Work, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition

Transcription

1 Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1

2 With the knowledge we got so far, we can handle the situation on the left but not the one on the right. 2

3 Main Goal: We will learn that: Work done by the net force acting on an object determines the exact change in the object s Kinetic Energy. 3

4 Work WW FF xx FF xx xx WW = NN mm JJ 4

5 Force and Displacement 5

6 Example 6.1 WW = FF xx xx FF xx = 650 NN xx = 4.3 mm 6

7 Example 6.2 WW = FF xx xx FF xx = FF cccccc θθ FF = 60 NN θθ = 35 xx = 45 mm 7

8 Force and Displacement are actually vectors!! 8

9 Definition of Scalar (Dot, Inner) Product of Two Vectors: AA. BB AA BB cccccc θθ P1: AA. BB = BB. AA P2: AA. BB + CC = AA. BB + AA. CC 9

10 AA.? BB + CC = AA. BB + AA. CC CC BB BB + CC AA AA. BB + CC AA. BB AA. CC 10

11 AA = AA xx ii + AA yy jj + AA zz kk BB = BB xx ii + BB yy jj + BB zz kk AA. BB = AA xx ii + AA yy jj + AA zz kk. BB xx ii + BB yy jj + BB zz kk = AA xx BB xx + AA yy BB yy + AA zz BB zz 11

12 As Force and Displacement are vectors, Work is better defined by: FF = FF xx ii + FF yy jj + FF zz kk FF θθ rr rr = xx ii + yy jj + zz kk WW FF. rr = FF rr cccccc θθ = FF xx xx + FF yy yy + FF zz zz 12

13 Example 6.3 FF = 1.2 ii jj MMMM θθ rr = 380 ii jj mm WW =?? θθ =?? 13

14 6.1 Got it? 14

15 Two objects are each displaced the same distance, one by a force F pushing in the direction of motion and the other by a force 2F pushing at 45 to the direction of motion. Which force does more work? a) F b) 2F c) they do equal work 2016 Pearson Education, Inc.

16 Two objects are each displaced the same distance, one by a force F pushing in the direction of motion and the other by a force 2F pushing at 45 to the direction of motion. Which force does more work? a) F b) 2F c) they do equal work 2F does 2 more work than F does. That's because 2F's component along the direction of motion is 2F cos 45, or 2F 2/2 = F Pearson Education, Inc.

17 Let go back to the 1D scenario: (just to make the notation simpler for the time being) WW FF xx xx 17

18 How do we calculate Work when the force is not constant along the path? 18

19 WW xx ii = FF xx xx ii xx xx xx 2 xx 1 NN NN NN WW xx ii = FF xx xx ii xx ii=1 ii=1 lim xx 0 ii=1 WW xx ii = xx 2FFxx WW = xx dddd xx 1 19

20 Stretching a Spring: FF ssssssssssss, xx xx = kk xx FF xx xx = + kk xx 20

21 What is the Work that needs to be done to stretch a spring: xx 2FFxx WW = xx dddd xx 1 FF xx xx = + kk xx xx 1 = 0 xx 2 = xx xx 2kk = xx dddd xx 1 = 1 2 xx2 xx = xx 2 = xx xx = xx 1 = 0 = 1 2 xx = 1 2 xx2 21

22 FF xx xx = + kk xx WW = 1 2 kk xx2 22

23 Example 6.4 xx ii = 11 mm xx ff = 22 mm xx = xx ff xx ii = 11 mm kk = 250 NN/mm FF xx xx = kk xx xx 2FFxx WW = xx dddd = 1 xx 1 2 kk xx2 23

24 Example 6.5 mm = 180 kkkk xx = 10 mm WW =?? FF ppppppp = FF ffffffffffffffff = FF xx xx μμ 0 = 0.17 aa = mm 2 vv = cccccccccccccccc FF xx xx = μμ kk nn = μμ 0 + aa xx 2 nn = mm gg xx 2FFxx WW = xx dddd xx 1 nn WW = mmmm μμ 0 xx + aa 3 xx3 24

25 6.2 Got it? 25

26 Three forces have magnitudes in newtons that are numerically equal to these quantities: (a) x, (b) x 2, and (c) x, where x is the position in meters. Each force acts on an object as it moves from x = 0 to x = 1 m. Notice that all three forces have the same values at the two endpoints namely, 0 N and 1 N. Which of the forces (a), (b), or (c) does the most work? a) x b) x 2 c) x 2016 Pearson Education, Inc.

