Mecânica Física-Classica

  • Upload
    amsm71

  • View
    217

  • Download
    0

Embed Size (px)

Citation preview

  • 8/8/2019 Mecnica Fsica-Classica

    1/208

    ! # %

    '

    !

    ) 0 24 5 7 09 4 @ AB C

    7 0 4

    E F G I Q S I V F Y S ` a F c d

    f g i q

    I

    r s s v

  • 8/8/2019 Mecnica Fsica-Classica

    2/208

    5 ) A

    ! # %

    0 1 0 4 5 6 8 @ B D 6 G @ 5 Q S@ U W 5 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    0 1 b d @ 5 6 G @ f @ h 8 i Q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b

    0 1 q s i Q 6 @ u v x@ Q W 5 D 6 G D 6 @ 5 6 G @ f @ h 8 i Q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q

    0 1 q 1 0 @ i U @ B D G D W i Q 6 @ U D W D i G D u v D Q 8 5 G D G @ G @ B i 5 B @ Q 8 i G @

    v B D D U 8 S

    W v D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q

    0 1 q 1 b @ i U @ B D G D W i Q 6 @ U D W D i G i B i B @ Q 8 i D Q v D U G @ v B D D U j

    8 S

    W v D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k

    0 1 q 1 q @ i U @ B D G D W i Q 6 @ U D W D i G D @ Q @ U 5 D G @ v B D D U 8 S

    W v D 1 1 1 k

    0 1 k n i 6 W 5 D G i U D U B Si Q 5 W i v Q 5 G 5 B @ Q 6 5 i Q D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    0 1 i 5 B @ Q 8 i v Q 5 G 5 B @ Q 6 5 i Q D G @ v B 6 5 6 8 @ B D W i Q 6 @ U D 8 5 i @ Q @ U 5 D

    @ U 6 v 6 i 6 5 W D i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 {

    0 1 i 5 B @ Q 8 i G @ v B @ @ W 8 U D i 6 i i @ @ 5 8 i G @ v B W D B i @ S@ W 8 U 5 W i @

    B D Q S@ 8 5 W i W i Q 6 8 D Q 8 @ 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    % %

    #

    b 1 0 s D B i G @ i U W D 6 W @ Q 8 U D 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 q

    b 1 b @ B i 6 G @ W D B i 6 G @ i U W D 6 W @ Q 8 U D 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    b 1 q Si U 5 8 D G @ v B D D U 8 S

    W v D 6 i U @ D u v D D W 8 v D v B D i U W D W @ Q 8 U D 1 0

    b 1 k 6 @ u v D W i @ 6 G @ B i 5 B @ Q 8 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    b 1 @ U @ 6 @ Q 8 D W D i U SD W D G @ U i U 5 @ G D G @ 6 G D 6 Si U 5 8 D 6 1 1 1 1 1 1 b q

    b 1 1 0 n i 8 @ Q W 5 D - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b k

    b 1 1 b n i 8 @ Q W 5 D -

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b

    b 1 1 q n i 8 @ Q W 5 D -

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b {

    b 1 D U U @ 5 U D G @ i 8 @ Q W 5 D W @ Q 8 U S

    v i

    b

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 b

    b 1

    S

    n U 5 8 D 6 5 B 5 8 D G D 6 Si U 5 8 D 6 @ W D G D 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q

    b 1 { @ 8 @ U B 5 Q D W D i G @ W i Q @ W 5 G D D Si U 5 8 D 1 1 1 1 1 1 1 1 1 1 1 q 0

    b 1

    S

    n U 5 8 D 6 W 5 U W v D U @ 6 @ 6 8 SD @ 5 6 @ 5 Q 6 8 SD @ 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q 0

    b 1 0 n U i @ B D G @ @ @ U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q b

    b 1 0 1 0 6 Si U 5 8 D 6 Q i U i @ B D G @ @ @ U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q q

    b 1 0 1 b @ U W @ 5 U D d @ 5 G @ @ @ U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q

    b 1 0 0 4 5 6 8 @ B D 6 5 Q SD U 5 i 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q

    5 5

  • 8/8/2019 Mecnica Fsica-Classica

    3/208

    b 1 0 b @ i U @ B D G i

    5 U 5 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q

    b 1 0 q s i 5 6 i @ 6 j 5 v 6 D i G @ D U 8 S

    W v D 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k 0

    b 1 0 q 1 0 6 @ W W D i @ W D G 5 @ U @ Q W 5 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k 0

    b 1 0 q 1 b 4 @ W W D i @ W D G @ v 8 @ U i U G 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k

    #

    #

    q 1 0 s i Q 6 @ U D W D i G D u v D Q 8 5 G D G @ G @ B i 5 B @ Q 8 i 1 1 1 1 1 1 1 1 1 1 1 1 1

    q 1 b s i Q 6 @ U D W D i G i B i B @ Q 8 i D Q v D U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b

    q 1 q s i Q 6 @ U D W D i G D @ Q @ U 5 D B @ W xD Q 5 W D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    %

    %

    % ! "

    k 1 0 d 5 D W i @ 6 s i i U G @ Q D G D 6 $ @ Q @ U D 5 D G D 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 {

    k 1 b ' U 5 Q W S

    5 i G ( @ B @ U 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b

    k 1 q u v D W i @ 6 G @ d D U D Q @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k

    k 1 k ' U 5 Q W S

    5 i G @ 0 D B 5 8 i Q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    k 1 s SD W v i G @ D U 5 D W i @ 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    k 1 4 5 6 8 @ B D W i B SD U 5 i 6 U D v 6 G @ 5 @ U G D G @ @ u v D W i @ 6 G @ v @ U 1 1 1 1 q

    k 1 u v D W i @ 6 G @ d D U D Q @ G i ' U 5 Q W S

    5 i G @ 0 D B 5 8 i Q 1 1 1 1 1 1 1 1 1 1 k

    k 1 1 0 i 5 B @ Q 8 i G @ v B D D U 8 S

    W v D @ B W i i U G @ Q D G D 6 W D U 8 @ 6 5 D Q D 6 1

    k 1 1 b ' D U 8 S

    W v D Q v B W D B i G @ i U W D 6 W @ Q 8 U D 5 6 1 1 1 1 1 1 1 1 1 1 1

    k 1 1c q SD u v 5 Q D G @ 8 h i i G U i G D Q D 6 @ B D 8 U 5 8 i @ 6 @ B B D 6 6 D 1 1 1

    k 1 { 4 i B D D i D U D Q 5 D Q i G @ v B D G @ U 5 D G D 8 i 8 D @ B i U G @ B D i 8 @ B l i 1 {

    k 1 v 8 5 5 W D G i U @ 6 G @ d D U D Q @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    k 1 0

    D U 5 D W D i 6 v 3 @ 5 8 D D 5 D W i @ 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 { b

    k 1 0 1 0 @ B i j U W i u v @ U i G D 6 @ B @ 6 W i U U @ D U Q v B D Q i 5 Q W 5 Q D G i { q

    k 1 0 0 s i i U G @ Q D G D 6 W S

    W 5 W D 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 { k

    k 1 0 0 1 0 @ B i 0 D U 8 S

    W v D Q v B W D B i G @ i U W D 6 W @ Q 8 U D 5 6 1 1 1 1 { k

    k 1 0 0 1 b @ B i b B i 5 B @ Q 8 i v Q 5 G 5 B @ Q 6 5 i Q D G @ G v D 6 D U 8 S

    W v D 6

    5 D G D 6 i U v B D B i D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 {

    k 1 0 b d @ 5 6 G @ W i Q 6 @ U D W D i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 {

    k 1 0 b 1 0 5 Q D U 5 xD Q W 5 D G @ @ U D Q 8 @ D 8 U D Q 6 i U B D W D i 7 9 A 7 9 C E 7 9 1 1 {

    k 1 0 b 1 b 5 Q D U 5 xD Q W 5 D G @ @ U D Q 8 @ D 8 U D Q 6 D W D i G i 6 5 6 8 @ B D W i B i v B

    8 i G i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 {

    k 1 0 b 1 q 5 Q D U 5 xD Q W 5 D G @ @ U D Q 8 @ D U i 8 D W D i G i 6 5 6 8 @ B D W i B i v B 8 i G i

    k 1 0 b 1 k s i Q 6 @ u v x@ Q W 5 D 6 G D v Q 5 i U B 5 G D G @ G i 8 @ B i 1 1 1 1 1 1 1 1 1 0

    k 1 0 q d D U D Q 5 D Q i G @ v B 6 5 6 8 @ B D Q D i @ W D G i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k

    k 1 0 k ' i 8 @ Q W 5 D G @ @ Q G @ Q 8 @ G D @ i W 5 G D G @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    k 1 0 n x@ Q G v i @ 6 S@ U 5 W i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

    ! R

    S%

    U

    W W

    1 0 4 5 6 8 @ B D W i B v B U D v G @ 5 @ U G D G @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    1 0 1 0 @ B i n x@ Q G v i 6 5 B @ 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 b

    1 b 4 5 6 8 @ B D W i B SD U 5 i 6 U D v 6 G @ 5 @ U G D G @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 q

    5 5 5

  • 8/8/2019 Mecnica Fsica-Classica

    4/208

    1 b 1 0 n x@ Q G v i G v i Q i D Q i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    1 b 1 b

    5 U D W i @ 6 G @ v B D B i S@ W v D 8 U 5 D 8 Si B 5 W D 5 Q @ D U @ 6 5 B S@ 8 U 5 W D 1 0 0

    #

    % %

    %

    %

    1 0 6 8 v G i G i B i 5 B @ Q 8 i G i W i U i U S

    5 G i @ 6 W i D G @ v B W i Q 3 v Q 8 i G @

    W i i U G @ Q D G D 6 5 Q G @ @ Q G @ Q 8 @ 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 k

    1 b U D Q 6 i U B D W i @ 6 i U 8 i i Q D 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0

    1 b 1 0 @ B i U i 8 D W D i Q v B D Q i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 b 0

    1 b 1 b 5 Q 8 @ U U @ 8 D W D i G D 8 U D Q 6 i U B D W D i U @ U @ 6 @ Q 8 D G D @ D B D 8 U 5 0 b b

    1 q

    x

    Q v i 6 G @ v @ U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 b q

    1 k @ i U @ B D G @ v @ U Q i B i 5 B @ Q 8 i G i W i U i U S

    5 G i 1 1 1 1 1 1 1 1 1 1 0 b

    1 i 8 D W i @ 6 Q 5 8 D 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 b

    1 i 8 D W i @ 6 5 Q Q 5 8 @ 6 5 B D 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 b

    1

    D U 5 D W D i G @ v B @ W 8 i U W i B i 8 @ B i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 q b

    1 { i U W D G @ s i U 5 i 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 q k

    % %

    %

    %

    1 0 i B @ Q 8 i D Q v D U @ @ Q @ U 5 D W 5 Q S@ 8 5 W D G i B i 5 B @ Q 8 i G v B W i U i W i B

    v B i Q 8 i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 k 0

    1 b f i W D i G @ 8 @ Q 6 i U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 k

    1 q i B @ Q 8 i G @ 5 Q S@ U W 5 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 k

    1 k i B @ Q 8 i 6 U 5 Q W 5 D 5 6 G @ 5 Q S@ U W 5 D j @ 5 i 6 U 5 Q W 5 D 5 6 G @ 5 Q S@ U W 5 D 1 1 1 1 0 k

    1 u v D W i @ 6 G @ B i 5 B @ Q 8 i G @ v B W i U i U S

    5 G i W i B v B i Q 8 i i 1 0 0

    1 1 0 i 5 B @ Q 8 i G @ v B W i U i U S

    5 G i 5 U @ W i B v B @ 5 i G @ 6 5 B @ 8 U 5 D 0 b

    1 i 5 B @ Q 8 i G @ v B W i U i U S

    5 G i j i U B D 5 6 B i d D U D Q 5 D Q i 1 1 1 1 1 0 k

    1 n $ 5 U i 6 W Si 5 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    1 1 0 s i Q 6 @ U D W D i G i B i B @ Q 8 i D Q v D U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    1 1 b s i Q 6 @ U D W D i G @ @ Q @ U 5 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    "

    # %

    %

    S%

    # %

    { 1 0 U D Q 6 i U B D W D i G @ d @ @ Q G U @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 q

    { 1 b u v D W i @ 6 G @ 0 D B 5 8 i Q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 k

    { 1 q @ B i 0 B i 5 B @ Q 8 i G @ v B D D U 8 S

    W v D Q v B W D B i G @ i U W D 6

    W @ Q 8 U D 5 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    { 1 k @ B i b D U 8 S

    W v D Q D i U @ D 8 5 5 6 8 D 6 v 3 @ 5 8 D D i W D B i @ @ W 8 U i j

    B D Q S@ 8 5 W i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    { 1 s i i U G @ Q D G D 6 W S

    W 5 W D 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    { 1 @ @ Q G x@ Q W 5 D Q i 8 @ B i G @

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    { 1 u v D W i @ 6 G @ 0 D B 5 8 i Q G @ v B U 5 Q W S

    5 i D U 5 D W 5 i Q D 1 1 1 1 1 1 1 1 1 0 {

    %

    S%

    %

    W

    5

  • 8/8/2019 Mecnica Fsica-Classica

    5/208

    W

    #

    %

    %

    0 1 0 ' U i U 5 @ G D G @ 6 G i 6 D U x@ Q 8 @ 6 5 6 G @ ' i 5 6 6 i Q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 k

    0 1 b u v D W i @ 6 G @ B i 5 B @ Q 8 i @ B 8 @ U B i 6 G i 6 D U x@ Q 8 @ 6 5 6 G @ ' i 5 6 6 i Q 1 1 1 0

    #

    U

    #

    #

    %

    S

    % %

    U

    #

    0 0 1 0 6 @ 5 6 G @ f @ h 8 i Q @ i U 5 Q W S

    5 i G D U @ D 8 5 5 G D G @ 1 1 1 1 1 1 1 1 1 1 1 0

    0 0 1 b 6 @ u v D W i @ 6 G @ D h @ @ i U 5 Q W S

    5 i G D U @ D 8 5 5 G D G @ 1 1 1 1 1 1 1 0

    0 0 1 q @ U 5 x@ Q W 5 D G @ 5 W @ 6 i Q j i U @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 k ' i 6 8 v D G i 6 G @ 5 Q 6 8 @ 5 Q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 f i W D i G @ i 6 @ U D G i U 5 Q @ U W 5 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 i U B v D W D i W i D U 5 D Q 8 @ Q v B @ 6 D W i 8 U 5 G 5 B @ Q 6 5 i Q D 1 1 1 1 1 1 1 1 1 0 { 0

    0 0 1 @ Q 6 i U @ 6 Q i @ 6 D W i G @ 5 Q i h 6 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 { b

    0 0 1 1 0 n 8 @ 8 U D j @ W 8 i U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 { b

    0 0 1 1 b @ 8 U D j @ W 8 i U W i Q 8 U D D U 5 D Q 8 @

    !

