Φυσική κατ. Γ΄ Λυκείου

επανάληψη

επανάληψη καλοκαίρι 2014

Φυσική κεφάλαιο 1ο αμείωτες ταλαντώσεις

κατεύθυνσης

Γ΄ Λυκείου

Φυσική Κατεύθυνσης Γ΄ Λυκείου

Οδηγίες επανάληψης Θα κάνετε τη πρώτη ουσιαστική σας επανάληψη στην ύλη που έχετε διδαχτεί. Για να την κάνετε σωστά και να έχει αποτέλεσμα ακολουθήστε πιστά τις παρακάτω οδηγίες. Θα διαβάσετε 1. Σχολικό βιβλίο:    

Παράγραφος 1-1: Εισαγωγή (σελ: 8) Παράγραφος 1-2: Περιοδικά Φαινόμενα (σελ: 8) Παράγραφος 1-3: Απλή Αρμονική Ταλάντωση (σελ: 9 έως 13) Παράγραφος 1-4: Ηλεκτρικές ταλαντώσεις (σελ: 14 έως 17)

2. Φυλάδιο «Κέντρο Μελέτης»: ενότητα: Αμείωτες Μηχανικές Ταλαντώσεις (σελ: 7 έως 37) 3.

Τις σημειώσεις θεωρίας από το τετράδιο σας Θα γράψετε τις απαντήσεις – λύσεις των παρακάτω:

4. Ερωτήσεις σχολικού βιβλίου 1.1 (σελ: 31) έως 1.16 (σελ: 33) 5. Ερωτήσεις πολλαπλής επιλογής φυλαδίου επανάληψης:  Ερωτήσεις πολλαπλής επιλογής σελ 153: 1 έως 10, 12 έως 15  Ερωτήσεις του τύπου Σωστό/Λάθος σελ 157: 1 έως 15, 22 έως 31  Ερωτήσεις αντιστοίχησης σελ 162: 1 έως 9  Ερωτήσεις ανοικτού τύπου σελ 172: 1 έως 14 6. Ασκήσεις σχολικού βιβλίου: 1.27 (σελ: 36) έως 1.31 7. Προβλήματα σχολικού βιβλίου: 1.37 (σελ: 38) έως 1.44 (σελ: 40), 1.46 (σελ: 40) έως 1.50 (σελ: 41) 8. Ασκήσεις – Ερωτήσεις φυλαδίου επανάληψης: (σελ: 177 και μετά) 35 έως 36, 41, 43, 46 έως 54, 56 έως 58, 63, 67, 69 β, 71 β, 73, 74 β 9. Προαιρετικά από το βιβλίο εκδόσεων Σαββάλα, Μαθιουδάκης – Παναγιωτακόπουλος, τις ερωτήσεις κλειστού τύπου στις σελίδες 66-77, 131-146. Σημείωση:

Στο φυλάδιο επανάληψης υπάρχει διαφορά στους συμβολισμούς:  Συχνότητα συμβολίζεται με v αντί με f  Πλάτος ταλάντωσης, μέγιστη ταχύτητα, επιτάχυνση και δύναμη με xo, υο, αο και Fo αντί των Α, υmax, αmax και Fmax  Κινητική και δυναμική ενέργεια ΕΚ και ΕΔ αντί των K και U

Καλές Διακοπές & Καλή σας μελέτη!

Οδηγίες Επανάληψης

2014 - 2015

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171

11. 2 1$ . / 02. !.3 .! 12.1 2 # 2 #" 0 . . .1 " 2. 2&1 " .! * 2. . 2&2 10 1# !2 1 0 2 1#$ )2 2. 2 # / 0 !2 . / 3 !0" 2 " 2 # 1# 20 012 . )1 01 " . . 2 12 $ 1020 12 " . * 0" 1# 2 1 * 2 " 2 " 2 # 1# 20 012 . )1 01 " 2 " /0 " 12 " 2 "

1. E =

ú +

.0 0

V P V 3. E = ⋅ − 1 ⋅ P V 4. E = ⋅ − 1 ⋅ P

2. E = ⋅ − 1 ⋅

ù

0

172

ü ü ùÿûü ÿ þ ü ü ù

ü!&2 10 " . 2 * 2* # 1. . 1 02. 0! / . . .3 !020 2! . .!./0 .2. 0! / + 10& 2. . 1 02. 2. 2&1 . . .3 !020 /* .!./0 .2.

3. . 1 02. i) !. 2. 2&1 ii) . .! 2. 2&1 . . .3 !020 . 1*12 . # . ! *10 . 0 20 10 . .! 2. 2&1 4. 0 . ) . 2 0 0 0 20 0 . .! 2. 2&1 . 2 $! 12 W = 0! 0 . ) 2 1 [ * 0 .2 2 02 .20* # 1 2 # . [ ′[ . . ! %020 2 " 0 1+10 " [ I W # I W . I W ) I W

. .2.1 0# 1020 2. / . ! .2. 2& .!. & 0 1+10& 5. # 0 3 1 2 " . " .! " 2. 2&1 " . .!.12 1020 !.3 2 02. 2 " 3 1 " 10 1# !2 1 0 2 $!)

6. 1 . 0 )! " ©.!$ 3 1 ª +" ! 3 2. 0 1+10 " [ I W # I W . I W

2 " . " .! " 2. 2&1 " )2. # !$0 .!$ 3 1 7. . 0 . / .3 ! 3 1 " 02. * i) . !# 1 " 2.$*2 2." ii) . !# 1 " 0 2 $# 1 " iii) 2.$*2 2." 0 2 $# 1 " 0 )" # * 1 0 # # 0 20 0 . .! 2. 2&1 8. . 0 . . . . . . . 1# +120 . 1 0 . ) . 2 0 0 2 . . !* 02. . ) 2 1 1 !! ." 2 # . 0 20 10 . .! 2. 2&1 9. # 0 12. 0! 0 . .3 ! " ù ) 2 0 .!2 2. . 2 0 3! 0 10. . /0 020 )2 2 1*12 . ." 0 .2 ! # 0 .2 / 0!1 0 20 0 . .! 2. 2&1 11. . . /0 020 )2 2 . ) 0 !0 " 0 ! 1 0" ! ( 10 " !0 . 0 20 10 . .! 2. 2&1 . . # 1020 2 0! / # 0 . ) .! ) 2. . 2&2 . . .3 !020 . .! /0 .

173

12. . / 2. # 0 . ) 0 !0 " 13. . # 0 0 ! 0 . 2 # . * .! * 2. . 2&2 2 !)2# 2 # . * .! * 2. . 2&2 0 ! 1 0 ! 0 . . # 1020 10 1# !2 1 0 2 . !# 1 [ 2 /# . 2 2 . 2 2 # 0 ! 0 . . 020 2 " !.3 " .!.12 10 " 2& 1# .!2 10& ü û = I [ ü = I [

. ( = I [ 10 ) / !. .

14. 2 !)2# 2 # . * .! * 2. . 2&2 0 ! 1 0 ! 0 . . # 1020 10 1# !2 1 0 2 $!) W 2 /# . 2 2 . 2 2 # 0 ! 0 . . 020 2 " !.3 " .!.12 10 " 2& 1# .!2 10& ü û = I W ü = I W ü = I W 10 ) / !. .

15. . 2. 2&1 02. . 0 0* 0! . 0 &2 3 #1.

16. . &" ! * 0 0 !. .2 . 0 02 1 # 0 2 3 #1. .! 2. 2&1 . .!.12 1020 !.3 10 1# !2 1 0 2 $!) 2 . !# 1 [ 2 # 2. . 2&2 # 0 20 0 3 #1. .! 2. 2&1 . / 3 !0" 2 " 2 # 1# 20 012 . )1 01 " . 1# 0! 1 .2. ! * 2 # . ) 2 0 2 2& .!. & . # + 17. 20$ ) 0 / .3 ! .! #1 0 !# )" 0 2 3 # 2. . 2+10 " 18. . . /+1020 !.3 10 1# !2 1 0 2 $!) 2 2 " 3 #1." .! " 2. 2&1 " 1# !.1 . ! * 20 . ) .#2 2 !.3 .! 12.1 19. . . .3 !020 .!./0 .2. 0 0 + # 0 + 2. 0 02 0 2 $!)

