6. Work, Energy & Power 功，能和功率 Forces that Vary 會變化的力 Kinetic Energy 動能 Power 功率

Direct application of Newton’s law can be infeasible. 直接使用牛頓定律可能不切實際。 Does the work of climbing a mountain depend on the route chosen? 爬一個山所作的功會隨路徑而改變嗎？ No 否 Direct application of Newton’s law can be infeasible. 直接使用牛頓定律可能不切實際。 這滑雪人的加速度隨坡度而變 這滑雪人的加速度固定 What’s the speed of these skiers at the bottom of the slope? 這兩個滑雪人抵達山坡底時的速率為何? Energy conservation to the rescue (Chap 7). 能量守恆來救駕 (第七章) simple 簡單 complicated 複雜

6.1. Work 功 Work W done on an object by a constant force F is rF = displacement along direction of F. 沿 F 方向的位移 Fr = force along direction of r . 沿 r 方向的力 Note: F need not be a net force. 注意：F 不需要是淨力 

Example 6.1. Pushing a Car 推一部車 The man pushes with a force of 650 N, moving the car 4.3 m. 這男人用 650 N 的力推，把車移動了4.3 m。 How much work he does? 他作了多少功？

Example 6.2. Pulling a Suitcase 拉一個行李箱 Woman exerts 60 N force on suitcase, pulling at 35 angle to the horizontal. 女人對行李箱用力 60 N，朝與水平成 35 的方向拉。 How much work is done if the suitcase is moved 45 m on a level floor? 如果行李箱在水平的地板上移動了 45 m，所作的功為何？ xF = x cos x

Work & the Scalar Product 功和純量(乘)積 Scalar = quantity specified by a single number 純量  由一個在任何座標系统 that is the same in every coordinate system. 都一樣的數字所代表的量 Scalar has no direction. 純量無方向 B Scalar (dot) product of vectors A & B : 向量 A & B 的純量(點)乘積： is a scalar 是個純量  2-D A BA = B cos 3-D = angle between F & r F & r 的夾角 Work is a scalar. 功是一個純量

Proof of y A  B A B x

Example 6.3. Tugboat 拖船 Tug boat pushes a cruiser with force 拖船推巡洋艦的力為 F = ( 1.2, 2.3 ) MN, displacing the ship by 使之移位 r = ( 380, 460 ) m. Find the work done by the tugboat. 求拖船所作功 Find the angle between F & r. 求 F & r 的夾角

6.2. Forces that Vary 會變化的力 力 位置

Tactics 策略 6.1. Integrating 作積分  Example 例 :  Since 因 we have 故 See Appendix A for integral table. 附錄 A 是一個積分表

Stretching a Spring 拉長一個彈簧

Example 6.4. Bungee Jumping 高空彈跳 Bungee cord is 20 m long with k = 11 N/m. 彈跳繩長 20 m，其 k = 11 N/m。 At lowest point, cord length is doubled. 在最低點時，繩長加倍。 How much work is done on cord? 對繩所作功為何？ How does work done in the last meter compare with that done in the 1st meter? 最後一米所作的功與第一米的比較怎樣？ (a) (b) 1st meter 第一米 Last meter 最後一米

Example 6.5. Rough Sliding 在粗糙地面上滑動 Workers pushing a 180 kg trunk across a level floor encounter a 10 m region where floor becomes increasingly rough. 工人在一片水平的地板上推一個 180 kg 的箱子時，遇到一 10 m長，越來越粗糙的區段。 There, k = 0 + a x2, with 0 = 0.17, a = 0.0062 m2 & x is the distance into the rough part. 其間， k = 0 + a x2, 0 = 0.17, a = 0.0062 m2 & x 是進入粗糙區段的距離。 How much work does it take to push the trunk across the region? 把箱子推過這區段所需的功為何？

Force & Work in 2- & 3- D 二和三維的力與功 Line integral 線積分 1-D 2-D 3-D

Work Done Against Gravity 對重力做的功 Only vertical displacement requires work against gravity 祇有垂直方向的位移需要對重力做功 W = m g h