    $ 1 1 1 1 1 1 0 { q

    0 0 1 1 q @ 8 U D j @ W 8 i U W i D U 5 D Q 8 @ %

    %

    %

    !

    %

    %

    $

    1 1 1 1 1 1 1 1 1 0 { k

    0 0 1 1 k @ Q 6 i U G @ b

    D

    i U G @ B W i Q 8 U D D U 5 D Q 8 @ W i D U 5 D Q 8 @ @ B 5 6 8 i 1 1 0 {

    0 0 1 1 ' U i G v 8 i @ 6 W D D U @ Q 8 U @ G i 5 6 k j @ W 8 i U @ 6 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 1 n 8 @ Q 6 i U B S@ 8 U 5 W i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 { s i Q 6 @ U D W D i G i k j @ W 8 i U (

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 { 1 0 n k j @ W 8 i U @ i W 5 G D G @ 0

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 { 1 b n k j @ W 8 i U u v D Q 8 5 G D G @ G @ B i 5 B @ Q 8 i (

    1 1 1 1 1 1 1 1 1 1 0

    0 0 1 { 1 q d 5 B 5 8 @ W SD 6 6 5 W i G D @ Q @ U 5 D2

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 q

    0 0 1 { 1 k @ B i W i 5 6 D i 8 i 8 D B @ Q 8 @ 5 Q @ SD 6 8 5 W D 1 1 1 1 1 1 1 1 1 1 1 0 q

    0 0 1 { 1 d @ 5 G @ 8 U D Q 6 i U B D W D i G @ (

    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 k

    0 0 1 { 1 ' D U 8 S

    W v D W i2 B 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    0 0 1 { 1 4 5 6 8 @ B D W @ Q 8 U i G @ B D 6 6 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    0 0 1 { 1 { s i 5 6 D i G @ G v D 6 D U 8 S

    W v D 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

    0 0 1 { 1r n 5 B 5 D U D 6 i v 8 i G v B D U @ D W W D i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 {

    0 0 1 6 @ v Q G D @ 5 G @ f @ h 8 i Q @ B @ W xD Q 5 W D @ D 8 5 5 6 8 D 1 1 1 1 1 1 1 1 0

    0 0 1 0 v Q 6 8 @ 8 U D j @ W 8 i U @ 6 @ B S

    6 5 W D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 b b

  • 8/8/2019 Mecnica Fsica-Classica

    6/208

    4 AB

    ) 0 24 5 7 0 4 )

    5 7 4 5 4

    " $ ' ) 1 2 "

    4 v i B i 6 D @ 5 6 8 x@ Q W 5 D G @ 6 5 6 8 @ B D 6 G @ U @ @ U x@ Q W 5 D i 6 6 5 6 8 @ B D 6 G @ 5 Q S@ U W 5 D Q i 6 u v D 5 6

    D 6 @ 5 6 G @ f @ h 8 i Q 6 D i SD 5 G D 6 1

    5 6 8 @ 8 C D 6 F H D @ R TD V W @ F

    S@ v B 6 5 6 8 @ B D @ B U @ D W D i D i u v D i @ 6 D W i S@ i B i S@ Q @ i

    @ 5 6 i 8 U Si 5 W i @ i 8 @ B i v Q 5 i U B @ 1 s i Q W U @ 8 D B @ Q 8 @ v B W i U i 5 U @ @ @ B U @ i v 6 i

    Q v B 6 5 6 8 @ B D G @ 5 Q S@ U W 5 D B D Q 8 S@ B j 6 @ 5 U @ @ @ B U @ i v 6 i i U v B 8 @ B i 5 5 B 5 8 D G i 1

    4 @ D S@ B G i 6 5 6 8 @ B D G @ 5 Q S@ U W 5 D W i Q 6 5 G @ U D G i @ 6 W i @ U B i 6 v B i v 8 U i u v @ 6 @ B i D

    @ B U @ D W D i D i U 5 B @ 5 U i W i B B i 5 B @ Q 8 i U @ W 8 5 S

    Q @ i @ v Q 5 i U B @ D 6 @ 5 6 G i B i 5 B @ Q 8 i

    Q @ 6 8 @ Q i i 6 5 6 8 @ B D 6 @ U D i D 6 B @ 6 B D 6 G i 6 5 6 8 @ B D G @ U @ @ U x@ Q W 5 D 5 Q 5 W 5 DF 1 @ @ U 5 x@ Q W 5 D

    B i 6 8 U D u v @ Q @ 6 8 @ 6 6 5 6 8 @ B D 6 Q D i 6 Si D 6 @ 5 6 G i B i 5 B @ Q 8 i G i W i U i 5 U @ 6 D i D 6

    B @ 6 B D 6 B D 6 u v @ i B @ 6 B i 6 @ @ U 5 W D D U D 8 i G D 6 D 6 @ 5 6 G D B @ W xD Q 5 W D 1 i G i 6 @ 6 8 @ 6

    6 5 6 8 @ B D 6 6 D i @ u v 5 D @ Q 8 @ 6 G i i Q 8 i G @ 5 6 8 D G D B @ W xD Q 5 W D 1

    ` V @ R W T

    d e

    @ g H F V D q F C @ u @ H F H D H D v F q @ q D y

    Q i 6 6 5 6 8 @ B D 6 G @ 5 Q S@ U W 5 D D 6 U i U 5 @ G D G @ 6

    G i @ 6 D W i @ G i 8 @ B i @ B W i B i 8 i G D 6 D 6 @ 5 6 G D B @ W xD Q 5 W D 6 D i D 6 B @ 6 B D 6 1

    B D W i Q 6 @ u v x@ Q W 5 D 5 B @ G 5 D 8 D Q D i @ 5 6 8 @ v B 6 5 6 8 @ B D G @ U @ @ U x@ Q W 5 D D 6 i v 8 i

    s i Q 6 5 G @ U @ B i 6 G i 5 6 6 5 6 8 @ B D 6 G @ 5 Q S@ U W 5 D @ 1 G @ 6 i W D j 6 @ W i B v B D @ i W 5 j

    G D G @ U @ D 8 5 D B @ Q 8 @ D W i Q i U B @ D v U D 0 1 0 6 W i i U G @ Q D G D 6

    @

    G @ v B

    i Q 8 i U @ 6 @ W 8 5 D B @ Q 8 @ Q i 6 6 5 6 8 @ B D 6 @

    @ 6 8 D i U @ D W 5 i Q D G i 6 @ D 6 U @ D W i @ 6

    C

    A

    C

    0 1 0

    5 6 8 i u v @

    A

    6 @ W i Q 6 5 G @ U D U B i 6 u v @ Q i 5 Q 6 8 D Q 8 @ 5 Q 5 W 5 D D 6

    i U 5 @ Q 6

    @

    G @ D B i 6 i 6 6 5 6 8 @ B D 6 W i 5 Q W 5 G @ B 1 @ u v D W D i 0 1 0 @ U 5 B @ D @ 5

    G @ 8 U D Q 6 i U B D W D i G @ $ D 5 @ v 1

    0

  • 8/8/2019 Mecnica Fsica-Classica

    7/208

    O

    O

    P

    rr

    S

    S

    5 v U D 0 1 0 ! # % ( (

    0 2 $ 3 4 5 '

    ` V @ 6 D @ V F q D @

    v B D D U 8 S

    W v D 5 U @ 6 i U @ @ D Q D i D W 8 v D B u v D 5 6 u v @ U i U W D 6 B D Q j

    8 S@ B i 6 @ v @ 6 8 D G i G @ U @ i v 6 i i v G @ B i 5 B @ Q 8 i U @ W 8 5 S

    Q @ i v Q 5 i U B @ j @ 5 G D

    5 Q S@ U W 5 D 1

    4 @ 3 D

    D i U W D 8 i 8 D u v @ D W 8 v D Q D D U 8 S

    W v D 1 4 @

    @ Q 8 D i D @ i W 5 G D G @

    G D D U 8 S

    W v D

    U

    S@ W i Q 6 8 D Q 8 @ 1

    7

    D 8 y R H F q D @

    6 @ 6 i U @ v B D D U 8 S

    W v D D W 8 v D B i U W D 6 D D U 5 D W D i G D 6 v D u v D Q 8 5 G D G @

    G @ B i 5 B @ Q 8 i i U v Q 5 G D G @ G @ 8 @ B i S@ 5 v D @D U @ 6 v 8 D Q 8 @ G D 6 i U W D 6 u v @

    D W 8 v D B Q @ D 1

    4 @

    S@ D u v D Q 8 5 G D G @ G @ B i 5 B @ Q 8 i G D D U 8 S

    W v D @

    D U @ 6 v 8 D Q 8 @ G D 6 i U W D 6

    u v @ D W 8 v D B Q @ D @ Q 8 D i

    G

    G

    B

    0 1 b

    D DA V W D @ V F q D @

    u v D Q G i G v D 6 D U 8 S

    W v D 6 5 Q 8 @ U D @ B D i U W D u v @ D U 5 B @ 5 U D @ @ U W @ Q D

    6 @ v Q G D S@ 5 v D @ B U D Q G @ D @ G 5 U @ W W D i B D 6 G @ 6 @ Q 8 5 G i i i 6 8 i @D i U W D u v @

    D 6 @ v Q G D @ @ U W @ Q D U 5 B @ 5 U D j @ 5 G D 6 i U W D 6 G @ D W W D i @ U @ D W W D i 1

    4 @

    !

    S@ D i U W D u v @ D D U 8 S

    W v D 0 @ @ U W @ 6 i U @ D D U 8 S

    W v D b @

    !

    S@ D i U W D

    u v @ D D U 8 S

    W v D b @ @ U W @ 6 i U @ D D U 8 S

    W v D 0 @ Q 8 D i

    !

    !

    B

    0 1 q

    F

    y Fs V C F q D @

    D G 5 W D i G D 6 i U W D 6 u v @ D W 8 v D B Q v B D D U 8 S

    W v D 6 @ v @ D 6 U @ U D 6 G @

    D G 5 W D i G @ @ W 8 i U @ 6 1

    b

  • 8/8/2019 Mecnica Fsica-Classica

    8/208

    f D i U B D 0 1 b D 6 @ v Q G D @ 5 G @ f @ h 8 i Q D 5 W D j 6 @ 8 D B S@ B u v D Q G i D B D 6 6 D

    G @ 5 Q S@ U W 5 D Q D i S@ W i Q 6 8 D Q 8 @ 1 f i @ Q 8 D Q 8 i 6 @ D B D 6 6 D G @ 5 Q S@ U W 5 D G D D U 8 S

    W v D

    S@ W i Q 6 8 D Q 8 @ 0 1 b i G @ U @ @ 6 W U @ @ U j 6 @ Q D i U B D B D 5 6 W i Q @ W 5 G D 1 4 v 6 8 5 8 v 5 Q G i

    U

    @ B 0 1 b 8 @ B i 6

    G

    U

    G

    B

    0 1 k

    D B S@ B W i Q W v S

    B i 6 u v @ D U 5 B @ 5 U D @ 5 S@ v B W D 6 i @ 6 @ W 5 D G D 6 @ v Q G D @C 5 1 f D

    @ U G D G @ 6 @

    @ Q 8 D i

    G

    G

    i v

    U

    W i Q 6 8 D Q 8 @ 1

    5 ' ' 2 " $ " $ 3 4 5 '

    # $ & ( $ 0 2 4 2 5 & 7 9 $ ( B 2 D5 F2 & 4 2 G H@ 2 P 7 Q S 4 2 C 4 $ 4 $ 0 & B S X

    0 $ 7 Q & 4 $ H 0U 2 3 a 2 ( Q b

    c

    5 H d 2 e

    D D g V D 6 F Hp F W g R 8 DA V u F g W iF g H F q y F R C @ H F H D H D 6 g u @ 6 D R C g H D y 6 F

    e

    Fs V C T

    d

    W y q F s

    6 @ D

    i U W D 8 i 8 D

    u v @ D W 8 v D 6 i U @ v B D D U 8 S

    W v D S@ Q v D @ Q 8 D i

    G u

    G

    @ D u v D Q 8 5 G D G @

    G @ B i 5 B @ Q 8 i W i Q 6 @ U D j 6 @ 1

    D 6 @ 5 6 G @ f @ h 8 i Q i G @ B i 6 D 5 Q G D U i D U u v @ @ 6 8 @ 8 @ i U @ B D S@ SD 5 G i D U D v B

    6 5 6 8 @ B D 5 6 i D G i G @ G v D 6 D U 8 S

    W v D 6 1 4 @ 3 D

    !