20. 02. $!) " 10 ." & " 0 )" 0 #" # 0 + 02. 0 02 0 2 $!) . !0 20 2 $!) 10 ." & " .2 2 / 1 .1 !./ 0 0! + #! & 21. . . 2. 2&1 02. 0 . . .1 # 0 / 1#$ )2 2. 0 )" 2. . 2&2 ù ) 2 0 .!2 2. . 0 . / 1#$ )2 2. 2 # 1#12 .2 " ." 0 .2 ! #

ü ü ùÿûü ÿ þ ü ü ù

174

22. . )20 ." 2. . 2&2 " # 0 20 0 0 . . .1 2. 2&1 ! 1 02. 10 .2 12.1 1# 2 1 * . 0 . 0 /!.1 2 " . )1 01 " 12 1# 2 1 ) 23. . # 30 02. 0 12 1 2 # 2 #" .2 2 1# 2 1 ) ! 3 ! 0" ! * 0 . ! # 0 . ) . 1*12 . 2 ! 1 02. 10 .2 12.1 1# 2 1 * 24. . &" ! * 0 0 !. .2 . 0 02 1 # 0 2 0 . . .1 2. 2&1 2 # 1#12 .2 " ." 0 .2 ! # . .!.12 1020 !.3 2 2 " 2 " 0 . . .1 " 2. 2&1 " 10 1# !2 1 0 2 1#$ )2 2. 2 # / 0 !2 . / 3 !0" 2 " 2 " 12. 0! " . )1 01 " . 1# 0! 1 .2. ! * 2 # . ) 2 0 2 .#2+ 2& . # + 25. ò ." )$ " 12!.2 &2+ ./ 0 0 © .ª õ2. !) 0 2. . 0! 10 . 3#!. $. )" / .2 0 2 #" 12!.2 +20" . ./ 1 # 0 0* 0!. + .2 26. 0 . ) . 2 0 0 0 20 0 . .! 2. 2&1 þ 0 1&1 2 " . !# 1 " [ . ) 2 1 1 !! ." 2 # 0 . [ = [ &W + 3

0 . . ) 2. .!. 2& 1# 3& 0 20 . .2 . + . 2 1# 12. /* . # 0 0! 0 12 . 2 0 0 1$*0 1$ 1 ) = −P& [

þ 3 1 2 " 2.$*2 2." # ! 0 2. 2 " 3 1 " 2 " . !# 1 " [ .2

þ 3 1 2 " 0 2 $# 1 " . ! 0 2. 2 " 3 1 " 2 " . !# 1 " [ .2 / þ 2.$*2 2. # . /* . ) 0 . 0 1# 3.1 27. þ !.3 .! 12.1 2 " . !# 1 " 10 1# !2 1 0 2 $!) . . 1 0 . ) . 2 0 0 # 0 20 0 . .! 2. 2&1 3. 02. 12 1$ . 0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . 2! 2 " 2.$*2 2." $0 2 12 2 2 # 2 " $! " 12 " V . V 2! 2 " 0 2 $# 1 " $0 2 12 2 2 # 2 " $! " 12 " V . V

x (m) 0,2

1 2 3 4 5 6 7 8

16

t (s)

-0,2

.

/ $! 12 V 2 2! 2 " 2.$*2 2." 0 . !)20! . ) 2 2! 2 " 2.$*2 2." 2 $! 12 V

$! 12 W V 2 2! 2 " 0 2 $# 1 " 0 . . =

175

28. þ !.3 .! 12.1 2 " 2.$*2 2." 10 1## !2 1 0 2 $!) . . 1 0 . ) . 2 #0 0 0 # 0 20 0 . .! 2. 2&1 3. 02. 12 1$ . 0" . ) 2 " .!. 2& ! 2 10 " 0 . 1&12 " 0" 0 . t (s) . .1 0" . .2 12 3 4 5 6 7 8 16 . " $! " 12 " V . V 2 . 2 0 0 / !$02. . ) 2 1 1 !! ."

- #0 2 # " $! " 12 " V . V 2 2!

2 " 0 2 $# 1 " 0 . 12

2 $! ) / 12 . . ) V $! V 2. / . *1 .2. # . ) 1# 12. /* .

0 . 1# !. . )!! . / 2 $! ) / 12 . $! V 2 . 2 0 0 0 2. ! " 2 1 1 !! ." 2 # .

29. þ !.3 .! 12.1 2 " 0 2 $# 1 " 10 1# !2 1 0 2 $!) . . 1 0 . ) . .0 2 0 0 # 0 20 0 . .! 2. 2&1 3. 02. 12 1$ .

0 . . ) 2. .!. 2& 1# 3& 0 20 (s) 2 4 6 8 10 12 14 16 32 / .3& 0 20 . .2 . " $! " 12 " V . V 2.$*2 2. 2 # . 2 0 # 0 . 1 0 / - .0 $! 12 W V 2 . 2 0 0 0 2. ! " 2 1 1 !! ." 2 # " $! " 12 " V . V 2 2! 2 " 2.$*2 2." 2 # . 2 0 # $0 2 12 2 2 # / þ 2.$*2 2. 2 # . 2 0 # 0 $! 12 . ! 02. . ) 2 0 1&1 # = # &W +

F 30. þ !.3 .! 12.1 2 " /* . " 10 1# !2 1 0 2 $!) . . 1 0 . ) . 2 0 0F0 # 0 20 0 . .! 2. 2&1 3. 02. 12 1$ .

0 . . ) 2. .!. 2& 1# 3& 0 20 t (s) 2 4 6 8 10 12 14 16 32 / .3& 0 20 . .2 . " $! " 12 " V . V 2.$*2 2. 2 # . 2 0 # 0 . 1 0 / - F0 $! 12 W V 2 . 2 0 0 0 2. ! " 2 1 1 !! ." 2 # " $! " 12 " V . V 2 2! 2 " 2.$*2 2." 2 # . 2 0 # $0 2 12 2 2 # / þ . !# 1 [ 2 # . 2 0 # . ) 2 1 1 !! ." 2 # 0 $! 12 . ! 02. . ) 2 0 1&1 [ = [ &W

ü ü ùÿûü ÿ þ ü ü ù

176

31. 1*12 . ." 0 .2 ! # 2 # 1$ .2 " 0 20k m 0 . .! 2. 2&1 2 #" [ $! 12 W . / !$02. . ) 2 1 1 !! ." 2 " * 0 ! " 2 .! 2 .20* # 1 . 0&! 1020 )2 . !# 1 [ 2 " ." . ) 2 1 1 !! ." 2 " 0 . 2 1# !2 1 2 # $!) # 0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 .

7 . $! 12 W = 0 2 $# 1 $0 . 0 ! 2 . = 0 .

2 þ 2.$*2 2. 2 " ." . ! 02. 0 12 . ) 2 0 1&1 # = # 1# &W 7 $! 12 W = /# . 0 ! 0 . 2 # 1#12 .2 " 0 . 1 0 2 2 2 # m / þ 0! / " 2 " 2. 2&1 " 2 # 1#12 .2 " / 02. . ) 2 0 1&1 = 2 . k 32. 2 !)2# 2 # . * .! * 2. . 2&2 0 ! 1 0 ! 0 . . . 2 12 $ 1020 0 . . ) 2 " 1# .!2 10 " i) ü û = I [ , ii) ( = I [ . iii) ü = I [ 0 2 !.3 2 " .! 12.1 . / . 1020 2 " . . 2 10 " 1."

(.

- x0

(

- x0

0

x0

0

x0

(

0

x0

- x0

33. ù )" .! )" 2. . 2&2 " 0 20 0 2. 2&1 2 #" [ ù 2 2 " 2. 2&1 " / .1 .120 2)20 . 0! / " 2. 2&1 " / .1 02.

2 2! 2 " 12 " /* . " 0 . .3 ! " / .1 02.

0 ! 0 . 2 # 1#12 .2 " 202!. .1 02. / 2 2! 2 " 12 " 2.$*2 2." 202!. .1 02.

0 . . ) 2. .!. & 1# 3& 0 20 . .2

177

34. 2 !)2# 2 # . * .! * 2. . 2&2 0 ! 1 0 ! 0 . . !# 1 2 " ." . ) 2 1 1 !! ." 2 " 10 1# !2 1 0 2 $!) / 02. . ) 2 0 1&1   x = x 0  &t +  . . . 2 12 $ 1020 0 .

2  . ) 2 " 1# .!2 10 " i) ( û = I W ii) ü = I W . LLL ü = I W

0 2 !.3 2 " .! 12.1 . / . 1020 2 " . . 2 10 " 1."

(

(.

t

(

t

35. 2 #" /* . *" .! *" 2. . 2&2 " ù . k ú / # 0 2 / . 0 ! 0 . 0 . m . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . 2. . 2&2 " 0 20 * .! 2. 2&1 (ù

/ # 2 #" 2! 2 " 12 " /* . " 0 . .3 ! " 12 2k m 2. . 2&2 ù 0 . / 1 2 # 2! # 2 " 2 12 " /* . " 0 . .3 ! " 12 2. . 2&2 ú

2. . 2&2 " 2. . 2+ 2. 0 2 / . 1#$ )(ú

2 2. / 2! 2 " 12 " 2.$*2 2." # ú 2 # 2. . 2&2 ú 0 . 3 ! " 0 . *20! . ) 2 2! 2 " 12 " 2.$*2 2." # ù 2 # 2. . 2&2 ù

36. . )" .! )" 2. . 2&2 " 2 # 1$ .2 " 0 20k m 0 2. 2&1 2 #" [ û .2 ! * 0 12. 0!) 2 2 " 2. 2&1 " . / .1 # 0 2 . 2 # 1+ .2 " 0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . þ 0! / " 2 " 2. 2&1 " / .1 02. þ 0 ! 0 . 2 # 1#12 .2 " / .1 02. # 2! # 2 " 12 " 2.$*2 2." 2 # 1+ .2 " 02. 1 0

/ 2! 2 " 12 " 0 2 $# 1 " 2 # 1+ .2 " # / .1 02.

t

178

ü ü ùÿûü ÿ þ ü ü ù

37. û .2 ! * 0 12. 0!) 2 2 " 2 " 2. 2&1 " . * 0 !0 *" . 2 . 2 # 13. ! / # 0 + 202!. .1 # 0 2 " 2 # .2 " 0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . þ 1#$ )2 2. 2 " 2. 2&1 " / .1 02.