GOT IT 懂嗎? 6.2. 3 forces have magnitudes in N that are numerically equal to 三個力的大小以 N 為單位時，數值等於 (a) x, (b) x2, (c) x, where x is the position in meters. x 是以米為單位的位置。 Each force moves an object from x = 0 to x = 1 m. 每個力都去把一個物體從 x = 0 移到x = 1 m 。 Note that each force has the same values at the end points, namely, 0 N & 1 N. 留意每個力在兩端點的值都一樣是 0 N & 1 N 。 Which force does the most work? 那個力作最多功？ Which does the least? 那個力作最少功？ (c) (b)

6.3. Kinetic Energy 動能 Kinetic energy 動能 : K is relative (depends on reference frame). K 是相對的 (依參考框而定) 。 K is a scalar (independent of coordinate system). K 是個純量。 Work-energy theorem : 功-能定理

Example 6.6. Passing Zone 超車路段 A 1400 kg car enters a passing zone & accelerates from 70 to 95 km/h. 一輛 1400 kg 的汽車進入超車路段後由 70 加速至 95 km/h 。 How much work is done on the car? 對車子所作的功為何？ If the car then brakes to stop, how much work is done on it? 如果車子之後剎車停下來，對它所作的功為何？ (a) b)

GOT IT 懂嗎? 6.3. For each situation, tell whether the net work done on a soccer ball is 為下列每一狀況，判斷所作的淨功是 positive, negative, or zero. 正的，負的，或是零。 Justify your answer using the work-energy theorem. 用功-能定理證明你的答案是對的。 You carry the ball to the field, walking at constant speed. 你以等速行走，把球帶到球場。 You kick the stationary ball, starting it flying through the air. 你把靜止的球踢到空中。 The ball rolls along the field, gradually coming to a halt. 球在場上滾動，慢慢停下來。 zero 零 (K=0) positive 正的(K>0) negative 負的(K<0)

Energy Units 能量單位 [ energy 能 ] = [ work 功 ] = J 焦耳 (SI 公制) CGS: Other energy units 其他的能量單位 : eV (electron-volt): used in nuclear, atomic, molecular, solid state physics. eV (電子伏特) : 用於核子，原子，分子，固態物理。 cal (calorie), BTU (British Thermal Unit): used in thermodynamics. cal (卡路里), BTU (英熱單位) : 用於熱力學。 kW-h (kilowatt-hours): used in engineering. kW-h (仟瓦-小時) : 用於工程。 See Appendix C 請看附錄 C

6.4. Power 功率 Average power 平均功率: (Instantaneous) power : (瞬間)功率 Example 6.7. Climbing Mount Washington 爬上華盛頓山 A 55 kg hiker makes the vertical rise of 1300 m in 2 h. 一個 55 kg 的行者用 2 h 完成那1300 m 的垂直距離 。 A 1500 kg car takes ½ h to go there. 一部 1500 kg 的汽車用 ½ h 就到了。 Neglecting loss to friction, what is the average power output for each. 忽略摩擦的消耗，每個的平均功率為何？ Hiker 行者: Car 汽車:

for constant power P 若功率 P 為定值  general case 廣義情況 Example 6.8. Yankee Stadium 洋基運動場 Each of the 500 floodlights at Yankee stadium uses 1.0 kW power. 洋基運動場內 500 盞泛光燈，每盞的功率是 1.0 kW 。 How much do they cost for a 4 h night game, if electricity costs 9.5 ₵ / kW-h ? 如果電費是 9.5 ₵ / kW-h ，一場 4h 的夜間賽事要多少錢？

Energy and Society 能量和社會 地熱，風力，太陽能 水力 生質 核子 煤 住宅 運輸 商業 油 氣 工業

Power & Velocity 功率和速度 Example 6.9. Bicycling 騎腳踏車 Riding a 14 kg bicycle at a steady 18 km/h (5.0 m/s), 騎着一部 14 kg 的腳踏車以 18 km/h (5.0 m/s) 的等速前進時， you experience a 30 N force from air resistance. 你受到的空氣阻力是 30 N 。 If you mass 68 kg, what power must you supply 如果你的質量是 68 kg ， on level ground. 在平地時 up a 5 slope. 上一個 5斜坡時 你需要的功率是多少？ Fair v v Fair Fg (a) (b)