    !

    D i U W D u v @ D D U 8 S

    W v D 0 b

    @ @ U W @ 6 i U @ D D U 8 S

    W v D b 0 1 D 8 @ U W @ 5 U D @ 5 G @ f @ h 8 i Q 8 @ B i 6 u v @

    !

    C

    !

    B

    0 1

    5 Q 8 @ U D Q G i 0 1 Q i 8 @ B i @ Q 8 U @ i 5 Q 6 8 D Q 8 @ @ i 8 @ B i 6

    v

    w w

    w

    !

    C

    !

    0 1

    i v D 5 Q G D

    v

    w w

    w

    G

    G

    !

    U

    !

    C

    U

    G

    B

    0 1

    Sv 8 5 B D @ U @ 6 6 D i i 5 i 8 5 G D 6 v 6 8 5 8 v 5 Q G i @ B 0 1

    !

    G

    G

    !

    U

    !

    @

    !

    G

    G

    U

    1 @ 0 1 i 8 @ B i 6

    !

    U

    !

    C

    U

    y

    w w

    w

    !

    C

    !

    C

    0 1 {

    v 6 D Q G i D G @ Q 5 W D i G @ u v D Q 8 5 G D G @ G @ B i 5 B @ Q 8 i

    U

    1 @ u v D W D i 0 1 {

    @ U 5 B @ i 8 @ i U @ B D G D W i Q 6 @ U D W D i G D u v D Q 8 5 G D G @ G @ B i 5 B @ Q 8 i D U D i 6 5 6 8 @ B D

    5 6 i D G i G @ G v D 6 D U 8 S

    W v D 6 u v @ 5 Q 8 @ U D @ B 1

    q

  • 8/8/2019 Mecnica Fsica-Classica

    9/208

    # $ & ( $ 0 2 4 2 5 & 7 9 $ ( B 2 D5 F2 & 4 & 0 & 0 $ 7 Q & 2 P 7 H d 2 ( 4 $

    H 0 2 a 2 ( Q b

    c

    5 H d 2

    4 @ 3 D

    i B i B @ Q 8 i D Q v D U G @ v B D D U 8 S

    W v D G @ B D 6 6 D @ u v D Q 8 5 G D G @ G @ B i 5 j

    B @ Q 8 i

    U @ D 8 5 D B @ Q 8 @ D i i Q 8 i

    0 1

    6 @ Q G i

    i U D 5 i @ W 8 i U G D i U 5 @ B

    @D D U 8 S

    W v D 1 n B i B @ Q 8 i G D i U W D

    u v @ D W 8 v D

    6 i U @ D D U 8 S

    W v D

    S@ G @ Q 5 G i i U

    B

    0 1 0

    ' i U i v 8 U i D G i W D W v D Q G i D D U 5 D W D i G @

    W i B i 8 @ B i i 8 @ B i 6

    G

    G

    G

    G

    U

    U

    C

    G

    G

    U

    B

    0 1 0 0

    f i 6 @ v Q G i B @ B U i D U 5 B @ 5 U D D U W @ D S@ Q v D i U u v @ @ U 5 B @ i U i G v 8 i @ W 8 i U 5 D

    G @ G i 5 6 @ W 8 i U @ 6 D U D @ i 6 1 6 6 5 B G @ 0 1 0 @ 0 1 0 0 6 v 6 8 5 8 v 5 Q G i

    G

    G

    U

    i U

    i 8 @ B i 6

    G

    G

    0 1 0 b

    i v

    G

    G

    0 1 0 q

    6 @ Q G i D B i 6 i 6 @ W 8 i U @ 6

    @

    G @ Q 5 G i 6 U @ D 8 5 D B @ Q 8 @ @D i U 5 @ B

    1 @ 0 1 0 q

    i 8 @ B i 6 i 8 @ i U @ B D G D W i Q 6 @ U D W D i G i B i B @ Q 8 i D Q v D U 1

    D D g V D 6 F H FU W g R 8 DA V u F g W iF g H g 6 g 6 D R C g F R 8 y q F V H D y 6 F

    e

    Fs V C T

    d

    W y q F

    6 @ i B i B @ Q 8 i

    G D 6 i U W D 6 8 i 8 D

    u v @ D W 8 v D 6 i U @ v B D D U 8 S

    W v D S@ Q v i @ Q 8 D i

    @ i

    B i B @ Q 8 i D Q v D U

    G D D U 8 S

    W v D W i Q 6 @ U D j 6 @ 1

    # $ & ( $ 0 2 4 2 5 & 7 9 $ ( B 2 D5 F2 C & 4 2 $ 7 $ ( S 2 4 $ H 0 2 a@ 2 C ( Q b

    c

    X

    5 H d 2

    s i Q 6 5 G @ U @ B i 6 i 8 U D D i U @ D 5 D G i @ D i U W D @ 8 @ U Q D

    6 i U @ D D U 8 S

    W v D u v D Q G i

    @ 6 8 D 6 @ G @ 6 i W D G i i Q 8 i 0 D U D i i Q 8 i b 1 ' @ D G @ Q 5 W D i G @ 8 U D D i 8 @ B i 6

    !

    v

    !

    B

    0 1 0 k

    4 @ W i Q 6 5 G @ U D U B i 6 D B D 6 6 D G D D U 8 S

    W v D W i Q 6 8 D Q 8 @ 6 v 6 8 5 8 v 5 Q G i

    i U

    G

    G

    U

    @

    i U

    U

    G i 8 @ B i 6

    !

    v

    !

    G

    G

    U

    B U

    G

    b

    v

    !

    G

    G

    U U

    G

    B

    k

  • 8/8/2019 Mecnica Fsica-Classica

    10/208

    D 6

    U U

    i i

    !

    b

    v

    !

    G

    G

    G

    b

    y

    b

    b

    !

    B

    u v D Q 8 5 G D G @ @ 6 W D D U

    S@ D @ Q @ U 5 D W 5 Q S@ 8 5 W D G D D U 8 S

    W v D 1 s i Q W v S

    B i 6 u v @

    !

    !

    0 1 0

    6 @ Q G i

    !

    D @ Q @ U 5 D W 5 Q S@ 8 5 W D G D D U 8 S

    W v D Q i 5 Q 6 8 D Q 8 @

    !

    b

    !

    b

    !

    B

    5 v D G D G @ 0 1 0 G 5 j Q i 6 u v @ i 8 U D D i U @ D 5 D G i 6 i U @ D D U 8 S

    W v D @ D 6 i U W D 6

    @ 8 @ U Q D 6 S@ 5 v D @D D U 5 D W D i G D @ Q @ U 5 D W 5 Q S@ 8 5 W D G D D U 8 S

    W v D 1

    4 @ i 8 U D D i

    !

    U @ D 5 D G i 6 i U @ D D U 8 S

    W v D S@ i B @ 6 B i u v D u v @ U u v @ 6 @ 3 D

    i W D B 5 Q i @ 6 W i 5 G i @ Q 8 U @ i i Q 8 i 0 @ b G 5 @ B i 6 u v @ D i U W D u v @ D W 8 v D 6 i U @ D

    D U 8 S

    W v D S@ W i Q 6 @ U D 8 5 D 5 1 @ 1

    0 1 0

    i v D 5 Q G D

    0 1 0

    i Q G @ S@ v B D v Q W D i @ 6 W D D U @ U @ U @ 6 @ Q 8 D i i @ U D G i U U D G 5 @ Q 8 @

    x

    ! C

    "

    x

    $ C

    %

    x

    '

    0 1 0 {

    x

    ! C

    "

    x

    $ C

    %

    x

    '

    B

    @ u v D W D i 0 1 0 @ U 5 B @ i D W 8 i G @ i 8 U D D i U @ D 5 D G i i U v B D i U W D W i Q 6 @ U D j

    8 5 D 6 i U @ v B D D U 8 S

    W v D D i i Q i G @ v B W D B 5 Q i @ W D G i 6 @ U Q v i 1 6 @ u v D W i @ 6

    0 1 0 @ 0 1 0 6 D i @ u v 5 D @ Q 8 @ 6 1

    ' D U D v B 6 5 6 8 @ B D W i Q 6 @ U D 8 5 i v B 6 5 6 8 @ B D 6 i U @ i u v D 6 Si D W 8 v D B i U W D 6

    W i Q 6 @ U D 8 5 D 6

    !

    v

    ! (

    G

    v

    !

    G

    !

    0 1 0

    6 @ Q G i

    !

    D @ Q @ U 5 D i 8 @ Q W 5 D Q i i Q 8 i 0 i Q 8 i b @

    (

    G

    G C

    "

    G

    "

    C

    %

    G

    %

    G

    B

    5 v D D Q G i 0 1 0 @ 0 1 0 i 8 @ B i 6

    !

    !

  • 8/8/2019 Mecnica Fsica-Classica

    11/208

    i v

    !

    C

    !

    C

    B

    0 1 b

    D D g V D 6 F H F W g R 8 DA V u F g W iF g H F D R D V 8 @ F H D y 6 F

    e

    Fs V C T

    d

    W y q F

    6 @ D 6 i U W D 6 u v @ D W 8 v D B

    6 i U @ v B D D U 8 S

    W v D 6 D i W i Q 6 @ U D 8 5 D 6 D @ Q @ U 5 D 8 i 8 D G D D U 8 S

    W v D 2 C

    W i Q 6 @ U D j 6 @ 1

    @ 6 5 Q D B i 6 i U

    2 C 0 1 b 0

    D @ Q @ U 5 D B @ W xD Q 5 W D G D D U 8 S

    W v D 1

    ' i G @ B i 6 6 @ B U @ v 8 5 5 D U i D W 8 i G @ D @ Q @ U 5 D B @ W xD Q 5 W D G @ v B 6 5 6 8 @ B D W i Q j

    6 @ U D 8 5 i 6 @ W i Q 6 @ U D U D U D i 8 @ U D @ 5 G @ B i 5 B @ Q 8 i G i 6 5 6 8 @ B D 6 @ @ 6 8 @ i U v Q 5 j

    G 5 B @ Q 6 5 i Q D 1

    s i Q 6 5 G @ U @ B i 6 v B D D U 8 S

    W v D u v @ G @ 6 W U @ @ v B B i 5 B @ Q 8 i v Q 5 G 5 B @ Q 6 5 i Q D 6 v j

    3 @ 5 8 D D @ Q D 6 D i U W D 6 W i Q 6 @ U D 8 5 D 6 1 6 v D @ Q @ U 5 D B @ W xD Q 5 W D S@

    2 C

    b

    C

    B

    0 1 b b

    @ 6 i @ Q G i @ 6 8 D @ u v D W D i @ B i U G @ B D

    8 @ B i 6

    G

    G

    b

    2

    i v 6 D @ Q G i u v @ G

    G

    G

    G

    G

    G

    2

    B

    0 1 b q

    5 Q 8 @ U D Q G i D Sv 8 5 B D @ u v D W D i @ Q 8 U @ i 5 Q 6 8 D Q 8 @

    @ i 8 @ B i 6

    v

    G

    2

    B

    0 1 b k

    5 2 4 " $ 5 1 " 1 )5 ' 2 5 ' $ ' 5 ' "

    @ Q @ U 5 D i 8 @ Q W 5 D G i i 6 W 5 D G i U D U B Si Q 5 W i S@ G D i U B D

    '

    b

    0 1 b

    i Q G @

    '

    S@ D W i Q 6 8 D Q 8 @ G @ @ D 6 8 5 W 5 G D G @ @ D G @ i U B D W D i G D B i D 1 4 v 6 8 5 8 v 5 Q G i

    0 1 b @ B 0 1 b k i 8 @ B i 6

    v

    G

    -

  • 8/8/2019 Mecnica Fsica-Classica

    12/208

    !

    v

    G

    0 -

    B

    @ W 8 v D Q G i D B v G D Q W D G @ D U 5 SD @ 5 6

    "

    '

    b 2

    G

    b 2

    '

    G

    "

    8 @ B i 6

    0

    v

    b 2

    '

    G

    "

    0

    "

    '

    D U W 6 5 Q

    "

    y

    0 1 b

    5 6 8 i u v @ G

    "

    0

    "

    D U W 6 5 Q

    "

    1 4 v i Q G i u v @ D U W 6 5 Q

    "

    @ 5 Q @ U 8 @ Q G i

    0 1 b i 8 @ B i 6

    D U W 6 5 Q

    "

    '

    b 2

    '

    6 5 Q

    '

    B

    0 1 b

    @ u v D W D i 0 1 b G @ Q @ D @ 5 G @ B i 5 B @ Q 8 i G i i 6 W 5 D G i U D U B Si Q 5 W i 1 U @ u v x@ Q W 5 D

    G D 6 i 6 W 5 D W i @ 6 S@ G D G D i U

    '

    @ D D B 5 8 v G @ G @ i 6 W 5 D W D i

    b2

    '

