Conservation of Energy. Chapter 7 of Essential University Physics, Richard Wolfson, 3 rd Edition

Transcription

1 Conservation of Energy Chapter 7 of Essential University Physics, Richard Wolfson, 3 rd Edition 1

2 Different Types of Force, regarding the Work they do. gravity friction 2

3 Conservative Forces BB WW cccccccc = FF cccccccc rr. ddrr AA = cccccccccccccccc (AA, BB) There are certain forces that their Work is always the same, regardless of the path of going from A to B. When this happens, we say that this type of force is conservative. 3

4 An Equivalent Definition for Conservative Forces: path 1 path 2 FF cccccccc rr. ddrr = 0 WW cccccccc = BB AA, pppppppp 11 FF cccccccc rr. ddrr + AA BB, pppppppp 22 FF cccccccc rr. ddrr = BB AA, pppppppp 11 FF cccccccc rr. ddrr BB AA, pppppppp 22 FF cccccccc rr. ddrr = 0 4

5 Examples of Conservative and Non-Conservative Forces: gravity conservative friction non-conservative 5

6 Work done by Gravity yy BB WW gggggggg = FF gggggggg rr. ddrr AA BB = mm gg dddd AA = mm gg yy BB yy AA = mm gg h All that matters for the amount of work done by gravity is the difference in height for the initial and final positions. 6

7 For Conservative Forces, we can write the following: BB WW cccccccc = FF cccccccc rr. ddrr = cccccccccccccccc AA = UU AA, BB = UU BB UU AA 7

8 Another way to look at conservative forces: BB WW cccccccc = FF cccccccc rr. ddrr = UU BB UU AA AA BB AA FF cccccccc rr. ddrr = UU BB UU AA FF eeeeee rr = FF cccccccc rr BB WW cccccccc = WW eeeeee = FF eeeeee rr. ddrr = UU BB UU AA AA 8

9 Gravitational Potential Energy WW cccccccc = UU BB UU AA yy WW gggggggg = mm gg yy BB yy AA UU gggggggg = mm gg yy 9

10 Example 7.1 mm = 55 kkkk UU gggggggg = mm gg yy yy 33 yy 59 UU 33 = 0 yy 1 UU ffffffffff = mm gg yy ffffffffff yy mm = 1 ffffffffff UU 59 = mm gg yy 59 yy 33 UU 1 = mm gg yy 1 yy 33 10

11 Spring Force, Another Conservative Force: BB WW ssssssssssss = FF ssssssssssss rr. ddrr AA xx = kk xx ddxx 0 = 1 2 kk xx2 UU = 1 2 kk xx2 = UU 0 Elastic Potential Energy 11

12 Example 7.2 kk = 120 kkkk/mm UU = 1 2 kk xx2 1 gg oooo gggggggggggggggg xx = 2 kk UU = 86 cccc 12

13 Here is the reason for the name Conservative Forces : Always: rr 2FFnnnnnn WW nnnnnn = rr. ddrr = KK rr 1 FF nnnnnn rr = FF cccccccc rr + FF nnnnnn cccccccc rr In the absence of Work by non-conservative forces: BB WW nnnnnn = AA FF cccccccc rr + FF nnnnnn cccccccc rr. ddrr = UU = KK KK + UU = 0 13

14 For Conservative Forces: or more generally, when the work done by non-conservative forces is zero KK + UU = 0 KK BB + UU BB = KK AA + UU AA EE KK + UU Mechanical Energy EE = KK + UU = 0 14

15 Consider that friction is negligible in the scenario below: FF nnnnnn = nn + FF gg 15

16 BB WW nnnnnn = AA nn + FF gg. ddrr = UU = KK KK BB + UU BB = KK AA + UU AA 1 2 mm vv oo 2 + mm gg yy oo = 1 2 mm vv2 + mm gg yy when friction is negligible 16

17 spring kk = 940 NN/mm xx oo = 25 cccc Example 7.4 vv =?? UU ss = 1 2 kk xx2 bullet mm = 38 gg KK bb = 1 2 mm vv2 vv oo = 0 mm/ss EE = UU ss + KK ss + UU bb + KK bb xx = 0 EE = UU ss + KK bb EE = 1 2 kk xx oo mm vv oo 2 = 1 2 kk xx mm vv2 17

18 Example 7.5 spring kk = 140 NN/mm xx ii,ss = 11 cccc yy ff,bb = h =?? UU ss = 1 2 kk xx2 UU bb = mm gg yy block mm = 50 gg yy ii,bb = 0 mm vv ii,bb = 0 mm/ss EE = UU ss + UU bb + KK bb KK bb = 1 2 mm vv2 xx ff,ss = 0 cccc vv ff,bb = 0 mm/ss EE = 1 2 kk xx ii,ss 2 + mm gg yy ii,bb mm vv ii,bb 2 = 1 2 kk xx ff,ss 2 + mm gg yy ff,bb mm vv ff,bb 2 18

19 EE KK + UU = 1 2 mm vv2 + UU vv 2 = 2 mm EE UU EE > UU 19

20 20

21 At the molecular level: 21

22 Example

23 xx WW cccccccc = FF xx xx ddxx = UU xx 0 00 dduu xx FF xx xx = dddd 23

24 When Non-Conservative Forces are present: BB WW nnnnnn = AA FF cccccccc rr + FF nnnnnn cccccccc rr. ddrr = KK UU EE iiiiii = KK Example: a constant friction force ff kk applied over a distance dd : ff kk dd = EE iiiiii 24

25 Example 7.6 spring kk xx oo xx ff = 0 xx = dd =?? EE iiiiii = ff kk dd block mm vv ii = 0 mm/ss vv ff = 0 mm/ss surface/block friction μμ KK + UU + EE iiiiii = 0 EE iiiiii = UU UU = 1 2 μμ mm gg dd = ff kk dd = kk xx oo 2 = 1 2 kk xx oo 2 kk xx2 25

26 What is actually the Internal Energy? KK + UU + EE iiiiii = 0 It is the sum of the kinetic, vibrational, and rotational energies of all molecules comprising the object(s). The motions of the molecules are random, as opposed to coherent motion of the macroscopic object. There is a distribution of velocities, vibrations, and rotations among the ensemble of molecules. Temperature is a measure of this internal (thermal) energy. 26