4. Newton's Laws 牛頓定律 The Wrong Question 問錯題 Newton’s 1st & 2nd laws 牛頓第一和第二定律 Forces 力 The Force of Gravity 重力 Using Newton’s 2nd Law 使用牛頓第二定律 Newton’s 3rd Law 牛頓第三定律

What forces govern the motion of the sailboard? Ans 答 : Human body : contact force 人體：接觸力 Wind & water : fluid pressure 風和水：流體壓力 Gravity: action-at-a-distance 重力：遠距作用 What forces govern the motion of the sailboard? 那些力主宰了帆板的運動？

… it always rises to its starting height … 4.1. The Wrong Question 問錯題 Q: Why do things move? 問：為甚麽東西會動? Aristotle (~350BC) : Because some forces are acting on them. 亞里士多德 ( ~公元前 350 ) ：因為有些力對它們作用。 Galileo 伽利略 (~1600) : Wrong question ( things move until stopped ). 問錯題 ( 東西受阻才不動 ) … it always rises to its starting height … …它每次都會升回原來的高度 … If a ball is released here … 如果把球在此放開 … The right question 問對題 : Why do moving things change direction? 為甚麽東西會改變方向? Newton: Because some forces are acting on them. 牛頓：因為有些力對它們作用。 … so this ball should roll forever. … 因此這球會一直在滾。

4.2. Newton’s 1st & 2nd laws 牛頓第一和第二定律 Fnet  0  v  Net force determines motion. 運動由淨力决定。 Newton’s 1st law of motion 牛頓第一運動定律 : A body at rest or in uniform motion remains so unless acted on by a nonzero net force. 靜止或在等速運動的物體會維持原狀，直至受到一個非零的淨力為止。 Fnet = 0  v = const 定值

GOT IT 懂嗎 ? 4.1. On a horizontal tabletop is a curved barrier that exerts a force on a ball, guiding its motion in a circular path. 在一個水平的桌面上，有一個彎曲的屏障。一個球在它的施力下，緣着一個圓形的路線走。 After the ball leaves the barrier, which of the dashed paths shown does it follow? 當球離開屏障時，它會順着那一條虛線走?

Newton’s 2nd Law 牛頓第二定律 (quantity of motion) 運動的數量 Momentum: 動量： Newton’s 2nd law of motion 牛頓第二定律 : The rate of change of momentum is equal to the net force. 動量的變化率等於淨力 For a constant mass : 質量固定時：

Mass, Inertia, & Force 質量，慣性，和力 Inertia: resistance to changes in motion. 慣性：對運動生變時的阻抗 1st law = law of inertia 第一定律 = 慣性定律 Operational definition of mass 質量的操作定義  Loaded truck has 載了貨的卡車 greater mass, 質量較大， more inertia, 慣性較多， less acceleration. 加速度較少。 1 N (Newton牛頓)  force required to give a mass of 1 Kg an acceleration of 1 m / s2. 給 1 Kg 質量 1 m / s2 加速度所需的力 1 N = 1 Kg m / s2. 1 Dyne 達因 = 1 g cm / s2.

Example 4.1. Accelerating Car 加速中的車子 A 1200 kg car accelerates constantly in a straight line from rest to 20 m/s in 7.8 s. 一部 1200 kg 的車子沿一直線用 7.8 s 由靜止穩定地加速至 20 m/s。 What is the net force on it? 它受的淨力為何？ b) What is the net force on it if it rounds a bend 85 m in radius at v = 20 m/s? 如果它以 v = 20 m/s 拐一個半徑 85 m 的彎，它受的淨力為何？

Inertial Reference Frames 慣性參考框架 Inertial Reference Frames: reference frame in which Newton’s 1st law works. 慣性參考框架：牛頓第一定律適用的參考框架。 Examples of non-inertial reference frames: 非慣性參考框架的例子： Accelerating planes. 加速中的飛機 Car rounding a curve. 轉彎中的車子 Merry-go-round. 旋轉木馬 Earth (rotating). 地球 (自轉中) Examples of inertial reference frames: 慣性參考框架的例子： Newton: Distant “fixed” stars. 牛頓：遙遠的 “固定” 星球 Einstein: General relativity. 愛因斯坦：廣義相對論