2! # 2 " 12 " 2.$*2 2." 2 # 13. ! / # / .1 02. þ 2 2 " 12. 0! " 0 . .3 ! " # 202!. .1 02. / þ 0 ! 0 . 2. 2&1 " 2 # 1#12 .2 " # 202!. .1 02.

38. û .2 ! * 0 12. 0!) 2 " 2 # .2 " . * 0 !0 *" . 2 . 2 # 13. ! / # 0 + # / .1 # 0 2 2 " 2 " 2. 2&1 " 0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . þ 0! / " 2 " 2. 2&1 " # / .1 02.

2! # 2 " 12 " 2.$*2 2." 2 # 13. ! / # # / .1 02. þ 2 2 " 12. 0! " 0 . .3 ! " / .1 02. / þ 0 ! 0 . 2. 2&1 " 2 # 1#12 .2 " # 202!. .1 02.

39. û .2 ! * 0 12. 0!) 2 " 2 # .2 " . * 0 !0 *" . 2 2 " 2 " 2. 2&1 " 0 + # / .1 # 0 2 . 2 # 13. ! / # 0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . þ 0! / " 2 " 2. 2&1 " / .1 02. þ 2 2 " 12. 0! " 0 . .3 ! " # / .1 02.

2! . 2 " 12 " 0 2 $# 1 " 2 # 13. ! / # / .1 02. / þ 0 ! 0 . 2. 2&1 " 2 # 1#12 .2 " # / .1 02.

40. 2 .! ) 2. . 2&2 2 # 1$ .2 " 0 2)" . ) 2 /* . 0 . .3 ! " − N[ 0 0! 0 . /* . . 2 12.1 " − E# ) # E 12. 0! . )1 01 " . # . 0 ! 2 2 " 2.$*2 2." 2 " ." P 0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . + . 2 2. . 2&2 . 1$*0 0 1&1 P. + N[ + E# = 2 " 2 " 2. 2&1 " 0 .22+ 02. !. 0 2 $!) ) " /* / ./ $ + 2 + 2 # 2 #" 0 . 12. 0!)" / $! ) / 12 . # . . 20 2. . . 0 & 0 . ! 1 2 2 # 2 #" $ .

12 1) 2 " 0 . 12. 0!)

k

m

b

41. ) 1 0 0 20 0 . .! 2. 2&1 2 #" [ = P . # " 1#$ )2 2." & = UDG V . 0&! 1020 )2 . !# 1 x 2 # # * 1 0 # . ) 2 1 1 !! ." 2 # 0 . 2 1# !2 1 2 # $!) # . ! %020 2 0 1&1 10 1# !2 1 0 2 $!) . 2 . !# 1 [ . / 02. )2 . W = 0 . . [ . # ! [ . # >ù . [ = W 6,

[ = W + (SI)]

179

42. 0 2. . 2 * 0 1*12 . ." 0 .2 ! # 0 2)" . ) 2 /* . 0 . .3 ! " − N[ 0 0! * i) . /* . . 2 12.1 " – b# ) # E 12. 0! . )1 01 " . # . 0 ! 2 2 " 2.$*2 2." 2 " ." P . ii) 0! / /* . ) = ) &W 12. 0! * 2 #" . 02. 2 " 1#$ )2 2."

0 . . ) 2. .!. 2& 1# 3& 0 20 / .3& 0 20 . .2 . + . 2 2. . 2&2 . 1$*0 0 1&1 ) &W − N[ − E# = P. ù 1#$ )2 2. 0 2 " 0! / " /* . " ) 0 . !)20! . ) 2 / 1#$ )2 2.

2 # 2. . 2&2 . .!$ 10 . .# 02. 1# 0$+" 2 2 " 2 " 2. 2&1 " 1# 0$+" . .# 02.

N þ / 1#$ )2 2. 2 # 2. . 2&2 0 . 1 0 = .

P / õ2. 0 . 0 < 2 1*12 . 2. . 2+ 02. 0 2 / 1#$ )2 2 2 # 43. ) 1 0 ." P 0 20 0 . .! 2. 2&1 þ . !# 1 2 # . ) 2 1 1 !! ." 2 # / 02. . ) 2 0 1&1 [ = [ &W + 3

. . . /0 020 )2 i) # = ± & [ − [ . ii) . = ± & # − # . . .!.12 1020 !.3 10 1# !2 1 0 2 . !# 1 [ i) 2 0 2 $# 1 2 # # * 1 0 # ii) 2 /# . 2 2 . 2 2 # 0 ! 0 . 10 ) / !. .

44. ) 1 0 0 20 0 . .! 2. 2&1 . 0! 0 . ) /* 1 0 . 2 " 2! $ " 2 # ù . ú # . $ # . )12.1 G = FP 0 2 / . 2.$*2 2. + . 2 02 .1 . ) 2 1 0 ù 12 ú . . 20 2. $! ) / 12 . W = V 02 2 !.1 2 # . ) 2 ú 2 # ) 1 0 $!0 02. $! ) / 12 . W = V . . 0! 10 . ) 2 1 0 % * 0 0 . 2 02 3 ! . !0 20 . 2 0! / 2 " 2. 2&1 " 2 2 " 2 " 2. 2&1 " >ù .

V FP@ 45. ) 1 0 0 20 0 . .! 2. 2&1 õ2. . !# 1 2 # $0 2 " [ [ 2.$*2 2 2 # $0 . 2 12 $0" 2 " # # . !0 20 . 2 0! / 2 " 2. 2&1 " 2 2 " 2 " 2. 2&1 " ü3.! [ = P [ = P # = P V # = P V >ù . .

7 =

[ − [ # − #

, 7 = V [ =

# [ − # [ # − #

, [ = P ]

180

ü ü ùÿûü ÿ þ ü ü ù

46. ) 1 0 0 20 0 . .! 2. 2&1 . . !# 1 2 # . ) 2 1 1 !! ." 2 # / 02. . ) 2 0 1&1 [ = [ &W + 3

. . # 1020 2 " 2 " 2& 0 0 + [ & 3 . &! 020 )2 . )12.1 2& . !. & 10& 2 # # * 1 0 # 0 . G P . . W = 0 . [ P . # = − P V . !0 20 2 $! 12 W = 2 0 2 $# 1 2 # # * 1 0 # . .!.12 1020 !.3 10 1# !2 1 0 2 . !# 1 [ . ) 2 1 1 !! ." 2 # 2 1# 12. /* . # .1 0 2. 12 # ) 1 0 . . 2 # 0 . m = 0,1 kg.

. = − P V 0# 0 .@ >ù . [ = P , & = UDG V , 3 =

47. ) 1 0 ." P NJ 0 20 0 . .! 2. 2&1 2 #" [ = P . 0! )/ # V . . !0 20 2 0 $ 12 $! ) / 12 . # . . 20 2. . . 02. 0 2 # ) 1 0 . ) 2 1 [ = P 12 1 [ = − P . / 02. )2 2 # ) 1 0 0! 0 . ) 2 1 [ * 0 i) ! " 2 02 .20* # 1 ii) ! " 2 .! 2 .20* # 1 )1 " 0 . !# )" 02. " 2 " ! " 2 # # * 1 0 # )2. .#2) 0! 0 . ) 2 " 10 " x1 . x2;

>ù .

L V ii) V − ⋅ − 1 ⋅ − 1 ]

48. ) 1 0 0 20 0 . .! 2. 2&1 .2 " 2 # . [ ′[ $!

12 W 2 # ) 1 0 / !$02. . ) 2 1 1 !! ." 2 # * 0 ! " 2 02 .20* # 1 2 " 2 " 2. 2&1 " 0 . [ = FP . 1#$ )2 2. +] . 0&! 1020 )2 . !# 1 [ 2 # # * 1 0 # . ) 2 1 1 !! ." 2 # 0 .

2 1# !2 1 2 # $!) # . . ! %020 2 0 1&1 2 " . !# 1 " [ 10 1# !2 1 0 2 $!) . ! 1/ ! 1020 2 12 2.$*2 2. .2 2!

2 # # * 1 0 # . 2 $! 12 W .2 2 . .#2) . . 2 10 .#2 2 2.$*2 2. . !+2 3 ! 02 2 $! 12 W = . ! 1/ ! 1020 2 12 0 2 $# 1 .2 2!

2 # # * 1 0 # . 2 $! 12 W .2 2 . 2 . 2 . !+2 3 ! 02 2 12 W = / . # 1020 2 1# . )12.1 # / #10 2 # ) 1 0 . ) 2 12 W = &" 2 12 W V >ù . [ = ⋅ − W 6,

P V V

P V , V / P@

181

49. ) 1 0 0 20 0 . .! 2. 2&1 þ . !# 1 2 # [ . ) 2 1 1 !! ." 2 # / 02. . ) 2 0 1&1 [ = W + 3 (SI). . + . 0" 2 " 2 " . !# 1 " [ /# . 2 # ü û 0 . 1 0 2 2 " " 2 # 0 ! 0 ." ü ;

. !0 20 2 2 2 " .!$ " 3 1 " 3 . / 02. )2 . 2 W = 0 . ü . = ü

0 [ ! . #

>ù . ± P P ]

50. ) 1 0 ." P = − NJ 0 20 0 . .! 2. 2&1 2 #" [0 = 0,2 m. $! 12 W = 0! 0 . ) 2 1 [ P * 0 .2 2 02 .

