C05._UsingNewtonsLaws
Transcription
C05._UsingNewtonsLaws
5. Newton's Laws Applications 牛頓定律的應用 1. 2. 3. 4. 5. Using Newton’s 2nd Law Multiple Objects Circular Motion Friction Drag Forces 使用牛頓第二定律 多個物體 圓形運動 磨擦 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 (a) The skier’s acceleration 滑雪人的加速度 (b) The force the snow exerts on him. 雪加於他的力 Fnet n Fg m a n 0 , ny nx Fg x m ax ny Fg y m a y a ax , 0 Fg m g sin , cos x: y 2 2 a ax 9.8 m / s 2 sin 32 5.193208... m / s 5.2 m / s n a y: Fg m g sin m ax x n y m g cos 0 n n y 65 kg 9.8 m / s 2 cos 32 540.2066... N 540 N 防熊措施 Example 5.2. Bear Precautions 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? Fnet T1 T2 Fg m a 0 T1 T1 cos , sin Fg 0 , m g T2 T2 cos , sin x : T1 cos T2 cos 0 y T2 y : T1 sin T2 sin m g 0 T1 x Fg T 17 kg 9.8 m / s 2 2 sin 22 a0 since T1 T2 T T mg 2 sin 222.3666... N 220 N 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? Fnet Fh n Fg ma0 Fh Fh , 0 n n sin , cos y n x: Fh n sin 0 y: n cos m g 0 x Fh Fh Fg a0 since Fg 0 , m g Fh n sin n mg 2 sin 60 kg 9.8 m / s tan 30 cos 339.4819... N 340 N mg cos Alternative Approach 另一作法 Net force along slope (x-direction) : 沿斜坡 (x-方向) 的淨力 Fh cos Fg sin 0 Fh Fg tan y n 60 kg 9.8 m / s 2 tan 30 340 N Fh x Fg m g sin 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 (a) greater than, 大, (b) less than, or 小,還是 (c) equal to 相同 the box’s weight? n x: T cos mg sin 0 T mg tan T Fg x Smaller smaller T 小些 T 小些 多個物體 5.2. Multiple Objects 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。 (a) What’s his acceleration? 他的加速度為何? (b) 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。 Frock Tr Fg r n mr a r ac ar a Fclimber Tc Fg c mc ac Tc Tr T Tr Tr , 0 Fg r 0 , mr g Tc 0 , Tc Fg c 0 , mc g Tr mr ar mr g n 0 Tc mc g mc ac n 0 , n ar ar , 0 ac 0 , ac T mr a mr g n 0 T mc g mc a T mr a mr g n 0 T mc g mc a a mc g mr mc 70 kg 9.8 m / s 2 940 kg 70 kg 0.679207... m / s 2 0.68 m / s 2 x x0 v0 t x x0 51 m 1 2 at 2 t 2 x x0 a v0 0 Tension 張力 整條都是 T = 1N throughout 2 51 m 0.679 m / s 2 12 s GOT IT 懂嗎 ? 5.2. What are 求下列各值 1N (a)the rope tension and 1N (b)the force exerted by the hook on the rope? 繩子的張力 掛勾施於繩子的力 5.3. Circular Motion 圓形運動 均勻圓形運動 Uniform circular motion 2 v 2nd law: Fnet m a m 第二定律: r 需有淨力才能轉變運動的方向。 力指向曲線的中心。 centripetal 向心 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。 T Fg m a T T cos , sin a a , 0 Fg 0 , m g x: y T cos m a 半徑是 L cos T T a mg sin T a cos g cot m v2 r x v Fg y : T sin m g 0 ar g L cot cos 道路工程 Example 5.6. Engineering a Road 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 ? n Fg m a n n sin , cos Fg 0 , m g y v2 x : n sin m r n y : n cos m g 0 25 m / s v2 tan r g 200 m 9.8 m / s 2 2 a Fg v2 a , 0 r x 0.318877... 0.32 17.74... 18 繞着圈轉 Example 5.7. Looping the Loop 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? n Fg m a n 0 , n Fg 0 , m g v2 a 0 , r v2 n m g m r Minimum speed n = 0 最低速率 v gr 9.8 m / s 6.3 m 2 7.9 m / s Conceptual Example 5.1. Bad Hair Day 醜髮日 What’s wrong with this cartoon showing riders of a loop-the-loop roller coaster? 這幅畫了一些坐雲霄車的人的卡通有甚麼毛病? From Eg. 由例 5.7: n + m g = m a = m v2 / r Consider hair as mass point connected to head by massless string. 把頭髮想成一粒以無質量的線附在頭上的質點。 Then 則 T + m g = m a where T is tension on string. 其中 T 為線的張力 Thus 故, T = n. Since n is downward, so is T. 因 n 朝下,故 T 亦然。 This means hair points upward 所以頭髮朝上 ( opposite to that shown in cartoon 與卡通所示相反 ). 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 摩擦的本質 摩擦增大 Frictional Forces 摩擦力 Pushing a trunk 推一個箱子 : 1.Nothing happens unless force is great enough. 除非用的力夠大,箱子不會動。 2.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 f s s n v0 s = coefficient of static friction 靜摩擦系數 f k k n Kinetic friction 動摩擦 v0 k = coefficient of kinetic friction動摩擦系數 k s 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: 已上臘的雪橇在乾的雪上 Body-joint fluid: 關節內的液體 k = 0.04 k = 0.003 Application of Friction Walking & driving require static friction. foot pushes ground 腳推地 ground pushes you 地推你 摩擦的應用 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 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) (a) determine the minimum stopping distance. 找出最短的停車距離。 (b) 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 a , 0 n Fg f f m a n 0 , n Fg 0 , m g n m a a v v 2 a x x0 2 v0 2 0 v02 x 2a n m (a) = s : (b) = k : f f n , 0 nm g 0 g x x v02 2s g v02 2k g 25 m / s 2 0.89 9.8 m / s 2 36 m 25 m / s 2 0.61 9.8 m / s 2 52 m 2 2 Application: Antilock Braking Systems (ABS) 應用 : 防鎖剎車系统 Skidding wheel kinetic friction 滑動的輪子 : 動摩擦 Rolling wheel static friction 滾動的輪子 : 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 ). v2 a ,0 r n Fg f f m a n 0 , n 摩擦力 Fg 0 , m g v2 s n m r 車從紙面 出來 v 曲線的中心 s r n m f f s n , 0 nm g 0 s r g (a) v 0.88 73 m 9.8 m / s 2 25 m / s 90 km / h (b) v 0.21 73 m 9.8 m / s 2 12 m / s 44 km / h Example 5.10. Avalanche! Storm dumps new snow on ski slope. 雪崩 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? n Fg f f m a n 0 , n a0 f f s n , 0 Fg m g sin , cos y n x: fs m g sin s n 0 y: n m g cos 0 tan s Fg x tan 1 s tan 1 0.46 25 拖着一個箱子 Example 5.11. Dragging a Trunk 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? n Fg f f T m a a0 n 0 , n f f k n , 0 Fg 0 , m g y x: n T fs x Fg T T cos , sin k n T cos 0 n T T k mg cos k sin y: T cos k n m g T sin 0 cos m g T sin 0 m g k cos k sin GOT IT 懂嗎? 5.4 Is the frictional force 摩擦力與 重量乘摩擦系數 比較是 (a) 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. Drag & Projectile Motion 阻力和拋體運動 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