    S@ U i i U W 5 i Q D @D U D 5 u v D G U D G D G D @ Q @ U 5 D G i i 6 W 5 D G i U 1 6 8 @ B @ 6 B i D W 8 i D B i 6

    @ Q W i Q 8 U D U U @ u v @ Q 8 @ B @ Q 8 @ Q i 6 G 5 @ U @ Q 8 @ 6 U D B i 6 G D S

    6 5 W D i U @ @ B i Q i @ 6 8 v G i

    G @ 8 i G i 6 i 6 @ Q Si B @ Q i 6 i Q G v D 8 Si U 5 i 6 1

    @ 6 W i D

    "

    @ 6 8 SD U @ D W 5 i Q D G D W i B D @ 6 W i D G D 6 W i Q G 5 W i @ 6 5 Q 5 W 5 D 5 6

    "

    5 B 5 W D U 5 D i D D U @ W 5 B @ Q 8 i G @ v B D D 6 @

    E

    D U W 6 5 Q

    "

    Q D @ u v D W D i 0 1 b

    b 2

    '

    6 5 Q

    '

    C E

    B

  • 8/8/2019 Mecnica Fsica-Classica

    13/208

    5 ' 5 ' $ ' 5 ' " $ "

    2 5 ' 1 " 5 ' 1 " 1 5 2 " 5

    @ Q @ U 5 D G @ v B D D U 8 S

    W v D u v @ 6 @ B i @ D i i Q i G @ v B D 5 Q D U @ W 8 5 S

    Q @ D 6 v 3 @ 5 8 D

    @D i U W D W i Q 6 @ U D 8 5 D S@ G D G D i U

    2

    b

    C

    B

    0 1 b {

    4 @ 3 D v B D v Q W D i W i Q 8 S

    Q v D U @ U @ 6 @ Q 8 D G D Q D v U D 0 1 b @ B v Q W D i G @ 1

    @ U @ 6 @ Q 8 D B i 6 @ D 5 Q D D U D @ D D i @ 5 i G i 6 D @ Q @ U 5 D B @ W xD Q 5 W D G D D U 8 S

    W v D 1

    E

    V(x)

    x x

    x

    x x1 2

    0

    4 5

    xx 3

    5 v U D 0 1 b #

    ! # %

    % ' ) 1 3

    5 7

    9%

    A

    #

    1

    D @ u v D W D i 0 1 b { @ G i U SD W i i G @ B i 6 8 5 U D U D 6 6 @ v 5 Q 8 @ 6 W i Q W v 6 i @ 6

    D

    @ 0 1 b { 8 @ B i 6 u v @ D @ Q @ U 5 D W 5 Q S@ 8 5 W D G D D U 8 S

    W v D S@ G D G D i U

    0

    b

    2

    B

    0 1 b

    @ Q @ U 5 D W 5 Q S@ 8 5 W D Q v Q W D i G @ U SD 6 @ U Q @ D 8 5 D i u v @ 5 B 5 W D u v @

    2 E

    B

    0 1 q

    f i U SD W i @ 5 6 8 @ B G i 5 6 5 Q 8 @ U D i 6 D U D i 6 u v D 5 6 D W i Q G 5 W D i 0 1 q Q D i S@

    6 D 8 5 6 @ 5 8 D

    G I

    !

    Q

    @ G I

    $

    T I

    B

    6 8 D 6 i Q D 6 6 D i i Q D 6 U i 5 5 G D 6 @ D D U 8 S

    W v D Q v Q W D 6 @ U SD @ Q W i Q 8 U D G D Q @ 6 8 @ 6

    5 Q 8 @ U D i 6 1

    {

  • 8/8/2019 Mecnica Fsica-Classica

    14/208

    D

    G 5 6 8 xD Q W 5 D G D U @ W 8 D 2 @D W v U D U @ U @ 6 @ Q 8 D D @ Q @ U 5 D W 5 Q S@ 8 5 W D 1 f i 6

    i Q 8 i 6

    !

    @

    $

    @ D S@ Q v D Q i i Q 8 i

    @ D S@ B SD 5 B D 1 n 6 i Q 8 i 6

    !

    @

    $

    6 D i W D B D G i 6 i Q 8 i 6 G @ U @ 8 i U Q i 1 f @ 6 8 @ 6 i Q 8 i 6 D @ i W 5 G D G @ G D D U 8 S

    W v D

    D Q v D j 6 @ 5 Q @ U 8 @ Q G i j 6 @ i 6 @ Q 8 5 G i G @ D W i U G i W i B D 6 v D i 6 5 W D i 1

    D

    B D D U 8 S

    W v D 6 v 3 @ 5 8 D@

    D @ Q @ U 5 D i 8 @ Q W 5 D U @ U @ 6 @ Q 8 D G D i U

    @ W i B

    @ Q @ U 5 D 2 i G @ 8 @ U G i 5 6 8 5 i 6 G @ B i 5 B @ Q 8 i 6 G @ D W i U G i W i B D 6 v D i 6 5 W D i

    ' i G @ G @ 6 i W D U j 6 @ @ Q 8 U @

    Q

    T

    !

    Q

    1 6 8 @ B i 5 B @ Q 8 i 6 Si S@ 5 B 5 8 D G i Q v B

    D G i 1 D U 8 S

    W v D i G @ D U i 5 B D U j 6 @ G i i Q 8 i

    !

    5 Q G D G i 5 Q Q 5 8 i

    T 1 f i i Q 8 i

    !

    D 6 v D @ i W 5 G D G @ D Q v D j 6 @ @ B v G D G @ 6 @ Q 8 5 G i 1

    D U 8 S

    W v D @ Q 8 D i D D 6 8 D j 6 @ 5 Q G @ Q 5 G D B @ Q 8 @ 1 6 8 @ B i 5 B @ Q 8 i S@ 5 5 B 5 8 D G i 1

    D U 8 S

    W v D B i @ j 6 @ Q i 5 Q 8 @ U D i I

    $

    Q

    1 6 8 @ B i 5 B @ Q 8 i S@ 5 B 5 8 D G i

    @ B D B i 6 i 6 D G i 6 1 s i Q 6 5 G @ U @ B i 6 v B D D U 8 S

    W v D u v @ D U 8 @ G D i 6 5 W D i

    Q i 6 @ Q 8 5 G i G i 6 W U @ 6 W @ Q 8 @ 6 1 6 v D @ i W 5 G D G @ Q i 6 @ Q 8 5 G i

    i 6 5 8 5 i G i @ 5 i G i 6 D v B @ Q 8 D D 8 S@ D i i Q 8 i

    6 @ v 5 G D B @ Q 8 @

    G 5 B 5 Q v 5 D 8 S@

    i 8 D D D v B @ Q 8 D U D 8 S@ @ D D U 8 5 U G @

    G 5 B 5 Q v 5 D 8 S@

    $

    i Q G @ 6 @ D Q v D @ D 6 6 D D 6 @ U Q @ D 8 5 D B i 5 B @ Q 8 i Q i

    6 @ Q 8 5 G i G i 6 G @ W U @ 6 W @ Q 8 @ 6 D 6 6 D Q G i D D U 8 S

    W v D @ i 6 i Q 8 i 6

    D 8 S@ i 8 D U D D 8 5 Q 5 U

    1

    S

    v B B i 5 B @ Q 8 i 5 B 5 8 D G i @ U 5 Si G 5 W i W i B

    @ U S

    i G i

    2 b

    v

    G

    2

    B

    0 1 q 0

    n @ U S

    i G i S@ v Q W D i G @ @ Q @ U 5 D G D D U 8 S

    W v D @ S@ 5 v D D i G i U i G i

    8 @ B i u v @ D D U 8 S

    W v D G @ B i U D D @ U W i U U @ U D G 5 6 8 xD Q W 5 D @ Q 8 U @

    @

    $

    1

    5 ( ' 5 $ ! 2 1 " 5 5 5 5 $

    2 Q " 5 )! 2 Q 1 2 5 " ' ) 2 5 2 5 ' " '

    @ q @ g 8 V F F s V D R W D " CE g R @ F R $ D W F R @ W 8

    e

    8 & ( 0 2

    D B i 6 D @ U i @ 6 8 v G i G i B i 5 B @ Q 8 i G @ v B @ @ W 8 U D i 6 i i @ @ 5 8 i G @ v B W D B i

    @ S@ W 8 U 5 W i @ v B W D B i B D Q S@ 8 5 W i W i Q 6 8 D Q 8 @ 6 1 4 v i B i 6 u v @ 8 @ B i 6 v B D U G @

    D W D 6 D U D @ D 6 @D G 5 6 8 xD Q W 5 D

    v B D G D i v 8 U D B i Q 8 D G D 6 G @ Q 8 U i G @ v B 8 v i i Q G @

    @ 5 6 8 @ i SD W v i @ 5 D G D 6 D v B D D 8 @ U 5 D G @ B i G i D @ 5 6 8 5 U v B W D B i @ S@ W 8 U 5 W i

    v Q 5 i U B @ G @ 5 Q 8 @ Q 6 5 G D G @ 4 6

    @ Q 8 U @ D 6 D W D 6 G @ D W i U G i W i B D v U D 0 1 q 1 6

    D W D 6 @ 6 8 D i W i i W D G D 6 @ Q 8 U @ i 6 i i 6 G @ v B B D Q @ 8 @ u v @ W U 5 D v B W D B i v Q 5 i U B @

    @ U @ Q G 5 W v D U D i D Q i G i D @ 1 4 v i B i 6 u v @ i 6 @ @ W 8 U i @ 6 G @ W D U D 7 @

    W i B @ W D B i 6 @ v B i 5 B @ Q 8 i D D U 8 5 U G D D W D G @ D 5 i W i B v B D @ i W 5 G D G @ B v 5 8 i

    @ u v @ Q D 1 n 6 @ @ W 8 U i @ 6 i G @ U D i 6 @ U 5 @ U 8 D G i 6 i U v B U i W @ 6 6 i i 8 i j @ S@ W 8 U 5 W i 1

    i U W D B D Q S@ 8 5 W D 6 @ Q G i @ U @ Q G 5 W v D U @D G 5 U @ W W D i G i B i 5 B @ Q 8 i G i @ @ W 8 U D i

    8

    @

    U

    9

  • 8/8/2019 Mecnica Fsica-Classica

    15/208

    d

    x

    y

    V

    0

    B

    5 v U D 0 1 q 7 # % !# % # % # % # # #

    Q D i U @ D 5 D 8 U D D i @ D @ Q @ U 5 D G i @ @ W 8 U D i S@ G D G D i U

    2

    b

    @

    6

    "

    b

    C

    @

    6

    "

    B

    0 1 q b

    Sv 8 5 B D D U W @ D U @ U @ 6 @ Q 8 D D @ Q @ U 5 D G i @ @ W 8 U D i Q i W D B i @ S@ W 8 U 5 W i 1 @ Q @ U 5 D

    B @ W xD Q 5 W D 0 1 q b S@ @ U @ 6 6 D @ B 8 @ U B i 6 G @

    @

    "

    @ G @ 5 G i @D @ 6 W i D G D i U 5 @ B

    G D @ Q @ U 5 D i 8 @ Q W 5 D S@ Q v D 5 6 8 i u v @ D U D

    "

    1 ' i G @ B i 6 8 U D Q 6 i U B D U

    @ 6 8 @ U i @ B D Q v B U i @ B D v Q 5 G 5 B @ Q 6 5 i Q D @ 5 B 5 Q D Q G i

    @ B v Q W D i G @

    "

    1 n

    W D B i @ S@ W 8 U 5 W i 6 Si i G @ D W @ @ U D U i @ @ W 8 U D i Q D G 5 U @ W W D i G i @ 5 i G i 6

    "

    1 n B i 5 j

    B @ Q 8 i Q D G 5 U @ W W D i G i @ 5 i G i 6 S@ D @ Q D 6 G @ 5 G i D i W D B i B D Q S@ 8 5 W i9

    1 6 6 5 B

    D W i B i Q @ Q 8 @ G D @ 5 G @ f @ h 8 i Q 6 @ v Q G i i @ 5 i G i 6 S@ G D G D i U

    G

    G

    @

    U

    9

    i v D 5 Q G D 5 6 8 i u v @9

    %

    x

    '

    G

    G

    @

    %

    B

    5 Q 8 @ U D Q G i D Sv 8 5 B D @ u v D W D i W i B D W i Q G 5 W D i

    U

    Q i i Q 8 i

    "

    i 8 @ B i 6

    G

    G

    @ %

    "

    "

    B

    0 1 q q

    5 Q D B @ Q 8 @ 6 v 6 8 5 8 v 5 Q G i D Sv 8 5 B D @ u v D W D i Q D @ U @ 6 6 D i 0 1 q b G @ 8 @ U B 5 Q D B i 6 D

    @ Q @ U 5 D B @ W xD Q 5 W D G i @ @ W 8 U D i D @ Q D 6 @ B v Q W D i G @

    "

    @

    2

    b

    C

    "

    @

    6

    "

    B

    v G i 6 @ D 6 6 D W i B i 6 @ D D U 8 S

    W v D @ @ W v 8 D 6 6 @ v B B i 5 B @ Q 8 i v Q 5 G 5 B @ Q 6 5 i Q D

    6 v 3 @ 5 8 D D v B i 8 @ Q W 5 D @ @ W 8 W 5 i

    b

    "

    @

    6

    "

    B

    0 1 q k

    0

  • 8/8/2019 Mecnica Fsica-Classica

    16/208

    4 @ 3 D

    "