20* # 1 0 + 2 $! 12 W = V 0! 0 . ) 2 / . 1 * 0 .2 2

.! 2 .20* # 1 . 0&! 1020 )2 . !# 1 [ 2 # # * 1 0 # . ) 2 1 1 !! ." 2 # 0 . 2 1# !2 1 2 # $!) # . . # 1020 2 0! / 2 " 2. 2&1 " . ! %020 . 2 2. 2&1 # 0 20 0 2 # ) 1 0 2 " 0 1+10 " 10 1# !2 1 0 2 $!) : i) 2 " . !# 1 " [ LL 2 " 2.$*2 2." # LLL 2 " 0 2 $# 1 " .

.2 2 $! ) / 12 . 2 " 1 " . ) W = $! W = V . !0 20 . 2 1# 12. /* . # 0 0! 0 12 # ) 1 0 : i) 2 + 1 LL 2 !     ≈ >ù .

V L [ =  W +  ii) # = 1#  W +  (SI)       iii) . = −  W +  6,

L − − 1 ⋅ V ii) − ⋅ − - ]   *51. . /* 1+ .2. ù . ú # /0 $ 0 2 1$ . 0 . 2 0y 2 . 2 . & 12 . 0 20 * .2. )!#3 . .! 2. 2&1 0 0! / V . 2 " +y0 \ = P 1+ . ú $0 . P NJ B . . !0 20 2 /* . # .1 0 2 1+ . ú 12 1+ . ù A y 12 " 10 " i) y = 0.

0 ii) y = – 0,25 m. iii) y = + 0,25 m. + . . 2 2 # 2 #" 2. 2&1 " 2 1+ . ú . 0 .2. 0 %0 2 1+ . ù )2. 0! / " 2 " 2. 2&1 " 0 -y0 . V . 0 . 12 1#$ )2 2. 2 " 2. 2&1 " . 2 . 2 1+ . ú /0 . 0 .2. 0 %0 2 1+ . ù )2. 2 2 " 2 " 2. 2&1 " 0 . P û 2. g = 10m/s2 . ≈ >ù .

L ± 1 LL ± 1 LLL ± 1 P þ] @

ü ü ùÿûü ÿ þ ü ü ù

182

B *52. . ! .2. )!#3 # /. * 0 .2 ! # 12. 0! " N = 1 P 0 . 120!0& 10 ! ) 2 0 / 0/ 2 ! 2 # 0 . 12. 0! 1# /0/0y0 m " / 1 " ù ." NJ & 12 / 1 M 0 . 2 02 1+ . ú ." P NJ 1*12 . 1 !! 0 # 0 2 1*12 . .2. )k

k !#3. ! " 2. 2& .2 \ = P . 2 .3 # 0 0 0* 0! y . . /0 020 )2 2 1+ . ú . 0 .2. 0 %0 2 / 1 ù . 0 . 2.$*2 2. . 0 2 $# 1 2 # 1+ .2 " ú 2 12 # 0 .2. 0 0 2 / 1 0 )1 *% " . 3 10 2 1+ . ú & . ) 2 1 12 . 0 .2. 0 0 2 / 1 þ . 2 12.1 2 # . !. 0&!0 2. . 0 2 . . J P V2. >ù .

0 .2. 0 0 12 1 \ = − P # = − P V , . = P V K P@

53. ) 1 0 ." P NJ 0 20 0 . .! 2. 2&1 2 #" [0 P 0 0! / V $! 12 W 2 # ) 1 0 0! 0 . ) 2 1 [ ⋅10-2 m * 0 .2 2 .! 2 .20* # 1 . . ! %020 2 0 1&1 2 " . !# 1 " [ 10 1# !2 1 0 2 $!)

7 $! 12 W = . !0 20 . 2 # ) 1 0

i) 2 /# . 2 # 0 ! 0 . ii) 2 2 2 # 0 ! 0 . ii) 2 !# ) 02. " 2 " ! " 2 # . 0&! 1020 )2 . !# 1 [ 2 # # * 1 0 # . ) 2 1 1 !! ." 2 # 0 .

2 1# !2 1 2 # $!) # . ≅   >ù . [ =  W +  6,

L ⋅ − - ii) ⋅ − - iii) ⋅ − 1 ]   54. ) 1 0 ." P NJ 0 20 0 . .! 2. 2&1 . 2 # 0 ! 0 . 0 . ü = ⋅ − - þ . !# 1 [ 2 # # * 1 0 # . ) 2 1 1 !! ." 2 # 0 . 2 1# !2 1 2 # $!) # . 0 2 $# 1 2 # . 1# / 02. 0 2 . !# 1 . ) 2 1 1 !! ." 2 # 0 2 1$ 1 . ± x 12 6,

. . !0 20 2 0! / . 2 2 " 2 " 2. 2&1 " . ! %020 2 0 1&1 2 " . !# 1 " [ 10 1# !2 1 0 2 $!) . .

W = 2 # ) 1 0 $0 ( û = ü . 0 2. .2 2 02 .20* # 1 >ù . =



 V , [ = P L [ =  W +  (SI) 

   ii) [ =  W +  (SI)]  

183

*55. + . ." P NJ 0 .!2 2. . ) 2 2& ! /. * 0 .2 ! # 12. 0! " N 1 P 0 .2 ! 0 . 120!0& 12 ! 3 . 0 #12 !. . ! 02. . 2 10 1$ 1 0 2 . 0 #12 !. . +" .#2)" . 0 . 0 0 12. 0! 2.$*2 2. 2! # # = P V 0 . 12 . 0 #12 !." 12. .2 0 . )2 . ù . 0 . 2 2 " 2 " 2. 2&1 " # . 0 20 10 . P . . 2 0 .2 ! 2. .!. !3& 0 . 2 ." 2 # ! #" 2 # 1+ .2 " 1# !.2 * 0 . 2 2 1+ . +120 2 0 .2 ! . $0 2 3#1 ) 2 # " ú . ! %020 2 0 1&1 2 " . !# 1 " y 2 " ." P . ) 2 1 1 !! ." 2 " 10 1# !2 1 0 2 $!) . 2 0! 2&1 . . 0&! 1020 )2 . !# 1 y 0 . 2 1# !2 1 2 # $!) # . t )2. . 0 #12 !." 12. .2 0 . 0&! 1020 2 .20* # 1 ! " 2. 2& 02 . J = P V >ù ù .

W + (SI)] P P ú \ =

56. + . ." P NJ 0 . /0 12 . ! /. * 0 .2 ! # 12. 0! " N 1 P . 1 !! 0 ) &" 3. 02. 12 1$ . ù . !* # 0 2 . . ) k 2 1 1 !! ." 2 " .2 2 / 0* # 1 2 # . k 2 # 0 .2 ! # .2 P ! " 2. 2& . 2 .3 # 0 0 0* 0! m . . /0 020 )2 2 1*12 . ." 0 .2 ! # . 0 20 10 . .! 2. 2&1 . . # 1020 y0 2 0! /) 2 " )1 0 . 12 /# . 0 ! 0 . 2 " 2. y 2&1 " )1 0 . 12 /# . 0 ! 0 . 2 # 0 .2 ! # / . ! %020 2 0 1&1 2 " . !# 1 " 2 " ." . ) 2 1 1 !! ." 2 " 10 1# !2 1 0 2 $!) . . W / !$02. . ) 2 1 \ P * 0 ! " 2 .! 2 .20* # 1 . 0&! 1020 )2 . !# 1 y 0 . 2 1# !2 1 2 # $!) # . g = 10 m/s2.

  >ù .

7 = V - - / \ =  W +  (SI)]

  k

k

1 2 57. + . ." P NJ 1 !! 0 & 10 0 ! m ) 2 0 0/ /0 12. !. /* ! ) 2 & /.y + 0 .2 ! & ) &" 3. 02. 12 1$ . . 0 .2 ! . ! 1 2. 12 3#1 ) 2 #" " . $ # x 12. 0! " N = 1 P . N = 1 P ù .k1 k2 !* # 0 2 . . ) 2 1 1 !! ." 2 " .2 m 2 / 0* # 1 2 # . 2& 0 .2 ! & . 2 .3 y # 0 0 0* 0! . . /0 020 )2 2 1*12 . ." 0 .2 ! & . 0 20 10 . .! 2. 2&1 . . # 1020 2 0! / )1 0 . 0 ! 0 . 2 " 2. 2&1 " . 2 2 " 0 . [ = P

>ù .

V -@

184

ü ü ùÿûü ÿ þ ü ü ù

58. + . ." P NJ 1 !! 0 1# /0/0 12. !. /* .2. )!#3& /. + 0 .2 ! & ) &" 3. 02. 12 1$ . 12. 0! " 2& 0 .2 ! & 0 . N1 1 P . N2 1 P ù . !* # 0 2 . . ) 2 1 1 !! ." 2 " .2 2 / 0* # 1 2 # . 2& 0 .2 ! & . 2 .3 # 0 0 0* 0! . . /0 020 )2 2 1*12 . . 0 20 10 . .! 2. 2&1 . . # 120 2 0! / ù 2 2 " 2 " 2. 2&1 " 0 . \0 P )1 0 . 12 2 0 ! 0 . 2 " ." P J P V2). . 0&! 1020 )2 12 1 # . 1 !! 0 2 0 .2 ! 12. 0! " k1 0 . 20 2& . 2 0 .2 ! 12. 0! " k2 0 . 1#1 0 !&

>ù .

