5. Newton's Laws Applications 牛頓定律的應用 Using Newton’s 2nd Law 使用牛頓第二定律 Multiple Objects 多個物體 Circular Motion 圓形運動 Friction 磨擦 Drag Forces 拖曳(阻)力
Why doesn’t the roller coaster fall from its loop-the-loop track? 為甚麼這雲霄車沒有從它的環路車軌掉下來? Ans. The downward net force is just enough to make it move in a circular path. 答: 向下的淨力剛好足夠使它作圓形運動。
5.1. Using Newton’s 2nd Law 使用牛頓第二定律 Example 5.1. Skiing 滑雪 A skier of mass m = 65 kg glides down a frictionless slope of angle = 32. Find 一個質量為 m = 65 kg 的滑雪人滑下一個無摩擦,傾角為 = 32 的山坡。求 The skier’s acceleration 滑雪人的加速度 The force the snow exerts on him. 雪加於他的力 x : y n a y : x Fg
Example 5.2. Bear Precautions 防熊措施 Mass of pack in figure is 17 kg. 圖中背包的質量為 17 kg 。 What is the tension on each rope? 每根繩子的張力為何? since x : y y : T2 T1 x Fg
Example 5.3. Restraining a Ski Racer 頂住一個滑雪選手 A starting gate acts horizontally to restrain a 60 kg ski racer on a frictionless 30 slope. 在一無摩擦,傾角 30 的斜坡上,一起賽閘從水平的方向頂住一個 60kg 的滑雪選手。 What horizontal force does the gate apply to the skier? 閘在水平方向施於選手的力為何? since y n x : y : x Fh Fg
Alternative Approach 另一作法 Net force along slope (x-direction) : 沿斜坡 (x-方向) 的淨力 y n Fh x Fg
GOT IT 懂嗎? 5.1. A roofer’s toolbox rests on a frictionless 45 ° roof, secured by a horizontal rope. 一個修理屋頂用的工具箱,由一條水平的繩索縛在一個無摩擦,斜角 45 ° 的屋頂上。 Is the rope tension 繩索的張力比箱子的重量 greater than, 大, less than, or 小,還是 equal to 相同 the box’s weight? n x : T Smaller smaller T 小些 T 小些 Fg x
5.2. Multiple Objects 多個物體 Example 5.4. Rescuing a Climber 拯救一個爬山人 A 70 kg climber dangles over the edge of a frictionless ice cliff. 一個 70 kg 爬山人吊在一片無摩擦的冰崖下 He’s roped to a 940 kg rock 51 m from the edge. 他的繩索縛在一塊 940 kg,離崖邊 51 m 的石頭上。 What’s his acceleration? 他的加速度為何? How much time does he have before the rock goes over the edge? 在石頭掉下冰崖之前,他還有多少時間? Neglect mass of the rope. 忽略繩索的質量。
Tension 張力 整條都是 T = 1N throughout
GOT IT 懂嗎 ? 5.2. What are 求下列各值 the rope tension and 繩子的張力 the force exerted by the hook on the rope? 掛勾施於繩子的力 1N 1N
5.3. Circular Motion 圓形運動 Uniform circular motion 均勻圓形運動 2nd law: 第二定律: centripetal 向心 需有淨力才能轉變運動的方向。 力指向曲線的中心。
Example 5.5. Whirling a Ball on a String 揮舞一個縛在繩子上的球 Mass of ball is m. 球的質量是 m。 String is massless. 繩子無質量。 Find the ball’s speed & the string tension. 求球的速率 & 繩子的張力。 x : y : y 半徑是 L cos T a x Fg
Example 5.6. Engineering a Road 道路工程 At what angle should a road with 200 m curve radius be banked for travel at 90 km/h (25 m/s)? 一條彎曲半徑為 200 m 的道路應該傾斜幾度來支援 90 km/h (25 m/s) 的車速 ? y x : y : n x a Fg
Example 5.7. Looping the Loop 繞着圈轉 Radius at top is 6.3 m. 頂點處的半徑是 6.3 m。 What’s the minimum speed for a roller-coaster car to stay on track there? 要雲霄車在該點維持在軌道上的最低速率為何? Minimum speed n = 0 最低速率
5.4. Friction 摩擦 Some 20% of fuel is used to overcome friction inside an engine. 大概 20% 的燃料是用來克服引擎內的摩擦。 The Nature of Friction 摩擦的本質 摩擦 摩擦增大