    @ 6

    6

    @

    %

    B

    f D v U D 0 1 k U @ U @ 6 @ Q 8 D B i 6 D v Q W D i @ B

    v Q W D i G @

    "

    W i U U @ 6 i Q G @ Q G i D v B D D U SD i D G @ S@ U 8 5 W @ Q i i Q 8 i

    "

    "

    "

    W i B i S@ SD W 5 G @ W i Q W v 5 U 6 @ U @ @ 6 W U @ @ U B i 6 0 1 q k Q D i U B D

    b

    "

    "

    b

    "

    B

    0 1 q

    4 v 6 8 5 8 v 5 Q G i2

    @

    "

    Q D @ u v D W D i 0 1 b k i 8 @ B i 6

    Uef

    y2y ay a

    0

    -m ya2 2

    0 /2

    5 v U D 0 1 k # ! # % % ' ) 3

    v

    G

    "

    "

    "

    "

    0

    v

    %

    0

    %

    %

    "

    "

    0

    0

    D U W 6 5 Q

    "

    "

    0 D U W 6 5 Q

    "

    "

    0

    5 Q D B @ Q 8 @ 6 D @ Q G i u v @ D U D

    "

    @ u v @ D U W 6 5 Q 0 q

    b i 8 @ B i 6

    "

    "

    0 C 6 5 Q

    C q

    b

    "

    0 W i 6

    B

    0 1 q

    @ 8 @ U B 5 Q D B i 6 5 Q 8 @ U D Q G i 0 1 q q

    "

    6 5 Q

    B

    0 1 q

    6 @ u v D W i @ 6 0 1 q @ 0 1 q G @ 8 @ U B 5 Q D B D 8 U D 3 @ W 8 Si U 5 D G i @ @ W 8 U D i U @ U @ 6 @ Q 8 D Q G i

    v B W 5 W i S

    G @ Q D 6 v D i U B D D U D B S@ 8 U 5 W D @ U v U D 0 1 1

    S

    G @ Q i 8 D U u v @ i D i U

    B SD 5 B i G @ S@

    "

    b

    "

    @ u v @ 6 @ b

    "

    E

    i @ @ W 8 U D i S@ D 6 i U 5 G i @ D D W D

    G @ W 5 B D Q D i 6 @ i 8 @ Q G i D 8 U D 3 @ W 8 Si U 5 D U @ U @ 6 @ Q 8 D G D @ B 0 1 1 n D i U G @

    "

    b 6

    @

    %

    i G @ U SD 6 @ U D 8 @ U D G i D U 5 D Q G i % i v 6 1

    0 0

  • 8/8/2019 Mecnica Fsica-Classica

    17/208

    0 2 4 6

    y

    2y a

    x/ 0 3 5

    5 v U D 0 1 # # # % % # A # # ! % # 1

    0 b

  • 8/8/2019 Mecnica Fsica-Classica

    18/208

    4 AB

    4

    )

    0 4C

    0 ) 5

    4 7C

    0 " 5 $ 5 1 2Q " 24 ' 1 "

    B W D B i G @ i U W D 6 W @ Q 8 U D 5 6 S@ W D U D W 8 @ U 5 D G i i U 5 Q D 6 G @ i U W D W i B D G 5 U @ W W D i

    G D 5 Q D u v @ v Q @ i W i U i Q i u v D D i U W D D W 8 v D @ i W i U i u v @ U i G v i W D B i G @

    i U W D 6 1 s i Q 6 5 G @ U D Q G i v B D D U 8 S

    W v D Q v B W D B i G @ i U W D 6 @ 8 @ U Q i i W D B i G @

    i U W D 6 W @ Q 8 U D 5 6 S@ v B W D B i G @ i U W D 6 Q i u v D D i U W D u v @ D W 8 v D 6 i U @ D D U 8 S

    W v D

    8 @ B D G 5 U @ W W D i G D 5 Q D u v @ v Q @ D D U 8 S

    W v D D v B i Q 8 i i i W @ Q 8 U i G i W D B i

    G @ i U W D 6 1 i U W D

    u v @ D W 8 v D Q D D U 8 S

    W v D 6 @ U SD G D i U B D

    "

    %

    B

    6 8 @ W D B i G @ i U W D 6 S@ W i Q 6 @ U D 8 5 i 6p @ i B Si G v i G @

    D @ Q D 6 G @ @ Q G @ U G D

    G 5 6 8 xD Q W 5 D D i W @ Q 8 U i G @ i U W D 6 5 1 @ 1

    B

    b 1 0

    D D g V D 6 F

    B W D B i G @ i U W D 6 W @ Q 8 U D S@ W i Q 6 @ U D 8 5 i 6 @ v B D G D 6 G v D 6 W i Q G 5 W i @ 6

    S@ @ U 5 W D G D

    0 1 D G 5 U @ W W D i G D i U W D S@ D i i Q i G D 5 Q D u v @ v Q @ D D U 8 S

    W v D D v B i Q 8 i i

    @ D U D Q G @ D 6 Si G @ @ Q G @ G D G 5 6 8 xD Q W 5 D G D i U 5 @ B @D D U 8 S

    W v D 1

    b 1 D i U W D G @ U 5 D G @ v B D v Q W D i i 8 @ Q W 5 D u v @ D @ Q D 6 G @ @ Q G @ G D G 5 6 8 xD Q W 5 D G i

    i Q 8 i i @D D U 8 S

    W v D 1

    0 q

  • 8/8/2019 Mecnica Fsica-Classica

    19/208

    6 8 D 6 G v D 6 W i Q G 5 W i @ 6 6 D i @ u v 5 D @ Q 8 @ 6 1 s i Q 6 5 G @ U @ B i 6 D W i Q G 5 W D i b

    !

    x@

    !

    C

    x@

    C

    $

    x@

    $

    $

    !

    x@

    $

    !

    G

    G

    x@

    i Q G @ W i Q 6 5 G @ U SD B i 6 u v @ S@ D @ Q D 6 v Q W D i G @ 1 4 v 6 8 5 8 v 5 Q G i

    !

    C

    C

    $

    b

    b

    !

    C

    C

    $

    ! 0 b q

    i 8 @ B i 6

    0

    G

    G

    $

    !

    x@

    G

    G

    W i Q 6 5 G @ U D Q G i u v @

    !

    G

    G

    Q D i S@ D @ W 8 D G i @ i 6 i B D 8 Si U 5 i Q i S

    Q G 5 W @ ! @ u v @

    $

    !

    x@

    !

    x@

    !

    C

    x

    C

    $

    x@

    $

    B

    5 Q D B @ Q 8 @ 8 @ B i 6

    G

    G

    B

    S@ v B D i U W D W i B D G 5 U @ W W D i G i U D 5 i @ W 8 i U u v @ v Q @ D D U 8 S

    W v D D i W D B i G @

    i U W D 6 @ W v 3 i B Si G v i D @ Q D 6 G @ @ Q G @ G D G 5 6 8 xD Q W 5 D D i W @ Q 8 U i G @ i U W D 6 1

    ' U i SD B i 6 u v @ D W i Q G 5 W D i b S@ @ u v 5 D @ Q 8 @@

    D W i Q G 5 W D i 0 1 n 5 Q @ U 6 i 8 D B S@ B S@

    @ U G D G @ 5 U i 1

    4 @ 3 D

    G D i U B D

    B

    n 8 U D D i @ @ B @ Q 8 D U U @ D 5 D G i i U @ 6 8 D i U W D G v U D Q 8 @ i G @ 6 i W D B @ Q 8 i

    S@ G D G i

    i U

    B

    4 v 6 8 5 8 v 5 Q G i

    i U G

    B

    0

    b

    0

    b

    G

    G

    i 8 @ B i 6

    G

    B

    0 k

  • 8/8/2019 Mecnica Fsica-Classica

    20/208

    n 8 U D D i @ @ B @ Q 8 D U

    D @ Q D 6 G @ @ Q G @ G i D i U 5 Q 5 W 5 D @ Q D G D D U 5 SD @

    D G 5 6 8 xD Q W 5 D D i W @ Q 8 U i G @ i U W D 6 @ i U 8 D Q 8 i

    S@ v B D i U W D W i Q 6 @ U D 8 5 D 1 Q 8 D i

    @ 5 6 8 @ v B D v Q W D i i 8 @ Q W 5 D 8 D u v @

    G G

    i v

    5 6 8 i u v @ G

    1 !

    ' U i SD B i 6 u v @ D 6 W i Q G 5 W i @ 6 0 @ b G i 8 @ i U @ B D 6 D i @ u v 5 D @ Q 8 @ 6 1

    D 8 D D i U D

    U i D U u v @ 6 @

    S@ W i Q 6 @ U D 8 5 i i B Si G v i G @

    D @ Q D 6 G @ @ Q G @ G @ i v u v @

    1 4 v i B i 6 u v @ Q D G D S@ W i Q @ W 5 G i D W @ U W D G D G @ @ Q G x@ Q W 5 D G @ y

    y G @ 1

    ' D U 8 5 B i 6 G i D W 8 i u v @

    S@ v B D i U W D W @ Q 8 U D

    b 1 b

    @

    G @ U 5 D G @ v B i 8 @ Q W 5 D

    B

    b 1 q

    v 8 5 5 W D Q G i @ 6 W D D U B @ Q 8 @ b 1 b @ b 1 q i U

    @ 5 v D D Q G i D B D 6 D 6 @ U @ 6 6 i @ 6

    i 8 @ B i 6

    B

    B

    b 1 k

    5 B i 6 D Q 8 @ U 5 i U B @ Q 8 @ u v @

    G

    @

    G

    B

    4 v 6 8 5 8 v 5 Q G i D 6 Sv 8 5 B D 6 U @ D W i @ 6 @ B b 1 k 8 @ B i 6

    G G

    i v D v Q W D i i 8 @ Q W 5 D D U 5 D D @ Q D 6 u v D Q G i D D U 5 SD @ D U 5 D 1 Q 8 D i S@ v Q W D i

    G @ 1 D 6 5 B 5 W D u v @

    x

    W i B i u v @ U S

    D B i 6 G @ B i Q 6 8 U D U 1

    # % '

    )

    2

    4

    2

    4 6

    # 2

    4 8 9 @

    )

    2

    4

    2

    4 6

    % ' 2

    4 8 9

    F I

    % Q S

    0

  • 8/8/2019 Mecnica Fsica-Classica

    21/208

  • 8/8/2019 Mecnica Fsica-Classica

    22/208

    O

    P

    F

    v

    5 v U D b 1 0 ! # 5 # # 9 % % $ # % # # % $ # #

    5 # # 9 % ! # 9 % A # ! # !

    % 7 %

    s i Q W v S

    B i 6 u v @ i B i B @ Q 8 i D Q v D U G @ v B D D U 8 S

    W v D 6 v 3 @ 5 8 D v B W D B i G @ i U W D 6

    W @ Q 8 U D 5 6

    x S@ W i Q 6 8 D Q 8 @ 1 4 @ Q G i

    v B @ W 8 i U 8 @ B i 6 Q D @ U G D G @ 8 U x@ 6 W i Q 6 j

    8 D Q 8 @ 6 G @ B i 5 B @ Q 8 i D 6 8 U x@ 6 W i B i Q @ Q 8 @ 6 G @

    1 n @ W 8 i U i 6 5 W D i G D D U 8 S

    W v D

    S@ W i Q 6 8 D Q 8 @ B @ Q 8 @ @ U @ Q G 5 W v D U D

    @ i U 8 D Q 8 i W i B i W i Q 6 @ u v x@ Q W 5 D

    G @ 6 @ U v B @ W 8 i U W i Q 6 8 D Q 8 @ @ U 8 @ Q W @ @ B 8 i G i 6 i 6 5 Q 6 8 D Q 8 @ 6 D v B D Q i

    @ U @ Q G 5 W v D U D i D Q i G D Si U 5 8 D 1

    4 @ i B i 5 B @ Q 8 i S@ D Q D U v B U i @ B D u v @ 5 Q 5 W 5 D B @ Q 8 @ @ U D v B U i @ B D D q

    G 5 B @ Q 6 i @ 6 D 6 6 D D 6 @ U v B U i @ B D D G v D 6 G 5 B @ Q 6 i @ 6

    0 " 2 5 $ 5 ' 5

    6 6 i v W i @ 6 G D 6 @ u v D W i @ 6 G @ B i 5 B @ Q 8 i

    ! 0

    b

    q

    8 U x@ 6 @ u v D W i @ 6 G 5 @ U @ Q W 5 D 5 6 G @ 6 @ v Q G D i U G @ B W i Q 8 x@ B W i Q 6 8 D Q 8 @ 6 G @ 5 Q 8 @ U D W D i

    G v D 6 i U @ u v D W D i 1 ' D U D G v D 6 G @ 6 8 D 6 i G @ U @ B i 6 @ 6 W i @ U i 6 W i j 6 @ Q i 6 G 5 U @ W 8 i U @ 6 G @

    u v @ D B i D Q i G D Si U 5 8 D 1 @ 6 8 D j Q i 6 v B U i @ B D D G v D 6 G 5 B @ Q 6 i @ 6 W i B

    k W i Q 6 8 D Q 8 @ 6 G @ 5 Q 8 @ U D W D i 1

    B D G @ 6 8 D 6 S@ D U D Q G @ D G i @ W 8 i U

    u v @ 8 D B S@ B

    6 @ B D Q 8 S@ B W i Q 6 8 D Q 8 @ @ i v 8 U D S@ D @ Q @ U 5 D B @ W xD Q 5 W D G i 6 5 6 8 @ B D u v @ 6 @ B D Q 8 S@ B