4.3. Forces 力 Examples of forces : 力的例子 : Pushes & pulls. 推和拉 Car collided with truck & stopped. 汽車撞上貨車然後停止 Moon circles Earth. 月球環繞地球 Person sitting on chair. 人坐椅上 Climber on rope. 繩上的爬山人 (action-at-a distance) (遠距作用) (contact force) (接觸力) (tension force) (張力) Fc = Fg T = Fg

The Fundamental Forces 基本力 Gravity: large scale phenomena 重力： 大範圍現象 Electroweak force 電弱力 Electromagnetic force: everyday phenomena 電磁力：日常現象 Weak (nuclear) force 弱(核)力 Strong (nuclear) force 強(核)力 電 磁 1025 1036 電磁 弱 1 1038 重力 強 電弱 大統力 一切的理論

4.4. The Force of Gravity 重力 m in f = m a is the inertia mass (same everywhere). f = m a 裏的 m 是慣性質量 (到處都一樣)。 Weight = force of gravity on mass: 重量 = 質塊所受重力 For a mass of 65 kg, 質量為 65 kg 的質塊: weight on Earth = (65 kg) (9.8 m/s2 ) = 640 N. 在地球上的重量 = (65 kg) (9.8 m/s2 ) = 640 N. weight on Moon = (65 kg) (1.6 m/s2 ) = 100 N. 在月球上的重量 = (65 kg) (1.6 m/s2 ) = 100 N. weight in outer-space = (65 kg) (0 m/s2 ) = 0 N. 在外太空的重量 = (65 kg) (0 m/s2 ) = 0 N. Caution: 小心： In daily use, weight is often measured in units of mass. 日常用法中，重量常以質量的單位表示 E.g., a person “weights” 65 kg. 例：某人 65 kg “重”。 Strictly speaking, m in w = m g is the gravitational mass. 嚴格說來， w = m g 裏的 m 是重力質量 Galileo experiment: mI = mG (coincidence) 伽利略實驗： mI = mG (巧合) Einstein: mI = mG (exact: gravity is geometry) 愛因斯坦： mI = mG (絕對：重力是幾何)

Weightlessness 無重狀態 same 同樣的 a These astonauts feel “weightless” because they are in freefall. 這些太空人覺得 “無重” ，因為他們在自由下墜。 Their weight is about 93% of that on surface. 他們的重量是在地面時的 93%。 same 同樣的 a

4.5. Using Newton’s 2nd Law 使用牛頓第二定律 Tactics 4.1. Free-Body Diagram 策略 4.1. 自由體圖 Identify object of interest & forces on it. 找出關鍵的物體和作用其上的力 Object  dot. 物體  點 Draw forces on object as vectors at dot. 把作用在物體的力畫成點上的向量 Identify all forces on object 找出作用在物體上所有的力 2. Elevator & content reduce to dot 電梯和載物簡化成點 Cable tension T 纜的張力 T Draw forces as vectors on dot 把力畫成點上的向量 Fg Gravity Fg 重力 Free-body diagram 自由體圖

Identify all forces on object 找出作用在物體上所有的力 2. Skier reduces to dot 滑雪人簡化成點 Force from ground n 由地來的力 Draw forces as vectors on dot 把力畫成點上的向量 n Gravity Fg 重力 Fg Free-body diagram 自由體圖

Example 4.3. Elevator 電梯 A 740 kg elevator accelerates upward at 1.1 m/s2. 一部 740 kg 電梯往上以 1.1 m/s2 加速。 Find the tension force on the cable (of negligible mass). 求繩纜 (質量可忽略 ) 的張力。

GOT IT 懂嗎 ? 4.4. How’s T compared to w = m g if the elevator 電梯在下列情況時，T 與 w = m g 比較是 moves upward starting at rest. 從靜止往上升。 decelerates to stop while moving upward. 上升時減速至停。 starts moving downward, accelerating from rest. 從靜止加速往下。 slows to stop while moving downward. 往下時減速至停。 moves upward with constant speed. 以等速往上。 greater 較大 less 較小 less 較小 greater 較大 equal 相等 a > 0 going up & accel 往上且加速