Frictional Forces 摩擦力 Pushing a trunk 推一個箱子 : Nothing happens unless force is great enough. 除非用的力夠大,箱子不會動。 Force can be reduced once trunk is going. 箱子動了之後,用的力可以減少。 Static friction 靜摩擦 s = coefficient of static friction 靜摩擦系數 Kinetic friction 動摩擦 k = coefficient of kinetic friction動摩擦系數 k : < 0.01 (smooth 光滑), > 1.5 (rough粗糙) Rubber on dry concrete : k = 0.8, s = 1.0 橡皮在乾的水泥上 Waxed ski on dry snow: k = 0.04 已上臘的雪橇在乾的雪上 Body-joint fluid: k = 0.003 關節內的液體
Application of Friction 摩擦的應用 Walking & driving require static friction. 走路和開車都需要靜摩擦 No slippage 沒有滑動 : Contact point is momentarily at rest 接觸點在瞬時間是靜止的 static friction at work 靜摩擦作用 ground pushes you 地推你 foot pushes ground 腳推地
Example 5.8. Stopping a Car 把車子停下來 k & s of a tire on dry road are 0.61 & 0.89, respectively. 一個輪胎在乾路上的 k & s 分別是 0.61 & 0.89。 If the car is travelling at 90 km/h (25 m/s), 如果車子的速率是 90 km/h (25 m/s) determine the minimum stopping distance. 找出最短的停車距離。 the stopping distance with the wheels fully locked (car skidding). 輪子完全鎖住時(車子在滑動)的停車距離。 (a) = s : (b) = k :
Application: Antilock Braking Systems (ABS) 應用 : 防鎖剎車系统 Skidding wheel 滑動的輪子 : kinetic friction 動摩擦 Rolling wheel 滾動的輪子 : static friction 靜摩擦
Example 5.9. Steering 轉軚 A level road makes a 90 turn with radius 73 m. 一條水平的路以半徑 73 m 做 90轉彎 What’s the maximum speed for a car to negotiate this turn when the road is 一輛汽車在下列路況轉此彎時,最高的車速為何? (a) dry 乾 ( s = 0.88 ). (b) covered with snow 積雪 ( s = 0.21 ). 摩擦力 車從紙面出來 曲線的中心 (a) (b)
Example 5.10. Avalanche! 雪崩 Storm dumps new snow on ski slope. 暴風在滑雪坡上倒了一層新雪 s between new & old snow is 0.46. 新和舊雪間的 s 是 0.46 What’s the maximum slope angle to which the new snow can adhere? 新雪能維持附着的最大坡度為何? y n x : y : fs x Fg
Example 5.11. Dragging a Trunk 拖着一個箱子 Mass of trunk is m. Rope is massless. Kinetic friction coefficient is k. 箱子的質量為 m。 繩子無質量。動摩擦系數為 k 。 What rope tension is required to move trunk at constant speed? 如果要箱子以等速移動,繩子的張力為何? y y : x : n T fs x Fg
GOT IT 懂嗎? 5.4 less than, 較小 (b) equal to 相等, or 或 (c) greater than 較大 Is the frictional force 摩擦力與 重量乘摩擦系數 比較是 less than, 較小 (b) equal to 相等, or 或 (c) greater than 較大 the weight multiplied by the coefficient of friction? Reason: Chain is pulling downward, thus increasing n. 原因: 鏈子朝下拉,故 n 變大。
5.5. Drag Forces 阻力 Drag force: frictional force on moving objects in fluid. 阻力:流體內物體移動時所受到的摩擦力。 Depends on fluid density, object’s cross section area, & speed. 與流體的密度及物體的截面積和速率有關。 Terminal speed: max speed of free falling object in fluid. 終端速率: 流體內自由落體的最高速率 Parachute 降落傘: vT ~ 5 m/s. Ping-pong ball 乒乓球: vT ~ 10 m/s. Golf ball 高爾夫球: vT ~ 50 m/s. Sky-diver varies falling speed by changing his cross-section. 高空跳傘人以改變他的截面積來改變他下墜的速率 Drag & Projectile Motion 阻力和拋體運動