V -@

k1

k1

m y0 m

k2

k2

y

59. + . ." P NJ 1 !! 0 & 10 0 ! k1 ) 2 0 0/ 1# /0/0 12. !. /* ! ) 2 & /. + 0 .2 ! & ) &" 3. 02. 12 1$ m k2 . 12. 0! " 2& 0 .2 ! & 0 . . 2 12 $. k1 1 P . N2 1 P ù . !* # 0 2 . . ) 2 1 1 !! ." 2 " .2 2 / 0* # 1 2 # . 2& 0 .2 ! & .2 [0 = 0,2 m x0 k1 . 2 .3 # 0 0 0* 0! . . /0 020 )2 2 1*12 . ." 0 .2 ! & m . 0 20 10 . .! 2. 2&1 . . #k2 1020 2 0! / )1 0 . 0 ! 0 . 2 " 2. 2&1 " . ! %020 2 0 1&1 2 " . !# 1 " 0 2 $!) . . W = . / !$02. . ) 2 1 [ = − P * 0 ! " 2 .! 2 .20* # 1 þ . !# 1 [ 0 . 2 1# !2 1 2 # $!) #

  >ù .

V - [ =  W +  (SI)] 

 60. û* /. 0 .2 ! . 0 12. 0! " N1 = 200 N/m k1 k2 . N2 1 P 1# / 2. 10 10 ! m . ! 2 # 1#12 .2 " # ! * 20 1# x / 02. . ) 2. 0 .2. )!#3 2 $ . 2 1# / 02. 0 1+ . ." P NJ x0 k1 k2 1*12 . 1 !! 0 & 10 0 ! ) m 2 0 0/ ) &" 3. 02. 12 1$ . ù x . !* # 0 2 . . ) 2 1 1 !! ."

2 " .2 2 / 0* # 1 2 # . 2& 0 .2 ! & .2 [0 = 0,2 m . 2 .3 # 0 0 0* 0! . . /0 020 )2 2 1*12 . ." 0 .2 ! & . 0 20 10 . .! 2. 2&1 . . # 1020 2 0! / )1 0 . 0 ! 0 . 2 " 2. 2&1 " 1 12) 0 2 " 2 " " 0 ! 0 ." 2 " 2. 2&1 " 0 . 2 0 ! 0 .

2 " ." )2. / !$02. . ) 2 1 [ P >ù .

V - @

185

61. û* /. 0 .2 ! . 0 12. 0! " k1 = 150 N/m . k2 = 300 N/m 1# / 2. 10 10 ! . ! 2 # 1#12 .2 " # ! * 20 1# / 02. . ) 2. 0 ! 3 . 2 1# / 02. 0 1+ . ." P NJ 1*12 . 1 !! 0 ) &" 3. 02. 12 1$ . ù . !* # 0 2 . . ) 2 1 1 !! ." 2 " .2 2 / 0* # 1 2 # . 2& 0 .2 ! & .2 \0 = 0,2 m . 2 .3 # 0 0 0* 0! . . /0 020 )2 2 1*12 . ." 0 .2 ! & . 0 20 10 . .! 2. 2&1 . . # 1020 2 0! / )1 0 . 0 ! 0 . 2 " 2. 2&1 " 1 12) 0 2 " 2 " " 0 ! 0 ." 2 " 2. 2&1 " 0 . 2 0 ! 0 . 2 " ." )2. / !$02. . ) 2

P J = P V

1 0 \ = −

>ù .

k1

k1

k2 k2 m y0 m

y

V - @

62. 0 " 02. )" * /! " ." P NJ .

0 ./ * / .2 " 6 = FP 0 .!2 2. . ) 2 0 0* 0! ! .2. )!#3 # /. * 0 .2 ! # 12.k k 0! " N 1 P ! 2 # 0 .2 ! # 120!0+ 02. 10 . ) 2 1 0 1*12 . 1 !! 0 0 2 * /! # 1 10 # !) # )2 2." y0 NJ m ! = . 2 0 .2 ! 0 # ü 2! #x m P 0 2 * /! .2. )!#3. ! " 2. 2& .2 y \ = FP . 2 .3 # 0 0 0* 0! ) &" 3. 02. 12 1$ . ! " 12 1 2 # 1#12 .2 " /0 . 2. # )% . . /0 020 )2 2 1*12 . . 0 20 10 . .! 2. 2&1 . . # 1020 2 0! / 02 )1 $!) . ) 2 12 # .3 0 0 0* 0! " * /! " . 0! 10 . ) 2 1 1 !! ." 2 # . !+2 3 ! . ! %020 2 0 1&1 2 " . !# 1 " \ 2 # # /! # . ) 2 1 1 !! ."

2 # 10 1# !2 1 0 2 $!) . . W = 0 . \ = + ⋅ − P . # . 0&! 1020 )2 . !# 1 \ 0 . 2 1# !2 1 2 # $!) # J = P V

 

V \ = ⋅ −  W +  (SI)] >ù .

V

 

186

ü ü ùÿûü ÿ þ ü ü ù

63. + . ." P NJ 1 !! 0 & 10 0 0 0 0/ ) &" 3. 02. 12 1$ . 0 .2 ! 0&!0 2. /. ) 0 12. 0! N 1 P þ & . 1 " 2 # 0 # 0 / # 0 . 3 ° ù . !* # 0 2 1+ . . ) 2 1 1 !! ." 2 # .2 2 / 0* # 1 2 # . 2 # 0 .2 ! # .2 [0 P . 2 .3 # 0 0 0* 0! . . /0 020 )2 2 1*12 . . 0 20 10 . .!

2. 2&1 . . # 1020 2 0! / õ2. 2 1+ . ! 1 02. 12 " 10 " [ = ± P . !0 20 i) 2 /# . 0 ! 0 . 2 " 2. 2&1 " ii) 2 /# . 0 ! 0 . 2 # 0 .2 ! # iii) 2 !# ) 02. " 2 " ! " 2 #

>ù .

V L - LL - LLL ± 1 1@

*64. . ÿ/. ) 0 .2 ! $0 3#1 ) " .

12. 0! N ) # 0 2 0 .2 ! 10 2 .

1 # #" . !0 20 2 12. 0! . 0 )" . ) 2. ) . 0 .2 ! . # ! * 2 # ! 1 * 0 /* . ) 2. ) . 0 .2 ! . . 1# .! * 0 2 " / .2 0 " 2& 1$ P 2& ÿ . ÿÿ . !0 20 2 ) 2& .P + 2& 1& 2& . +120 2. /* 1#12 .2. . 2. . 2+ 2. 0 2 / . 0! / ! ) 2 0 0/ 0 . 0

y k

m 3

x y

k

m 3

1

x

(I)

m1

(Iÿ

2 m2

>ù . ⋅ N

P = ] P

*65. û* 1 0 . ù . ú ! 1 2. 12 / ! ) 2 0 0/ . . $ # 02. * 2 #" = P & 12 !. # 2. 0 + 0 !0 . 0 2. $&! " 2! # ) 1 0 ." P = ⋅ − NJ 2 / $02. . ) 2. 1 0 . ù . ú 0 2 " /# 0 " # $ # 2!. )$ = × $1 . )% = × %1 . 2 12 $. 12 6,

. . /0 020 )2 2 # ) 1 0 0 20 0 . .! 2. 2&1 ù 2 # ) 1 0 0! 0 . ) 2 1 0 ù 0 2.$*2 2. 2! # #ù P V 0 . 2 2! 2 " 2.$*2 2 " 2 # )2. 0! 0 . ) 2 1 0 ú >ù . ' 1 P # % = P V ]

187

*66. 0 " * /! " *% #" þ FP 1 !! 0 0 2 2 # .2. )!#3 12 / .$&! 12 0 !2 3 0 . /* # !+ 0 # )2 20" !1 = 13,6 g/cm3 . !2 = 1 g/cm3 ) &" 3. 02. 12 1$ . þ # )y0 H 2 2. 2 # # * 2 # # /! # 0 . ! 7,2 g/cm3 ü 2! # 0 .2. )!#3. ! " 2. 2& 2 * /! . ) 2 1 1 !! ." 2 # .2 \0 = 1 cm . 2 .3 # 0 0 0* 0! . . /0 020 )2 * /! " . 0 20 10 . .!y !1 2. 2&1 . . # 1020 2 0! / . ! %020 2 0 1&1 2 " . !# 1 " 2 # # /! # . ) 2 1 1 !! ." 2 # 10 1# !2 1 0 2 $!) . . W 0 . \ = + FP . 0&! 1020 )2 . !# 1 \ 0 . 2 1# !2 1 2 # $!) # 2! " 0&! * 2. . 0 2 0" . J = P V   >ù .

V \ = −  W +  (SI)]   67. û 1 " ." NJ 0 . 1# /0/0 " 12 & ! .2.m )!#3 # /. * 0 .2 ! # 12. 0! " N 1 P 2 # # 2 ! 120!0+ 02. 10 . ) 2 1 0 2 # /. / # ù ) *% " K P & . ) 2 / 1 .3 02. . 10 0 0* 0! .