Conceptual Example 4.1. At the Equator 在赤道 When you stand on a scale, the scale reading shows the force it pushes up to support you. 當你站上秤盤時，秤的讀數顯示它支撐你所用的力。 If you stand on a scale at Earth’s equator, is the reading greater or less than your weight? 如果你站在一個位於地球赤道的秤盤上，秤的讀數比你的重量大還是小? Ans 答 : You’re in uniform circular motion so that the net force on you is centripetal. 你正在等圓周運動中，所以你受的淨力向心。 m a W = m g Fscale

Parabolic Flight 拋物線形航線

4.6. Newton’s 3rd Law 牛頓第三定律 Newton’s 3rd law: Action = Reaction 牛頓第三定律 : 作用力 = 反作用力 Forces of 3rd law pair act on different objects. 第三定律成對的力作用在不同的物體上。 Hence, they don’t cancel each other out. 所以他們不會自相抵消。 Book pushes on hand with force FBA. 書以力 FBA 推手。 Rocket pushes on gas. 火箭推氣體。 Hand pushes on book with force FAB. 手以力 FAB 推書。 Gas pushes on rocket. 氣體推火箭。

Horse-Cart dilemma 馬-車疑難 These forces constitute an equal but opposite pair, but they don’t act on the same object so they don’t cancel. 這些力組成相等但相反的一對；但它們不是作用在同一物體上，因此它們不會自相抵消。 The forward force from the road on the horse is greater than the backward force from the cart so the net force & hence acceleration are forward. 路施於馬的向前力大於車施於馬的向後力，因此淨力及加速度都向前。

Example 4.4. Pushing Books 推書本 2 books lie on frictionless horizontal surface. 兩本書放在一片無摩擦力的水平面上。 You push with force F on book of mass m1 , 你用 F 力推質量為 m1 的書； which in turn pushes on book of mass m2. 它再推往質量為 m2 的書。 What force does the 2nd book exert on the 1st ? 第二本書加於第一本書的力為何? Acceleration of books : 書本的(共同)加速度： Net force on 2nd book : 第二本書受到的淨力 : Force exerted by 2nd book on the 1st 第二本書加於第一本書的力

GOT IT 懂嗎 ? 4.5 Is the net force on the larger block 施於較大質塊的淨力是 greater than 大於 2 N , equal to 等於 2N , or 還是 less than 小於 2 N ?

Normal force n : contact force acting normal to contact surface. Not a 3rd law pair 不是第三定律的成對力 Normal force 正力 Fnet  0 3rd law also applies to non-contact forces such as gravity. 第三定律亦適用於非接觸力，例如重力。

Measuring Force 力的測量 fsp = 0 fsp = fwall < 0 fsp = fwall > 0 Force can be measured using Newton’s 3rd law. 力可用第三定律測量。 Ideal spring (Hooke’s law) : 理想彈簧 (虎克定律) : k = spring constant 彈簧常數 fsp = 0 fsp = fwall < 0 Works only in inertial frame. 祇能用於慣性框架。 fsp = fwall > 0

Example 4.5. Helicopter Ride 坐直升機 Helicopter rises vertically. 直升機垂直上升 A 35 kg bag of concrete sits in it on a spring scale of k = 3.4 kN/m. 其內一個 k = 3.4 kN/m 的彈簧秤上放了一個 35 kg 的水泥袋子。 By how much the spring compresses when helicopter is at rest (a) 當直升機靜止時， when it’s accelerating upward at 1.9 m/s2. (b) 當直升機往上加速 1.9 m/s2 時， 彈簧壓縮了多少? (a) (b)

GOT IT 懂嗎 ? 4.6 Exmaple 4.5 Helicopter rises vertically. 直升機垂直上升 A 35 kg bag of concrete sits in it on a spring scale of k = 3.4 kN/m. 其內一個 k = 3.4 kN/m 的彈簧秤上放了一個 35 kg 的水泥袋子。 By how much the spring compresses when helicopter is at rest (a) 當直升機靜止時， when it’s accelerating upward at 1.9 m/s2. (b) 當直升機往上加速 1.9 m/s2 時， 彈簧壓縮了多少? Continued from Example 4.5 繼續例題 4.5 : Would the answer to (a) change if the helicopter were moving upward with constant speed? 如果直升機以等速往上升，小題 (a) 的答案會改變嗎 ? Would the answer to (b) change if the helicopter were moving downward but still accelerating upward? 如果直升機往下降但仍然往上加速，小題 (b) 的答案會改變嗎 ? No 不 No 不