27 Three forces have magnitudes in newtons that are numerically equal to these quantities: (a) x, (b) x 2, and (c) x, where x is the position in meters. Each force acts on an object as it moves from x = 0 to x = 1 m. Notice that all three forces have the same values at the two endpoints namely, 0 N and 1 N. Which of the forces (a), (b), or (c) does the most work? a) x b) x 2 c) x x does the most work. You can see this by plotting these two functions from x = 0 to x = 1 and comparing the areas under each. The case of x is intermediate Pearson Education, Inc.

28 Three forces have magnitudes in newtons that are numerically equal to these quantities: (a) x, (b) x 2, and (c) x, where x is the position in meters. Each force acts on an object as it moves from x = 0 to x = 1 m. Notice that all three forces have the same values at the two endpoints namely, 0 N and 1 N. Which of the forces (a), (b), or (c) does the least work? a) x b) x 2 c) x 2016 Pearson Education, Inc.

29 Three forces have magnitudes in newtons that are numerically equal to these quantities: (a) x, (b) x 2, and (c) x, where x is the position in meters. Each force acts on an object as it moves from x = 0 to x = 1 m. Notice that all three forces have the same values at the two endpoints namely, 0 N and 1 N. Which of the forces (a), (b), or (c) does the least work? a) x b) x 2 c) x x 2 does the least. You can see this by plotting these two functions from x = 0 to x = 1 and comparing the areas under each. The case of x is intermediate Pearson Education, Inc.

30 1D xx 2FFxx WW = xx dddd xx 1 FF xx xx xx 1 xx dddd xx 2 3D rr 2 rr 1 FF rr rr ddrr rr tt = trajectory rr 2FF rr 2 WW = rr. ddrr = FFxx rr dddd + FF yy rr dddd +FF zz rr 1 rr 1 rr dddd 30

31 Work and Kinetic Energy Work done by the net force acting on an object. FF nnnnnn, xx = mm ddvv xx dddd dddd = vv xx dddd xx 2FFnnnnnn, xx dddd xx 1 xx 2 ddvv xx = mm xx 1 dddd vv xx dddd vv 2mm = vvxx ddvv xx vv 1 = mm 1 2 vv xx 2 vv xx = vv 2,xx vv xx = vv 1,xx = 1 2 mm vv 2,xx mm vv 1,xx 2 31

32 rr 2FFnnnnnn WW nnnnnn = rr. ddrr rr 1 = 1 2 mm vv 2,xx mm vv 1,xx mm vv 2,yy mm vv 1,yy mm vv 2,zz mm vv 1,zz 2 = 1 2 mm vv 2,xx 2 + vv 2,yy 2 + vv 2,zz 2 vv 1,xx 2 + vv 1,yy 2 + vv 1,zz 2 = 1 2 mm vv 2 2 vv 1 2 = KK 2 KK 1 KK 1 2 mm vv2 32

33 Work done by the net force acting on an object determines the exact change in the object s Kinetic Energy. WW nnnnnn = KK rr 2FFnnnnnn WW nnnnnn rr. ddrr rr 1 KK 1 2 mm vv2 33

34 Example 6.6 mm = 1400 kkkk WW nnnnnn, 12 =?? vv 1 = 70 kkkk/h WW nnnnnn, 23 =?? vv 2 = 95 kkkk/h vv 3 = 0 kkkk/h WW nnnnnn = KK WW nnnnnn, 12 = 1 2 mm vv 2 2 vv 1 2 WW nnnnnn, 23 = 1 2 mm vv 3 2 vv

35 Instantaneous Power created by a particular force: ddrr = vv dddd rr 2FF WW rr. ddrr rr 1 tt 2FF = rr. vv dddd tt 1 PP dddd dddd = FF rr. vv PP = WW tt = NN mm = JJ ss ss WW 35

36 Work from Instantaneous Power PP dddd dddd tt 2PP WW = tt dddd tt 1 if power is constant over the time period: WW = PP tt 36