13. ! / ." P NJ 2 1# ! *02. 0 2 / 1 02& . .12 þ / ! 0 . 2 " ! *1 " 0&!0 2. . 0 2 . M . . # 1020 2 2 " 2 " 2. 2&1 " 2 # 1#12 .2 " . ! %020 2 0 1&1 2 " . !# 1 " 2 # 1#11& .2+ .2 " . ) 2 1 1 !! ." 2 # . . W / 02. \ . # k . 0&! 1020 )2 . !# 1 \ 0 . 2 1# !2 1 2 # $!) # J = P V

h

>ù . ⋅ − P \ = ⋅ − W + (SI)] 68. 3. !. ." P NJ .3 02. . 10 0 0* 0!. . ) *% " K = P & . ) / 1 ." NJ " 1 !! 0 1# /0/0 " 12 . ! .2. )!#3 # /. * 0 .2 ! # 12. 0! " N 1 P ) &" 3. 02. 12 1$ . þ 13. !.

1# ! *02. 02& 0 2 / 1 . / ! 0 . 2 " ! *1 " 0 . . 0 2 . 02 2 ! *1 2 " 0 2 / 1 13. !. 3 0 10 *% " K = P . !0 20 . 2. 2!. 2& 2.$#2 2& 2 " 13. !." . 2 # / 1 # . 1&" 02 2 ! *1 2 2 " 2 " 2. 2&1 " 2 # / 1 # 2 !# ) 02. " 2 " ! " 2 # / 1 # )2. ! 1 02. 12 " . !. 0" 10 " 2 " 2! $ " 2 # J = P V

>ù .

P V P V P ± @

y m h1 h2 M

k

y

ü ü ùÿûü ÿ þ ü ü ù

188

69. * " * " ." 1 !! 0 1# /0/0 " 12 . ! .2. )!#3 # /. * 0 .2 ! # 12. 0! " N 2 # # 2 ! 0 . 120!0& 10 ! 3 ò . . ." P 0 2. .2. )!#3. ! " 2. & 0 2.$*2 2. #0 ) &" 3. 02. 12 1$ . . . / . 0! 2 * . 0 !$02. 0 2.$*2 2. # = # þ / ! 0 . 1 " 2 # .2 " 1. 12 * 0 . . 0 2 . )1 0 .

2 2 " 2 " 2. 2&1 " # . 0 20 10 * " . 13 + 02. . .! . . 12 2! ." 2 # * # )1 0 .

12 0! 2&1 .#2 2 2 " 2. 2&1 " 2 # 1#11& .2+ .2 " >ù . \ = − P#

N#

PJ + ]

\ = N0 N 0 + P J

k

M #0 m

70. ù ) 2 !#3 0 # 0 # 0 / # & ." 1 " 3 = 120!0+ 02. / . k 1 # /. * 0 .2 ! # 1+ . ." s P = NJ . 2 1*12 . 1 !! 0 & m2 12 0 0 0/ ù ) 2 1 2 # 0#0 # 0 / # 0 2. ! " 2. & m1 3 1+ . ." P = NJ . .!$ " 2.$*2 2." # = P V # $0 2 / 0* # 1 2 # . 2 # 0 .2 ! # ) &" 3. 02. 12 1$ . þ .!$ . )12.1 2& 1& 2& 0 . V P . 12. 0! 2 # 0 .2 ! # N = 1 P . 1+ .2. 1# ! * 2. 02& . / ! 0 . 2 " ! *1 " 0 . . 0 2 . ù )1 0 . 2 2 " 2 " 2. 2&1 " 2 # 1+ .2 " P )2. ! *1 0 . 0 .12 ú õ2. ! *1 0 . .12 . !0 20 . 2 2 " 2 " 2. 2&1 " 2 # 1#11& .2+ .2 " 2 12 .!. )!3&1 2 # 0 .2 ! # .2 2 / ! 0 . 2 " 2. 2&1 " 2 # 1#11& .2+ .2 " J = P V

>ù ù P ú .

P

P 0 # 1 @

71. + . ." P = NJ 0 . 1# /0m1 m2 k /0 12 . ! ! ) 2 # /. * 0 .2 ! # 12. 0! " N 1 P 0 -x0 x0 . 0 20 0 . .! 2. 2&1 2 #" [ = P .2 " 0 # ! ) 2 # 0 / # õ2. 2 1+ . / !$02. . ) 2 1 1 !! ." 2 # 1# ! *02. 02& 0 . 2 1+ . ."

P = NJ þ ! *1 0 . 02& . / ! 0 2 " 0 . . 0 2 . . !0 20 2 2 " 2 " 2. 2&1 " # 0 20 0 2. . ! *1 0 . . 0 .12 .12 >ù .

P P@

189

*72. 1+ . 2 # 1$ .2 " ." P = NJ !0 2 . 0 20 10 . .! 2. 2&1 ! ) 2 / 0/ 0 . 0 . 2. 0 .2 ! . /. 0 h 12. 0! " N1 1 P . N2 1 P ü 2! # 0 k1 k2 2 1+ . . ) 2 1 1 !! ." 2 # 12 1 1 x0 P . 2 $! 12 W 2 .3 #y 0 0 0* 0! .#2)$! . . ) *% " K & . ) 2 1 1 !! ." .3 02. . 10 0 0* 0!. 1+ . ." P2 = 0,44 kg. . . !0 20 2 *% " K +120 2 1+ . . 1# . 2 10 2 )2. / !$02. . !+2 3 ! . ) 2 1 1 !! ." 2 # ù 2. 1+ .2. . 1# ! * 2. 02& . .12 12 1 . !0 20 2 2 " 2 " 2. 2&1 " 2 # 1#11& .2+ .2 " þ / ! 0 . 2 " ! *1 " 0 . !. * ! . . 2 12.1 2 # . !. . 0 2 .

J = P V =

>ù .

P P@

73. û 1 " ." 0 = NJ 0 . 120!0& " 12 & ! .2. )!#3 # /. * 0 .2 ! # 12. 0! " N = 1 P 2 # # 2 ! 0 . 120!0& 10 ! ) 2 / 0/ & 12 / 1 02. . # ." P = NJ . . 12 0 2 1102. .2. )!#3. ! " 2. & 0 2.$*2 2. # = P V . !0 20 k . 2 2! 2 " 2.$*2 2." # . 2 / 1 " 2 2 " 2 " 2. 2&1 " 2 # / 1 # 2 12 /# . 0 ! 0 . 2 " 2. 2&1 " / 2 12 /# . 0 ! 0 . 2 # 0 .2 ! # J = P V

>ù .

P V P - / -@ *74. . ! .2. )!#3 # /. * 0 .2 ! # 12. 0! " N = 1 P 0 . 120!0& 10 ! 3 2 ! 2 # 0 . 1# /0/0 1+ . ." P = NJ 2 0 20 0 . .! 2. 2&1 2 #" \ = P . 12 # 0&! * 0 &" .!$ 2 # $!) # W = 0 + 2 1+ . ! 1 02. 12 1) 2 # 2 #" .20!$) 0 ! " 2 1 1 !! ." 2 # 1# ! *02. 02& 0 1+ . 1 " ." # $0 . 2 02 2.$*2 2. . !0 20 2 " 0 1+10 " # 0! ! 3 # 2 1 0 1+ .2 " . ! *1 0 . . 0 .12 .12 þ / ! 0 . 2 " ! *1 " 0 . . 0 2 . ." \′\ 0 . .2. )!#3 " 0 3 ! ! " 2. 2& þ . !# 1 . ) 2 1 1 !! ." . 2 2. 2&1 0 . 2 1# !2 1 2 # $!) # J = P V .     >ù . \ =  W +  , \ = − + W + W (SI) \ =  W +  (SI)]    

ü ü ùÿûü ÿ þ ü ü ù

190

*75. + . ." m = 1 kg 1 !! 0 1# /0/0 12 . ! .2. )!#y 3 # /. * 0 .2 ! # 12. 0! " k = 100 N/m 2 # # 2 !

120!0+ 02. 10 ! 3 ú . 2 1 " ." 0 2 1 0 2. .2. )!#3. ! " 2. & . 1# ! *02. 0 2.$*2 2. 2! # # = P V 02& k . .12 0 2 1+ . 1 2 $! 12 t0 = 0 . . !0 20 2 2 " 2 " 2. 2&1 " 2 # 1#11& .2+ .2 " 02 )1 $!) . ) 2 12 2 " ! *1 " t0 = 0, 2.$*2 2. 2 # 1#11& .2+ .2 " . /0 120 . !+2 3 !

1 m . !0 20 . 2 $! ) / 12 . 2 # 0!&2 .2 "

i) 2 ! 2 " /* . " 2 # 0 .2 ! # ii) 2 + 1 2 # ! #" #0 iii) 2 + 1 2 " /* . " 2 # 0 .2 ! # m / . !0 20 2 !# ) 02. " 2 " ! " 2 # 1#11& .2+ .2 " i) . 1&" 02 2 ! *1 ii) )2. ! 1 02. 12 " . !. 0" 10 " 2 " 1 " 2 # . 0&! 1020 )2 . !# 1 \ . ) 2 1 1 !! ." . 2 2. 2&1 0 . 2 1# !2 1 2 # $!) # J = P V

>ù .