    W i Q 6 8 D Q 8 @ 6 @ i W D B i G @ i U W D 6 S@ W i Q 6 @ U D 8 5 i 1 6 Sv 8 5 B D 6 G v D 6 W i Q 6 8 D Q 8 @ 6 D D U @ j

    W @ B Q D 5 Q 8 @ U D W D i G D 6 @ u v D W i @ 6 G @ B i 5 B @ Q 8 i @ G @ Q @ B i2 U @ @ B i D i 6 5 W D i

    5 Q 5 W 5 D G D D U 8 S

    W v D 1

    6 W i @ B i 6 i @ 5 i G i 6

    %

    D U D @ i D i @ W 8 i U

    @ 5 Q 8 U i G v 5 B i 6 D 6 W i i U G @ Q D G D 6

    i D U @ 6 Q i D Q i

    "

    W i 6

    "

    6 5 Q

    B

    b 1

    6

    4

    4 ! #$ % % ( % 1 3 ( 4 ! % 74 9 3 4 $ A

    S B

    $

    4 D 3 HP #$ %

    4

    Q R T VQ

    7% X %

    4 ( %

    ! 4 A

    X $ A $

    Q

    4

    VQ c

    4

    Q e VQ S

    0

  • 8/8/2019 Mecnica Fsica-Classica

    23/208

    4 @ 3 D B @ 2 U @ 6 @ W 8 5 D B @ Q 8 @ D U D Q G @ D G i @ W 8 i U B i B @ Q 8 i D Q v D U @ D @ Q @ U 5 D

    B @ W xD Q 5 W D

    y

    U

    y

    "

    "

    B

    b 1

    2

    0

    b

    C

    "

    C

    B

    b 1

    @ u v D W D i b 1 U @ U @ 6 @ Q 8 D D W i B i Q @ Q 8 @

    $

    G i B i B @ Q 8 i D Q v D U 5 6 8 i u v @ @ 6 j

    W i @ B i 6

    x

    '

    1 4 v 6 8 5 8 v 5 Q G i b 1 @

    W i 6

    6 5 Q

    "

    6 5 Q C

    W i 6

    Q D 6 @ U @ 6 6 i @ 6 b 1 @ b 1 i 8 @ B i 6

    b 1 {

    2

    b

    C

    C

    B

    b 1

    @ u v D W D i b 1 { W i Q 8 S@ B D @ 5 G D 6 SD U @ D 6 1 4 D @ B i 6 u v @ S@ W i Q 6 8 D Q 8 @ 1 n 6 5 Q 5 W D G i

    S

    6 5 W i G i 6 @ v Q G i B @ B U i G D @ u v D W D i b 1 { W D W D U i D Q D 5 6 D Q G i D v U D b 1 b 1 4 @ 3 D

    O

    P

    Q

    d

    5 v U D b 1 b ! #&

    D i 6 5 W D i G D D U 8 S

    W v D Q i 5 Q 6 8 D Q 8 @ @ D 6 v D i 6 5 W D i Q i 5 Q 6 8 D Q 8 @ C G 1 f i

    5 Q 8 @ U D i G @ 8 @ B i G i U D 5 i @ W 8 i U

    D U U @ v B D SD U @ D

    0

    b

    0

    b

    B

    SD U @ D D U U 5 G D i U v Q 5 G D G @ G @ 8 @ B i

    G

    0

    b

    b 1 0

    0 {

  • 8/8/2019 Mecnica Fsica-Classica

    24/208

    S@ G @ D W i U G i W i B b 1 { v B D W i Q 6 8 D Q 8 @ G i B i 5 B @ Q 8 i

    G

    b

    B

    b 1 0 0

    ' i G @ B i 6 @ Q v Q W 5 D U D @ 5 G D 6 SD U @ D 6 G i 6 @ v 5 Q 8 @ B i G i i U D 5 i @ W 8 i U G D D U 8 S

    W v D

    U @ D 8 5 D B @ Q 8 @ D i W @ Q 8 U i G @ i U W D 6 D U U @ SD U @ D 6 5 v D 5 6 @ B 5 Q 8 @ U D i 6 G @ 8 @ B i 5 v D 5 6 1

    6 8 D @ 5 S@ W i Q @ W 5 G D @ D 6 @ v Q G D @ 5 G @ @ @ U Q i W D 6 i G i i 8 @ Q W 5 D U D S

    8 5 W i 1

    5 B i 6 Q i @ Q 8 D Q 8 i u v @ @ D S@ B D 5 6 @ U D S@ SD 5 G D D U D u v D u v @ U W D B i G @ i U W D 6

    W @ Q 8 U D 5 6 B @ 6 B i u v @ Q D i 6 @ 3 D W i Q 6 @ U D 8 5 i 1

    D @ u v D W D i b 1 W i Q W v S

    B i 6 u v @ D @ Q @ U 5 D W 5 Q S@ 8 5 W D @ B W i i U G @ Q D G D 6 i D U @ 6 S@

    G D i U B D

    0

    b

    C

    0

    b

    b 1 0 b

    @ 6 8 D Q G i D U 5 B @ 5 U D @ D 6 @ v Q G D D U W @ D 5 D G D 6 U @ 6 @ W 8 5 D B @ Q 8 @ D i B i 5 B @ Q 8 i

    U D G 5 D @ 8 U D Q 6 @ U 6 i G D D U 8 S

    W v D 1 B W i i U G @ Q D G D 6 i D U @ 6 D @ i W 5 G D G @ G D D U 8 S

    j

    W v D S@

    U

    x@

    C

    x@ b 1 0 q

    6 @ Q G i x

    @ B

    W D G D i Q 8 i @ x

    i @ W j

    8 i U v Q 5 8 SD U 5 i @ U @ Q G 5 j

    W v D U D x

    @ D i Q 8 D Q G i

    Q i 6 @ Q 8 5 G i G i 6 xD Q v i 6

    W U @ 6 W @ Q 8 @ 6 W i Q i U B @ D

    v U D D i D G i 1

    @ Q 5 Q G i

    @

    b 1 0 b i G @ @ 6 W U @ @ U j 6 @ Q D i U B D

    b

    C

    b

    B

    4 v 6 8 5 8 v 5 Q G i

    i U

    @ U @ u 1 b 1 { Q D @ u v D W D i b 1 0 b 8 @ B i 6

    0

    b

    C

    b

    B

    b 1 0 k

    D U W @ D

    S@ @ U D B @ Q 8 @ W D B D G D @ Q @ U 5 D i 8 @ Q W 5 D W @ Q 8 U S

    v D i U u v @ D i U W D

    i 8 5 G D W D W v D Q G i i Q @ D 8 5 i G i U D G 5 @ Q 8 @ G @ 6 8 D @ Q @ U 5 D i 8 @ Q W 5 D S@ G D G D i U

    G

    G

    b

    $

    B

    4 v 6 8 5 8 v 5 Q G i

    i 8 @ B i 6

    u v @ S@ 5 v D @D i U W D W @ Q 8 U S

    v D

    Q v B U @ @ U @ Q W 5 D u v @ U i G D W i B v B D @ i W 5 G D G @

    D Q v D U

    5 v D D i D i U 5 Q 6 8 D Q 8 xD Q @ i G @

    G

    G

    1 f i @ Q 8 D Q 8 i S@ 5 B i U 8 D Q 8 @ Q D i @ 6 u v @ W @ U

    0

  • 8/8/2019 Mecnica Fsica-Classica

    25/208

    u v @ G @ D W 8 i D @ Q @ U 5 D i 8 @ Q W 5 D W @ Q 8 U S

    v D S@ v B D D U W @ D G D @ Q @ U 5 D W 5 Q S@ 8 5 W D

    G D D U 8 S

    W v D D D U 8 @ W i U U @ 6 i Q G @ Q 8 @ @D W i B i Q @ Q 8 @ 8 U D Q 6 @ U 6 D @ U @ Q G 5 W v D U @ B

    W D G D 5 Q 6 8 D Q 8 @ D i U D 5 i @ W 8 i U G i B i 5 B @ Q 8 i 1

    i 6 6 5 5 5 G D G @ G @ @ U 5 B 5 U D @ Q @ U 5 D W 5 Q S@ 8 5 W D D @ Q D 6 @ B v Q W D i G D D U 5 SD @

    U @ G v i U i @ B D G i @ 6 8 v G i G i B i 5 B @ Q 8 i G @ v B D D U 8 S

    W v D Q v B W D B i W i Q 6 @ U j

    D 8 5 i G @ i U W D 6 W @ Q 8 U D 5 6 D v B U i @ B D v Q 5 G 5 B @ Q 6 5 i Q D G @ v B D D U 8 S

    W v D 6 v 3 @ 5 8 D

    D i i 8 @ Q W 5 D @ @ W 8 5 i 5 1 @ 1

    G

    G

    G

    B

    4 v 6 8 5 8 v 5 Q G i b 1 0 k Q D @ u v D W D i b 1 @ U 5 W D B i 6 u v @

    2

    0

    b

    C

    b

    C F b 1 0

    W i U U @ 6 i Q G @ D i U i @ B D v Q 5 G 5 B @ Q 6 5 i Q D G @ v B D D U 8 S

    W v D 6 v 3 @ 5 8 D @D @ Q @ U 5 D i j

    8 @ Q W 5 D @ @ W 8 5 D

    b

    C 2 b 1 0

    3 SD @ 6 8 v G D G i D Q 8 @ U 5 i U B @ Q 8 @ @ U @ u v D W D i 0 1 b b @ 6 @ v 5 Q 8 @ 6 1 Sv Q 5 W D G 5 @ U @ Q W D S@

    u v @ D D U 5 SD @ G I C T I @ Q i U i @ B D v Q 5 G 5 B @ Q 6 5 i Q D G

    Q

    T C T I 1 @

    b 1 0 @ 0 1 b q 8 @ B i 6

    G

    G

    2

    B

    b 1 0

    s D W v D Q G i D D U 5 D W D i G @ G u v D Q G i D U 5 D G @ v B D u v D Q 8 5 G D G @ G D D U 8 5 U G D

    @ u v D W D i b 1 {

    G

    G

    @ 6 v 6 8 5 8 v 5 Q G i @ B b 1 0 i 8 @ B i 6

    G

    b

    2

    B

    b 1 0 {

    5 Q 8 @ U D Q G i b 1 0 { @ Q 8 U @ v B i Q 8 i 5 Q 5 W 5 D W i B W i i U G @ Q D G D 6

    @ v B i Q 8 i

    @ Q S@ U 5 W i G @ 8 @ U B 5 Q D B i 6 D @ u v D W D i G D Si U 5 8 D G D D U 8 S

    W v D

    v

    G

    b

    2

    B

    b 1 0

    6 8 @ 5 Q 8 @ U D i G @ 6 @ U 5 Q 8 @ U D G i @ B 8 @ U B i 6 G @ v Q W i @ 6 W i Q @ W 5 G D 6 D U D i 6 6 @ v 5 Q j

    8 @ 6 i 8 @ Q W 5 D 5 6

    B

    b

  • 8/8/2019 Mecnica Fsica-Classica

    26/208

    s i Q 6 5 G @ U @ B i 6 U 5 B @ 5 U i i W D 6 i

    W i B

    i v

    1 @ W 8 v D Q G i D

    B v G D Q W D G @ D U 5 SD @ 5 6

    A

    b

    G

    0

    G

    i 8 @ B i 6 G @ b 1 0

    v

    G

    b 2

    b 1 b

    n D 6 6 i 6 @ v 5 Q 8 @ W i Q 6 5 6 8 @ @ B @ @ W 8 v D U v B D Q i D B v G D Q W D G @ D U 5 SD @ 5 6 G @ B i G i

    D i 8 @ U B i 6 v B 5 Q 8 @ U D G D i U B D

    v

    G

    "

    0

    "

    B

    ' D U D 5 6 6 i @ 6 W U @ @ B i 6 D v Q W D i 5 Q 8 @ U D Q G D @ B b 1 b Q D i U B D

    b 2 b

    b 2 C

    C

    i v D 5 Q G D

    0

    C

    W i B

    b 2

    C

    B

    4 v 6 8 5 8 v 5 Q G i @ B b 1 b

    v

    G

    0

    @ @ @ W 8 v D Q G i v B D Q i D B v G D Q W D G @ D U 5 SD @ 5 6

    A

    "

    C

    G

    "

    G

    8 @ B i 6 Q D B @ Q 8 @

    v

    G

    "

    0

    "

    D U W W i 6

    "

    y

    B

    6 W i @ Q G i

    G @ B i G i u v @ D U W W i 6

    "

    i 8 @ B i 6

    D U W W i 6

    " "

    W i 6

    b 0

  • 8/8/2019 Mecnica Fsica-Classica

    27/208

    i v

    0

    C

    W i 6

    B

    b 1 b 0

    i 8 D U @ B i 6 @D @ u v D W D i b 1 b 0 u v D Q G i @ 6 8 v G D U B i 6 i U i @ B D G @ @ @ U 1 G @ j

