. (B):   0. Which pair, acting as the only forces on the object, results in static equilibrium? 那一對力，單獨作用在物體上，就可以達成平衡？ Explain why the others don’t. 解釋為甚麼其他都不成。 (C) (A): F  0. (B):   0.

12.2. Center of Gravity 重心 Total torque on mass M in uniform gravitational field : 在均勻的重力場內，質塊 M 上的總力距 Center of gravity = point at which gravity seems to act 重心 = 重力似乎施於的那點 CG does not exist if net is not  Fnet . net 不  Fnet 則重心不存在。 for uniform gravitational field 均勻的重力場

Conceptual Example 12.1. Finding the Center of Gravity 求重心 Explain how you can find an object’s center of gravity by suspending it from a string. 說明如何用繩子吊起來就可以找到物體的重心。 2nd pivot 第二個支點 1st pivot 第一個支點

GOT IT 懂嗎? 12.2. The dancer in the figure is balanced; that is, she’s in static equilibrium. 圖中的舞孃是平衡的；也就是說，她在靜力平衡狀態中。 Which of the three lettered points could be her center of gravity? 三個有編碼的點中，那一個是她的重心？

12.3. Examples of Static Equilibrium 靜力平衡的例子 All forces co-planar: 所有的力都共面 2 eqs in x-y plane x-y 面上兩個方程 1 eq along z-axis z-軸上一個方程 Tips : choose pivot point wisely. 祕訣： 選支點要聰明。

Example 12.2. Ladder Safety 梯子的安全使用 A ladder of mass m & length L leans against a frictionless wall. 一質量為 m ，長度為 L 的梯子靠在一無磨擦的牆上。 The coefficient of static friction between ladder & floor is . 梯子與地板的靜磨擦系數為  。 Find the minimum angle  at which the ladder can lean without slipping. 求梯子不滑倒的最小角度  。 Fnet x : n2  y Fnet y :  Choose pivot point at bottom of ladder. 支點取在梯子底 z : mg n1  x fS = n1i   0    90

Example 12.3. Arm Holding Pumpkin 拿著南瓜的手臂 Find the magnitudes of the biceps tension & the contact force at the elbow joint. 求雙頭肌張力和手肘接觸力的大小。 雙頭肌 Fnet x : Fnet y : 肱骨 Pivot point at elbow. 支點在手肘 手肘支點 z : y T  Fc 80 x mg Mg ~ 10 M g

GOT IT 懂嗎? 12.3. A person is in static equilibrium leaning against a wall. 一個人靠在牆上達成靜力平衡。 Which of the following must be true: 下面那些是對的 There must be a frictional force at the wall but not necessarily at the floor. 牆一定要有磨擦力，地板可不必。 There must be a frictional force at the floor but not necessarily at the wall. 地板一定要有磨擦力，牆可不必。 There must be frictional forces at both floor and wall. 牆和地板都一定要有磨擦力。 Need frictional force to balance normal force from wall. 需要磨擦力來平衡牆的正力。

Application: Statue of Liberty 自由神像 落成品 愛菲爾的設計 Sculptor Bartholdi : lasting as long as the pyramids. 雕刻家爸掃地：跟金字塔一般長命。 Deviation from Eiffel’s plan resulted in excessive torque. 修改愛菲爾的設計做成過份的力距。 Major renovation was required after only 100 yrs. 才一百年就要大修。

12.4. Stability 穩定性 Stable equilibrium: Original configuration regained after small disturbance. 穩定平衡： 經過小干擾後能恢復原狀。 Unstable equilibrium: Original configuration lost after small disturbance. 不穩定平衡： 經過小干擾後不能恢復原狀。 Stable equilibrium 穩定平衡 Unstable equilibrium 不穩定平衡

Equilibrium 平衡 : Fnet = 0. V at global min V 為整體最小值 Stable穩定 Unstable 不穩定 V at local max V 為局部最大值 Neutrally stable 中性穩定 V = const 定值 Metastable 介穩定 V at local min V 為局部最小值

Metastable equilibrium : PE at local min 介穩定平衡：位能局部最小值 Stable equilibrium : PE at global min 穩定平衡： 位能整體最小值

Example 12.4. Semiconductor Engineering 半導體工程 A new semiconductor device has electron in a potential U(x) = a x2 – b x4 , 一個新的半導體元件把電子放在位勢 U(x) = a x2 – b x4 中， where x is in nm, U in aJ (1018 J), a = 8 aJ / nm2, b = 1 aJ / nm4. 此處 x 的單位是 nm， U 的是 aJ (1018 J) ， a = 8 aJ / nm2, b = 1 aJ / nm4. Find the equilibrium positions for the electron and describe their stability. 求電子的各個平衡位置，及其穩定性。 Equilibrium criterion : 平衡條件： equilibria 各平衡點  or Metastable 介穩定 x = 0 is (meta) stable (介)穩定 x = (a/2b) are unstable 不穩定

Saddle Point 鞍點 Equilibrium condition 平衡條件 stable 穩定 Saddle point 鞍點 unstable 不穩定 unstable 不穩定

GOT IT 懂嗎? 12.4 Which of the labeled points are stable, metastable, unstable, or neutrally stable equilibria? 下面有標簽的點是在穩定，介穩定，不穩定，還是中性穩定平衡中？ U U N M S