P

−   V iii)  −  V V L - LL

  / L ± LL ± @

76. ù ) 2 !#3 0 # 0 # 0 / # & ." 10&" k 3 o 0 .!2 2. /. ) 0 .2 ! 12. 0! " N = 1 P x m1 # m2 . 12 2& 0 0* 0! ! 2 # 1# / 02. 1+ . ."

P = NJ 1*12 . 1 !! 0 & 10 0 0 3 xï 0/ ò . . ." P = NJ 0 2. ! ) 2 . 0 2.$*2 2. 2! # # = P V . 1# ! *02. . .! . . 02& . .12 0 2 1+ . ." P 1#11& 2& . /0 . . / . . !0 20 2 2 " 2 " 2. 2&1 " 2 # 1#11& .2+ .2 " ,0&! * 0 .!$ 2! 1 " 2 # $!) # W = 2 12 2 " ! *1 " . . [ ′[ 0 2 .20* # 1 # 3. 02. 12 1$ . . ! %020 2 0 1&1 2 " . !# 1 " 2 # 1#11& .2+ .2 " . ) 2 1 1 !! ." 2 # . 0&! 1020 )2 . !# 1 [ .

2 2. 2&1 0 . 2 1# !2 1 2 # $!) # 02 )1 $!) . ) 2 12 W = 2.$*2 2. 2 # 1#11& .2+ .2 " /0 02. . !+2 3 ! / . !0 20 2 !# ) 02. " 2 " ! " 2 # 1#11& .2+ .2 " .2 2 / 0* # 1 2 # 0 # 0 / # i) . 1&" 02 2 ! *1 ii) )2. ! 1 02. 12 " . !. 0" 10 " 2 " 1 " 2 # J = P V



 >ù .

P [ =  W +  (SI) V

  / L ± LL ± @

191

77.

* " ." P = NJ 1 !! 0 2 02 " & 10 0 ! ) 2 0 0/ 2 . .2. )!#3 /!. 2 # * # 0 . /0 . ! /. * ! ) 2 # 0 .2 ! # 12. 0! " N = 1 P 2 # # ! 0 . /0 10 12. 0!) 1 0 .2. )!#3 # 2 $ # 0 .2 ! ! 1 02. 12 3#1 ) 2 # " 2 . . 2 .2. )!#3 /!. 2 # * # 0 . /0 0 .12 ) . . .! " . 2 $0 )! !.*10&" ) = 1& 2 # .2 " .1 * 0 12 1+ . /* . .2 2 / 0* # 1 2 # . 2 # 0 .2 ! # . 0 3 ! 2 2 . +120 2 0 .2 ! . 0 * 02. 2! 2 " /* . " 02. 02. 0 2 0 # 1 [ 2 # 0 .2 ! # 1* 3& . 0 2 0 1&1 ) = + [ (SI). . !0 20 . 2 /# . 0 ! 0 . 2 # 0 .2 ! # 2 12 # ) 02. 2 .

2 2 " 2 " 2. 2&1 " # . 0 20 10 2 1*12 . 0 .2 ! * " 02 )1 $!) . ) 2 12 W = # ) 02. 2 . . 0! 10 * " . ) 2 1 1 !! ." 2 # . !+2 3 ! . 0&! 1020 . [ ′[ 0 .!$ 2 1 1 !! ." 2 # * # . 02 3 ! 0 0 .2 2 . 2 0 .2 ! 0 * 02.

>ù .

- P V ]

78. ù ) 0 !0 " $0 0! / 7 )2. 2. . 2+ 02. 12 0 3 0 . 2 " + " . . . 0 . 0! / " 2. 2&1 " 2 # 0 !0 *" . 2 02.3 ! # 0 10 *% " K . ) J 2 0 3 0 . 2 " + " ) # J = ( J 0 . 2 2! 2 " 0 2 $# 1 " 2 " .!*2 2."

12 0 3 0 . 2 " + "

. 02. 0 2 " ! 0 . 0 3 ! # 0 12 " 2 # .2 " 2 # 0 !0 *" +120 . 0 20 0 2. . 2+10 " 0! )/ # 7 )2. 02.30! 0 10 *% " K >ù .

7 ± @ *79. 2 . ! 0 .12 * .2 " #" 2 0&! * 0 $&! " . ! 1/ 02. 13. ! / * # ) # 9 . # )2 2." ! ! 2 # .2 " ! 1/ 02.

12 # . / $0 # 2 0! $0 0!) # )2 2." ! # ! # > ! . . 2 13. ! / ! 1 2. # 1 . 1. 12 0!) . 2 1*12 . . 13. ! / 1 !! 0 ù . !* # 0 2 13. ! / . ) 2 1 1 !! ." 2 # +120 2 20 2& . . 1$ .2 10 0 2 .!$ / 0* # 1 ! & . 3 . 2 .3 # 0 0 0* 0! . . /0 020 )2 2 13. ! / . 0 20 10 . .! 2. 2&1 . !0 20 2 # )2 2. 2 # * # . / 2. = P 7 = V J = P V ! # = NJ P ≈ ! " /0 . 2. # )%

NJ 9 ! # → ! = ] >ù . ' = ! # − !

J ⋅ ! =

P

+ J7

192

ü ü ùÿûü ÿ þ ü ü ù

80. ù ) 0 !0 " . 20 0 2. . ) &2 ) . 0 .12 ) . 0 2 ." ." #" = P 2 . ! 2 # .2 " / 02. 13. ! / ." P = NJ 2 0 .

3 !2 1 0 3 !2 T = & ! 2 # .2 " 0 . . ) 2. 1# /0/0 0 ! 3 0 !0 " ! 1 02. 10 0! $ # 0 !.20 .2. )!#3 0 " 0 2! ) 0/ 2.1 " ( 2 " ." 2 2! $0 2 2 . 2 +120 . 1$*0 (T < PJ þ

3 ! 2 # ( !0 . 0 )!! . 2 !! 2 # J .!#2 ) 0/ 2 0&! * 0 0 1 " 0 " ù . !* # 0 2 13. ! / . ) 2 1 1 !! ." 2 # +120 2 . . 1$ .2 10 0 2 .!$ 2 # / 0* # 1 ! & . 3 . 2 .3 # 0 0 0* 0! . . /0 020 )2 2 13. ! / . 0 20 10 . .! 2. 2&1 . . 2 " /* .

20# * 10 " 2 # ( . û . 12+ # 0 0 !. .2 )2 2 0 !0 " 0 20 0 2. 2&1 0 0! / 7 = V

. 2 . .20* # 1 2 # .2. )!#3 # 0/ # ( . 0 0! / 7 = V )2. .

20* # 1 2 # ( . 2 12!.30 . !0 20

i) 2 2! 2 " 0 2 $# 1 " 2 " .!*2 2." J 12 2) # 02. 2 0 !. .

ii) 2 2! 2 " 2.1 " ( 2 # 0 2! * 0/ # P 1 PJ + (T PJ − (T >ù . ' =

L J = LL ( = ⋅ '′ =

& V 81. ù ) 0 !0 " 2 # # 2 1 /0! 13. ! / $0 . P = NJ 0 20 0 . ! 2. 2&1 10 V . )1 0 . 2 " 2 # .2 " 2 # 0 !0 *" ù 12 13. ! / 0 2)" . ) 2 2 " .!*2 2." .1 0 2. . . .2. )!#3

12. 0! /* . ) . ) . 0/ /# 0& 2)20 2 0 !0 " 0 20 0 2!0 " !0 " 2. . 2+10 " 10 V . !0 20 2 12. 0! /* . ) . û 2. J = P V . ≈

>ù .

P ) )!! 2 # ! #"@ 82. û* . 0 !0 2 . . #" = P ! 2. . ) 2 / 1 0 . 13. ! / . 2& 0 !0 + $ # 0" . . 2. .2 2 #" 1$ .2 # 0 2 .2. )!#3 ! " & 0" 3 = R . = R . 2 12 $. /* 0" .3 2. 2.#2)$! . 0 0* 0!0" . ! *1 2 #" 0 . 0 .12 . 0 1 0 . 0 !+2 ! *1 . .2 . !0 0 ) " 2& . + +120 . 0 2! 2. 2. 0 !0 1# 0$+" $! 2 " .!$ " & 0" 3 . . !0 20 2 0! / 2& 1# ! *10& 12 .!. & 0! 2&1 ù 2 12 10 " /0 . 2. # )% û 2. J = P V 1# 3 . 1#

>ù .

2 1 1 !! ." = = V ]

193

*83. 02. ) 13. ! / . * 0 !0 *" $0 . P NJ . 0 . 3 !2 1 0 3 !2 T ± & . 12 0 . /0 2 13. ! / 0 . &2 ) 0 .12 ) . 0 2 ." ." . $0 " = P 0 !0 " ! 1 02. 10 0! $ ) # 0 2)" . ) 2 0/ .!*2 2." 2 0&! * 0 0 " 0 !.20 . . 0 "

1 0 2! 12.2 ) 0/ ( )!! 2 # . [ ′[ 2 $0 2! ( =

& . . 1$0/ 1020 2 " /# 0 " # .1 * 2. 12 13. ! / )2. .#2) 1 !! 0 . /0 020 )2 12 .2 12.1 1 !! ." 2 # 13. ! / # 2 . 1$ .2 0 0 2 .2. )!#3 & . 3 . 2 . 1$*0 033 = ü 2! # 0 2 13. ! / . ) 2 1 1 !! ." 2 # +120 2 . . 1$ .2 10 0 2 / 0* # 1 # 0 $0 )2. 2 13. ! / 1 !! *10 ! & . . 2 .3 # 0 0 0* 0! . /0 020 )2 2 13. ! / . 0 20 10 . .! 2. 2&1 . . !0 20 2 0! /) 2 ".