    U 5 D W D i G @ b 1 b 0 6 v i @ u v @ 6 @ 2

    @ Q 8 D i

    b y 2 y

    B

    S@ B G 5 6 6 i D W i Q 6 8 D Q 8 @

    Q D @ Q @ U 5 D i 8 @ Q W 5 D

    i G @ 6 @ U i 6 5 8 5 D i v Q @ D 8 5 D 1 5 Q 8 @ U U @ 8 D W D i

    G D @ u v D W D i b 1 b 0 G @ @ Q G @ G i 6 5 Q D G @

    W i B i 6 @ U 5 D G @ @ 6 @ U D U 1 4 @

    U @ U @ 6 @ Q 8 D v B i 8 @ Q W 5 D U @ v 6 5 i 6 @

    S@ v B i 8 @ Q W 5 D D 8 U D W 8 5 i 1

    ' D U D i i 8 @ Q W 5 D

    U i W @ G @ B i 6 G @ v B B i G i 6 @ B @ D Q 8 @ 1 5 Q 8 @ U D W D i

    G @ b 1 0 Q i W D 6 i G i i 6 W 5 D G i U D U B Si Q 5 W i -

    '

    @ Q i @ B v G D Q W D 6

    G @ D U 5 SD @ v B i v W i G 5 @ U @ Q 8 @ 6 1 f @ 6 8 @ W D 6 i 8 @ B i 6

    v

    G

    b

    2 -

    @ @ @ W 8 v D Q G i D B v G D Q W D G @ D U 5 SD @ 5 6

    A

    0

    G

    G

    b 1 b b

    i 8 @ B i 6

    v

    G

    b 2 -

    v

    G

    '

    C b 2

    B

    4 i B D Q G i @ 6 v 8 U D 5 Q G i v B 8 @ U B i 5 Q G @ @ Q G @ Q 8 @ G @ G @ B i G i D W i B @ 8 D U v B

    u v D G U D G i @ U @ 5 8 i 3 v Q 8 D B @ Q 8 @ W i B i 6 Sv 8 5 B i 6 G i 5 6 8 @ U B i 6 G i U D G 5 W D Q G i i 8 @ B i 6

    v

    G

    '

    i v

    v

    G

    0

    w

    W i B

    2

    '

    B

    b 1 b q

    @ W 8 v D Q G i v B D 6 @ v Q G D B v G D Q W D G @ D U 5 SD @ 5 6

    "

    2

    G

    "

    b G

    b 1 b k

    b b

  • 8/8/2019 Mecnica Fsica-Classica

    28/208

    8 @ B i 6

    0

    b

    v

    G

    "

    0

    "

    0

    b

    D U W W i 6

    "

    y

    B

    b 1 b

    4 @ @ 6 W i @ U B i 6

    8 D u v @

    "

    8 @ B i 6 5 Q @ U 8 @ Q G i b 1 b

    "

    W i 6 b

    b 1 b

    i v D 5 Q G D 6 v 6 8 5 8 v 5 Q G i b 1 b b @ b 1 b k @C B b 1 b

    0

    2

    C

    2

    '

    W i 6 b

    B

    b 1 b

    @ u v D W D i b 1 b U @ U @ 6 @ Q 8 D v B D @ 5 6 @ W i B D i U 5 @ B G @ W i i U G @ Q D G D 6 Q i W @ Q 8 U i

    G D @ 5 6 @ 1

    # % !

    % # ! # $ % ( 1 2 4 $ ( $ B C $ B $ E 2 5 F ( % $ $ % B %

    ( $ $ % B % C 2 B S C ( ( % B ! % 2 b 2 P 4 $ $ % B %

    4 $ 1 e g $ ( 4 $ 1 C % $ $ i C 2 % $ $ ( ) p r p t 3 r

    s i Q W v S

    B i 6 u v @ v B D D U 8 S

    W v D 6 v 3 @ 5 8 D D i i 8 @ Q W 5 D -

    G @ 6 W U @ @ v B D

    Si U 5 8 D @ S

    8 5 W D W i B i U 5 @ B G @ W i i U G @ Q D G D 6 Q i W @ Q 8 U i 5 1 @ 1 i W @ Q 8 U i G @ i U W D 6 6 5 8 v D j

    6 @ Q i W @ Q 8 U i G D Si U 5 8 D 1 @ b 1 b 8 D B S@ B 6 @ i 8 S@ B i 5 B 5 8 @ 5 Q @ U 5 i U G D @ Q @ U 5 D

    G D D U 8 S

    W v DI 1 n B i 5 B @ Q 8 i 6 Si 6 @ i G @ U @ D 5 D U 6 @

    2

    '

    E i v 2 E

    '

    b 1 b {

    5 6 8 i u v @ i U D G 5 W D Q G i i 8 5 G i @ B b 1 b G @ @ U SD 6 @ U i 6 5 8 5 i 1

    0 u 1 ' " 2 " 5 1 )" v 2 " $ x

    1 5 1 $ " $

    $ " )5 1 "

    D 6 @ u v D W i @ 6 b 1 { @ b 1 0 i 5 i 6 6 S

    @ G @ 8 @ U B 5 Q D U D Si U 5 8 D G @ v B D D U 8 S

    W v D

    6 v 3 @ 5 8 D D v B W D B i G @ i U W D 6 W @ Q 8 U D 5 6 W i Q 6 @ U D 8 5 i 1 f i @ Q 8 D Q 8 i D @ u v D W D i b 1 0

    6 Si i U 6 5 W i Q 8 S@ B B v 5 8 D 5 Q i U B D W D i 6 i U @ i 8 5 i G @ Si U 5 8 D 6 u v @ D D U 8 S

    W v D i G @

    G @ 6 W U @ @ U 1 D Q SD 5 6 @ G i 6 U SD W i 6 G D @ Q @ U 5 D i 8 @ Q W 5 D @ @ W 8 5 D G D D U 8 S

    W v D

    C

    b

    b 1 b

    @ B v Q W D i G D W i i U G @ Q D G D S@ 6 v W 5 @ Q 8 @ D U D W D 6 6 5 W D U i B i 5 B @ Q 8 i G D D U 8 S

    W v D

    G @ D W i U G i W i B D 6 v D @ Q @ U 5 D 1

    6 8 v G D U @ B i 6 i 6 6 @ v 5 Q 8 @ 6 W D 6 i 6

    '

    y

    '

    $

    '

    b

    '

    B

    b 1 q

    b q

  • 8/8/2019 Mecnica Fsica-Classica

    29/208

    B W D G D W D 6 i U @ U @ 6 @ Q 8 D U @ B i 6 D v Q W D i @ B v Q W D i G @ @ @ 6 8 v G D U @ B i 6 D

    i 6 6 5 5 5 G D G @ G D D U 8 S

    W v D 8 @ U G 5 @ U @ Q 8 @ 6 D i U @ 6 G @ @ Q @ U 5 D 2 1 n B i 5 B @ Q 8 i 6 Si

    6 @ U SD i 6 6 S

    @ 6 @

    2 E b 1 q 0

    @ U 5 W D Q G i j 6 @ i 6 5 Q D G @ 5 v D 6 @ D D U W @ D

    D @ Q @ U 5 D W 5 Q S@ 8 5 W D U D G 5 D i U Q v D 1

    a & Q $ 7 5 ST 2 P d

    r

    E

    E

    E

    E

    - k/r

    L / 2mr

    r r r rr

    1

    2

    3

    4

    5

    22

    1 3 0 4

    2E = 0

    Uef(r)

    O

    5 v U D b 1 q ! # % % 5 % # ' ! # ' ) ! 3 # %

    -

    v Q W D i U @ U @ 6 @ Q 8 D G D Q D v U D b 1 q S@ i 6 5 8 5 D u v D Q G i A i U u v @

    y

    b

    y

    y

    '

    y 6 @ A

    B

    b 1 q b

    f i 5 B 5 8 @ A T

    y

    '

    y

    y

    b

    y A T b 1 q q

    @ i U 8 D Q 8 i S@ Q @ D 8 5 i 1 n B S

    Q 5 B i G @ S@ G @ 8 @ U B 5 Q D G i B 5 Q 5 B 5 D Q G i

    @ B U @ D W D i D

    G

    $

    C

    '

    B

    b 1 q k

    b k

  • 8/8/2019 Mecnica Fsica-Classica

    30/208

    n B S

    Q 5 B i @ U 5 W D j 6 @ D U D

    '

    '

    b

    B

    b 1 q

    S

    5 B i U 8 D Q 8 @ 6 D 5 @ Q 8 D U u v @ u v D Q G i i i 8 @ Q W 5 D @ @ W 8 5 i U @ G v j 6 @ D i i j

    8 @ Q W 5 D @ D D U 8 S

    W v D Q v Q W D 6 @ Q 8 @ i @ @ 5 8 i U @ v 6 5 i u v D Q G i 6 @

    D U i 5 B D G i W @ Q 8 U i G @ i U W D 6

    1

    s i Q 6 5 G @ U @ B i 6 i 6 G 5 @ U @ Q 8 @ 6 W D 6 i 6 5 Q G 5 W D G i 6 Q D v U D

    5 2 2

    !

    D @ Q @ U 5 D S@ i 6 5 8 5 D 1 D U 8 S

    W v D 8 @ B v B D Si U 5 8 D 5 Q Q 5 8 D 1

    D U 8 S

    W v D i G @ U SD 5 U G i 5 Q Q 5 8 i Q i 6 @ Q 8 5 G i G @ G @ W U @ 6 W @ Q 8 @ Q i i Q 8 i

    D 6 v D @ i W 5 G D G @ D Q v D j 6 @ i Q 8 i G @ U @ 8 i U Q i @ D D U 8 S

    W v D W i Q 8 5 Q v D U SD i 6 @ v

    B i 5 B @ Q 8 i Q i 6 @ Q 8 5 G i G @ W U @ 6 W @ Q 8 @ D 8 S@ D i 5 Q Q 5 8 i 1

    !

    S@ D W i i U G @ Q D G D

    U D G 5 D G i i Q 8 i G @ D U i 5 B D W D i B SD 5 B D G i W @ Q 8 U i G @ i U W D 6 1

    5 5 2 2

    S@ v B W D 6 i 6 @ B @ D Q 8 @ D i W D 6 i @ 6 8 v G D G i @ B 5 1

    5 5 5 2 2

    $

    D D U 8 S

    W v D 8 @ B v B D Si U 5 8 D 5 B 5 8 D G D @ Q 8 U @ i 6 i Q 8 i 6 G @

    W i i U G @ Q D G D 6 U D G 5 D

    $

    @

    1 n 6 i Q 8 i 6

    $

    @

    6 D i i Q 8 i 6 G @

    U @ 8 i U Q i U @ 6 @ W 8 5 D B @ Q 8 @ i 6 i Q 8 i 6 G @ D U i 5 B D W D i B SD 5 B D @ D D 6 8 D B @ Q 8 i

    B SD 5 B i G i W @ Q 8 U i G @ i U W D 6 1

    5 2 2

    D D U 8 S

    W v D 8 @ B v B D Si U 5 8 D W 5 U W v D U W i B

    1 f i 8 D U u v @ Q @ 6 8 D 6

    W i Q G 5 W i @ 6 D @ i W 5 G D G @ U D G 5 D

    S@ Q v D 1 D U 8 S

    W v D D @ Q D 6 8 @ B @ i W 5 G D G @

    D Q v D U

    1 n i Q 8 i

    i 8 S@ B j 6 @ B 5 Q 5 B 5 D Q G i @ B i U G @ B D

    G

    G

    B

    @ 6 8 D @ u v D W D i i 8 @ B i 6

    G

    G

    $

    b 1 q

    W i B

    G

    G

    @

    1 @ u v D W D i b 1 q @ U 5 B @ i D W 8 i

    G @ i W D B i G @ i U W D 6 8 @ U G @ @ @ U W @ U v B D i U W D 6 i U @ D D U 8 S

    W v D u v @ @

    8 U D Q 6 B 5 8 D D D W @ @ U D W D i W @ Q 8 U S

    @ 8 D Q @ W @ 6 6 SD U 5 D D U D D D U 8 S

    W v D 6 @ B D Q 8 @ U Q D

    Si U 5 8 D

    1

    2 2 i B i 5 B @ Q 8 i Q D i S@ i 6 6 S

    @ i U u v @ 2

    1 n B i 5 B @ Q 8 i

    6 Si S@ i 6 6 S

    @ D U D2

    E2

    S

    G @ Q i 8 D U u v @ @ B 8 i G i 6 i 6 W D 6 i 6 i G @ B i 6 G @ 8 @ U B 5 Q D U D @ Q @ U 5 D W 5 Q S@ 8 5 W D U D G 5 D

    b W D W v D Q G i D G 5 6 8 xD Q W 5 D G D U @ W 8 D 2 2

    @D W v U D 1

    f D v U D b 1 k @ 6 8 SD U @ U @ 6 @ Q 8 D G D v B D Si U 5 8 D 5 B 5 8 D G D 2 2

    $

    6 @ Q G i

    !

    @

    U @ 6 @ W 8 5 D B @ Q 8 @ i 6 i Q 8 i 6 G @ D U i 5 B D W D i B SD 5 B D @ D D 6 8 D B @ Q 8 i B SD 5 B i G i

    b

  • 8/8/2019 Mecnica Fsica-Classica

    31/208

    W @ Q 8 U i G @ i U W D 6 1

    S

    Q i @ Q 8 D Q 8 i 5 B i U 8 D Q 8 @ Q i 8 D U u v @ D i U B D G D Si U 5 8 D 6 Si i G @

    6 @ U G @ 8 @ U B 5 Q D G D 5 Q 8 @ U D Q G i D @ u v D W D i G D Si U 5 8 D b 1 0 " 1 i U SD W i G @ 6 Si

    i G @ B i 6 W i Q W v 5 U u v @ i B i 5 B @ Q 8 i G D D U 8 S

    W v D W i B @ Q @ U 5 D 2

    $

    @ 6 8 SD W i Q Q D G i

    D i @ 6 D W i 5 B 5 8 D G i @ D 6 W 5 U W v Q @ U x@ Q W 5 D 6

    !