J = P V

>ù 7 =

P

( T +P J

→7=

V]

84. þ 0! / " ." 3 #1." .! " 2. 2&1 " 0 . . 2 2 " 2 " . # 0 2 0 02 ) ) . = . H − W ) # 12. 0! 1)2 2. . . /0 020 )2 ) " /* / ./ $ + 2 + 2 # 2 #" 2 " 2. 2&1 " 0 . 12. 0!)"

02 . ) 1 = !0 " 2. . 2+10 " # / .! * W = V 2 2 " 2 " 2..

2&1 " 0 . 1 0 . !0 20 2 2 " 2 " 2. 2&1 " )2. # . ) .

= !0 " 2. . 2+10 " . . >ù .

Q = H 7 = 12. ] . Q +

85. 2 " ." 3 #1." .! " 2. 2&1 " . # 0 2 0 02 ) ) . = . H − W ) # . 2 .!$ ) 2 " . 12. 0! 1)2 2. . . 0 )1 $!) 2 2 " 2 " 2. 2&1 " . 0 . =

ù . 0 ! 2. 2&1 0 2 " 0 22&1 2 " " 0 ! 0 ." ü 2 " 2. 2&1 " 0 . . !0 20 2 0 2 " 02. 2 # 2 #" 2 " 2. 2&1 " Q >ù .

± 0 22&1 @

86. 0 0 0* 0! .! ) 2. . 2&2 0 0! 0 /* . . 2 12.1 " ) ± E# ) # E 12. 0! . )1 01 " . . /0 020 )2 . 12. 0! E $0 /0" kg/s.

G( $! )" !# )" 02. " 2 " $. " 0 ! 0 ." 2 # 2. . 2&2 0 . = − E# .

GW

194

ü ü ùÿûü ÿ þ ü ü ù

87. 2 " 2 " 3 #1." 2. 2&1 " .! * 2. . 2&2 0 + 02. 0 02 0 2 $!) 1* 3& . 0 2 0 1&1 . = . H − W $! 12 W = 0 ! 0 .

2 # 2. . 2&2 0 . (

(

)1 0 . 0 ! 0 . 2 # 2. . 2&2 2 $! 12 W = W

. 02 )1 $!) W 0 ! 0 . 2 # 2. . 2&2 . 0 ( =

>ù .

W =

Q

ü ]

88. $!) " 10 ." & " 0 )" !./ 0 0! * # * 0 .

. 02 )1 $!) . $ # / .1 .120 2. 2& .!$ + #! &

1 12) 0 2 " 2& .!$ + #! & . $0 / .1 .120 10 $!) W 7 >ù .

@ *89. 2 / 2. 2 # 1$ .2 " . &

1 ù + ù+ +þ þý /0 $ # & . 2

û 12.1 þ $ ! / " $0 ú ú " . P & . 2 12.1 5 . 12 1 2 " 1# / 02. 0 2 . ! ! ) 2 # /. * 0 .2 ! #

12. 0! " N 2 # # 2 ! ú ú x 0 . 1# /0/0 10 12. 0!) 1 0 x þ M M1 ý .2. )!#3 # 2 $ # þ ! / " !0 0 . 2 0 .2 ! $0 2 3#1 ) 2 # " ! 02. * ! / # . . & + /0 # !$0 þ ) / 2. ! 1 02. 10 0! $ ) # 0 !.20 0 " . 2 ) 0/ 2.1 " % / . ) 2 " û 0 . .!$ . 2)" ù . !* # 0 2 ! / ! " 2. /0 .2 . ) &" 3. 02. 12 1$ . . 2 .3 # 0 0 0* 0! 2 $! 12 W0 = 0. . . . /0 020 )2 ! / " . 0 20 10 . .! 2. 2&1 . .!.12 1020 !.3 10 1# !2 1 0 2 $!) 2 . !# 1 2 " ! / # . ) 2 1 1 !! ." 2 " . 0&! 1020 2 2 1# !2 1 2 " . !# 1 " x 0 2 $!) $! 12 W = .3 # 0 0 0* 0! 2 ! / . ) 2 1 [ .0 . 2.#2)$! . 0 # 0 2 / . ) 2 û i) 1 . 0 20 10 2+!. ! / " . / . 1020 2 . 2 1 1." ii) . .!.12 1020 !.3 10 1# !2 1 0 2 $!) 2 . !# 1 2 " ! / # . ) 2 1 1 !! ." 2 " 2

 N  % >ù . ' N [ = .  ] W +  L 3 #1. .! 2. 2&1 E =  5  P

195

*90. 2 . ! /. * ! ) 2 # 0 .2 ! # 12. 0! " N 1 P 0 . 1# /0/0 1+ . ." P NJ 2 !0 . 0 2. & 10 ! ) 2 0 0/ ! 2 # 0 .2 ! # 120!0+ 02. 10 . ) 2 1 0 ù . !* # 0 2 1+ . . ) 2 1 1 !! ." 2 # .2 2 / 0* # 1 2 # . 2 # 0 .2 ! # .2 x0 = 0,2 m . 2 .3 # 0 0 0* 0! ) & 2! + 2 2 " 2 " 2. 2&1 " 0 .22+ 02. .2 02 . ) 0 ! 2. 2&1 . . 0 . / 1#$ )2 2. 2 # 2. . 2&2 )1 0 ! 0 . .3. !0 2. . ) 2 2. . 2&2 1& 2 # ! # 2& 2! + 12 / ! 0 .

2 " !+2 " 0! )/ # )1 0 ! 0 . ! 0 . 02.30! 0 12 2. . 2&2 1& 2 # ! # 0 &20! " .! " /* . " 10 $!) W V +120 . 0 20 0 . 0 &20" 2. . 2+10 " 0 1#$ )2 2. 0;

>ù . +] - -@

91. ò . 1 0 . ) . 2 0 0 ." m !0 . 0 2. ) & 2 " .!#2 " " 2 " + " 1. 10 ! ) 2 1& . $&! " 2! ) &" 3. 02. 12 1$ . ù . !* # 0 2 . 2 0 0 . ) 2 1 1 !! ."

2 # .2 x . 2 .3 # 0 0 0* 0!

2 # 0 )2 2 x 0 . * !) 10 1* ! 1 0 2 . 2 . R 2 " + " . . . /0 020 )2 2 . 2 0 0 . 0 20 10 . .! 2. 2&1 . # 1020 2 0! / 2 " 2. 2&1 " þ + 0&!0 2. . 2 0 " 13. !.. û 2. ù 2 . + " 5 NP . g0 = 10 m/s2.

PJ 5 >ù . ' =

7 = → 7 = V ] J 5

y x m

0

R

x ú 3

K

y 92. ü . &12) )2 2 2! 2 " 0 2 $# 1 " 2 " .!*2 2." . ) 2 2! 2 " + " $! 2 0 3 0 2 " 02. x 02. 0 2 . )12.1 r . ) 2 2! 2 " + " 1* 3& .

U A 0 2 ) J = J ) # J 2 2! 2 " 0 2 $# 1 " 0 5 x 2 " .!*2 2." 12 0 3 0 . 2 " + " . 5 . 2 . 2 " 3 B + " 2 # 0 )2 $ # 0 . 0 . 1 !. . 12 + ) &" 3. 02. 12 1$ . 0 . ) . 2 0 0 ù ."

R m, !0 . 0 2. $&! " 2! .2 " 2 " 1 !. K ." ù . !* # 0 2 . 2 0 0 .2 x . ) 2 1 1 !! ." 2 # . 2 .3 # 0 0 0* 0! . . . /0 020 )2 2 . 2 0 0 . 0 20 10 . .! 2. 2&1 . . # 1020 2 0! /) 2 " )1 . 2. 0! / " 2 " 2. 2&1 ", . 1 !. . 0! *10 . ) 2 2! 2 " + " þ + 0&!0 2. . 2 0 " 13. !. . 2 ." R = 6400 km . g0 = 10 m/s2.

PJ 5 >ù . ' = 7 = → 7 = V / .@ J 5

196

ü ü ùÿûü ÿ þ ü ü ù

*93. þ 3 #1. .! 2. 2&1 1#12 .2 " ." 0 .2 ! # 0! ! 302. . ) 2 N − .

0 1&1 [ .1# &W ) # . = . H − W . & = P 2 1*12 . 0 0! 0 /* . . )1 01 " . 2 " 2.$*2 2." / ./ ) ± E# ù . .!.12 1020 !.3 2 1# !2 1 [ I W 2 ) / !. . .

N ú ù . )1 01 0 . * ! +120 << . /0 020 )2 : P i) + 0 ! 0 . 2 # 2. . 2 * 0 # 1#12 .2 " !0 . !.30 0 2 !3

ü = N. H − W

ii) 2 ) " /* / ./ $ + 2 + 2 " " 0 ! 0 ." # / .3 ! # $! .2 ( . 0! / T 0 . 1 " ! " Q = H 7 ( Q +