    @

    1 G 5 6 W v 6 6 D i u v @ i 5 @ 5 8 D

    r r1 2

    5 v U D b 1 k

    !

    # % C #

    $

    D U D i i 8 @ Q W 5 D 2 $ - S@ SD 5 G D D U D u v D u v @ U i 8 @ Q W 5 D G @ 6 G @ u v @

    D

    A T y y A B D 5 6 G @ D D U u v @ 0

    1

    D

    A A T B D 5 6 G @ D D U u v @ 0

    1

    D

    f i W D 6 i D U 8 5 W v D U G @ -

    D D U 8 S

    W v D 8 @ U SD v B D Si U 5 8 D

    5 5 @ U Si 5 W D 6 @ 2 2

    !

    5 5 D U D Si 5 W D 6 @ 2 2

    5 5 5 @ S

    8 5 W D 6 @ 2 2

    $

    5 W 5 U W v D U 6 @ 2 2

    1

    # % !

    % # C ( 2 B 4 C 2 % 4 B C ! # $ F b % B F9 2 P 4 $ C B 2 1 S C 2 5

    b$ i 2 C 4 $ b

    2 3 % ( 4 C b

    1 2 5 C $ 4 $ C B 2 P 2 3 # ! $ ( " $ r

    a & Q $ 7 5 ST 2 P d

    8

    @ U @ 6 @ Q 8 D B i 6 Q D v U D b 1 i i 8 @ Q W 5 D @ @ W 8 5 i W i U U @ 6 i Q G @ Q 8 @ D i i 8 @ Q W 5 D

    -

    '

    1 n U @ 6 @ Q 8 @ @ 6 8 v G i S@ @ u v 5 D @ Q 8 @ D i @ 6 8 v G i u v @ 6 @ U 5 D @ 5 8 i 6 @

    5 y y A B D 5 6 G @ U @ 6 6 D u v @ 0

    u v D Q G i A T 1

    5 5 A T B D 5 6 G @ U @ 6 6 D u v @ 0

    u v D Q G i A 1

    b

  • 8/8/2019 Mecnica Fsica-Classica

    32/208

    - k/r

    L / 2mr2

    2

    3

    r

    E

    E

    E

    E = 0

    E

    rr

    1

    2

    3

    4

    5

    2 0 3O r

    r1

    Uef(r)

    5 v U D b 1 ! # % % 5 % # ' ! # ' ) ! 3 # %

    -

    W i Q G 5 W D i 5 5 B 5 W D u v @ A i U D i U @ 6 i 6 5 8 5 i 6 u v D Q G i A T i U u v @

    i 8 @ U B i

    S@ B D 5 i U u v @ y

    '

    $ y u v D Q G i A 1 D W i Q G 5 W D i 5 5 U @ 6 v 8 D u v @

    A T u v D Q G i A i 5 6 D D U W @ D

    '

    $ S@ B D 5 i U @ B D i U D 6 i v 8 i

    u v @

    b

    Q @ 6 8 @ W D 6 i 1 S@ B G 5 6 6 i 5 6 8 i u v @ U 5 B @ 5 U i W U @ 6 W @ @ G @ i 5 6

    G @ W U @ 6 W @ D v Q W D i 8 @ B v B B SD 5 B i

    G

    $

    C

    q

    '

    q

    '

    B

    6 8 v G D U @ B i 6 6 @ v 5 G D B @ Q 8 @ D 6 U i U 5 @ G D G @ 6 G D 6 G 5 @ U @ Q 8 @ 6 Si U 5 8 D 6 u v @ D D U j

    8 S

    W v D 8 @ B W i Q 6 i D Q 8 @ D 6 v D @ Q @ U 5 D

    5 2 2

    !

    Si U 5 8 D 5 Q Q 5 8 D Q D i 5 B 5 8 D G D 1 D U 8 S

    W v D i G @ G @ 6 i W D U j 6 @ G @ 6 G @

    D 8 S@ T 1

    5 5 2 2

    6 @ D D U 8 S

    W v D @ 6 8 SD @D G 5 6 8 xD Q W 5 D

    @ D B D Q 8 S@ B j 6 @ 6 @ B U @ D

    @ 6 8 D G 5 6 8 xD Q W 5 D G i W @ Q 8 U i G @ i U W D 6 G @ 6 W U @ @ Q G i v B D Si U 5 8 D W 5 U W v D U 1 6 v D

    @ i W 5 G D G @ U D G 5 D S@ Q v D 1 4 @ Q i @ Q 8 D Q 8 i D D U 8 S

    W v D Q D i @ 6 8 SD @D G 5 6 8 xD Q W 5 D

    G D i U 5 @ B D i G 5 @ U @ Q 8 @ 6 v W @ G @ 1 4 @ @ D @ 6 8 SD D

    G @

    @ 6 @

    D D 6 8 D G i W @ Q 8 U i G @ i U W D 6 @ D W i Q 8 5 Q v D U SD D D D 6 8 D U j 6 @ D 8 S@ D i 5 Q Q 5 8 i 1 f i

    @ Q 8 D Q 8 i 6 @ @ D 6 @ D U i 5 B D G v B i Q 8 i W i B

    i i u v @ @ D D 8 5 Q 3 D @ 6 8 @

    i Q 8 i @ D W D U SD D W 5 U W v D U @ B 8 i U Q i G @

    @D G 5 6 8 xD Q W 5 D

    i U u v @ Q @ 6 8 @ i Q 8 i

    b

  • 8/8/2019 Mecnica Fsica-Classica

    33/208

    D @ i W 5 G D G @ U D G 5 D S@ Q v D @ D i U W D @ @ W 8 5 D u v @ D W 8 v D Q D G 5 U @ W W D i U D G 5 D

    8 D B S@ B S@ Q v D 1 f D i @ i G @ U SD 6 @ U 8 U D Q 6 B 5 8 5 G D v B D D W @ @ U D W D i U D G 5 D u v @

    D i U 5 v @ D D D 6 8 D U j 6 @ G @

    1 4 @ @ D @ 6 8 SD D

    G @

    @ D W D U SD Q @ 6 8 D

    U @ 5 D i D 8 S@ D 8 5 Q 5 U i i Q 8 i

    B D Q 8 @ Q G i j 6 @ D W 5 U W v D U @ B 8 i U Q i G @

    D

    v B D G 5 6 8 xD Q W 5 D

    5 Q G @ Q 5 G D B @ Q 8 @ 1

    5 5 5 2 2

    $

    i B i 5 B @ Q 8 i 6 Si S@ i 6 6 S

    @ 6 @

    @

    $

    1 4 @

    i

    B i 5 B @ Q 8 i S@ 5 B 5 8 D G i

    1 4 @

    $

    D Si U 5 8 D S@ 5 Q Q 5 8 D E

    $

    6 @ Q G i 5 B 5 8 D G D 5 Q @ U 5 i U B @ Q 8 @ 1

    5 2 2

    6 Si S@ i 6 6 S

    @ B i 5 B @ Q 8 i Q D U @ 5 D i E E

    !

    Si U 5 8 D 5 B 5 8 D G D 1

    2 2

    6 5 8 v D W D i @ u v 5 D @ Q 8 @ D 5 1

    s i Q W v S

    B i 6 u v @ D i W i Q 8 U SD U 5 i G i @ @ B i D Q 8 @ U 5 i U i B i 5 B @ Q 8 i G D D U 8 S

    W v D S@

    6 @ B U @ i 6 6 S

    @ u v D u v @ U u v @ 6 @ 3 D D @ Q @ U 5 D G D D U 8 S

    W v D 1

    a & Q $ 7 5 ST 2 P d

    6

    6 8 @ S@ v B W D 6 i D 6 8 D Q 8 @ G 5 @ U @ Q 8 @ G i 6 i v 8 U i 6 G i 5 6 D Q 8 @ U 5 i U @ 6 Q D i 5 U @ B i 6 @ Q W i Q 8 U D U

    Si U 5 8 D 6 5 5 B 5 8 D G D 6 1 @ U @ 6 @ Q 8 D B i 6 Q D v U D b 1 i i 8 @ Q W 5 D @ @ W 8 5 i W i U U @ 6 i Q j

    G @ Q 8 @ D i i 8 @ Q W 5 D F -

    '

    1 v D Q G i A D @ Q @ U 5 D i 8 @ Q W 5 D

    Uef(r)

    r

    E

    E

    r r rO

    kr /2

    L /2mr 22

    1

    0

    1 0 2

    2

    5 v U D b 1 ! # % % 51 % # ' ! # ' ) ! 3 #

    -

    W @ Q 8 U S

    v D U @ D @ W @

    b

    A T @ Q u v D Q 8 i u v @

    '

    b A 1 ' @ i W i Q 8 U SD U 5 i

    b {

  • 8/8/2019 Mecnica Fsica-Classica

    34/208

    u v D Q G i A T

    '

    b A T @

    b

    A 1 W v U D G D @ Q @ U 5 D i 8 @ Q W 5 D

    @ @ W 8 5 D 8 @ B v B B S

    Q 5 B i D U D

    8 D u v @

    G

    '

    '

    B

    4 @ Q G i D @ Q @ U 5 D W 5 Q S@ 8 5 W D G D D U 8 S

    W v D v B D u v D Q 8 5 G D G @ i 6 5 8 5 D i G @ Q G i u v D Q G i

    B v 5 8 i 6 @ U Q v D D @ Q @ U 5 D B @ W xD Q 5 W D 2

    b C Q v Q W D i G @ U SD 6 @ U 5 Q @ U 5 i U

    D

    i D i U B S

    Q 5 B i G @ 1 Q D 5 6 D U @ B i 6 @ B 6 @ v 5 G D D v B D 6 G D 6 U i U 5 j

    @ G D G @ 6 G D Si U 5 8 D G @ 6 W U 5 8 D i U v B D D U 8 S

    W v D W i B @ Q @ U 5 D B @ W xD Q 5 W D 2

    @ 2

    D

    2

    D D U 8 S

    W v D G @ 6 W U @ @ v B D Si U 5 8 D W 5 U W v D U G @ U D 5 i

    1 n W @ Q 8 U i

    G @ i U W D 6 S@ i W @ Q 8 U i G D Si U 5 8 D @ D @ i W 5 G D G @ U D G 5 D G D D U 8 S

    W v D

    S@ Q v D 1

    D

    2

    !

    D D U 8 S

    W v D G @ 6 W U @ @ v B D Si U 5 8 D 5 B 5 8 D G D 6 @ Q G i

    !

    @

    U @ 6 j

    @ W 8 5 D B @ Q 8 @ D 6 G 5 6 8 xD Q W 5 D G @ D U i 5 B D W D i B SD 5 B D @ G @ B D 5 i U D D 6 8 D B @ Q 8 i 1

    f @ 6 8 @ 6 i Q 8 i 6 D @ i W 5 G D G @ U D G 5 D D Q v D j 6 @ 1 D U 8 S

    W v D 6 Si i G @ U SD 6 @ U @ Q j

    W i Q 8 U D G D Q v B D U @ 5 D i G i @ 6 D W i W i B U @ @ Q G 5 G D @ Q 8 U @ D 6 W 5 U W v Q @ U x@ Q W 5 D 6 G @

    U D 5 i

    !

    @

    1 i U SD W i Q D i i G @ U @ B i 6 8 5 U D U B D 5 6 Q @ Q v B D 5 Q j

    i U B D W D i 6 i U @ D i U B D G D Si U 5 8 D u v @ D D U 8 S

    W v D G @ 6 W U @ @ 1

    5 B i 6 Q i

    @ Q 8 D Q 8 i Q i D U SD U D i D Q 8 @ U 5 i U u v @ D @ u v D W D i G @ Si U 5 8 D @ u 1 b 1 b U @ U @ j

    6 @ Q 8 D v B D @ 5 6 @ @ 6 8 D Q G i i W @ Q 8 U i G @ i U W D 6 Q i W @ Q 8 U i G D @ 5 6 @ 1 6 6 5 B

    !

    S@ i 6 @ B 5 j @ 5 i B @ Q i U G D @ 5 6 @ @

    i 6 @ B 5 j @ 5 i B D 5 i U 1

    6 Si U 5 8 D 6 G @ 6 W U 5 8 D 6 i U D U 8 S

    W v D 6 6 v 3 @ 5 8 D 6 D i i 8 @ Q W 5 D D U B Si Q 5 W i @ 6 S@ U 5 W i 6 D i

    8 i G D 6 5 B 5 8 D G D 6 u v D u v @ U u v @ 6 @ 3 D D @ Q @ U 5 D G D 6 D U 8 S

    W v D 6 @ D @ Q D 6 u v D Q G i D

    D U 8 S

    W v D 8 5 @ U i B i B @ Q 8 i D Q v D U Q v i

    i G @ U SD D 6 6 D U @ i W @ Q 8 U i G @

    i U W D 6 G @ 5 G i @D D v 6 x@ Q W 5 D G D D U U @ 5 U D G i i 8 @ Q W 5 D W @ Q 8 U S

    v i 1

    0

    " 1 1 ( 1 " $ 5 ! ' 2 4 " 2 4 ' 1 )

    5

    b b

    B W @ U 8 D 6 W i Q G 5 W i @ 6 D D U U @ 5 U D G @ i 8 @ Q W 5 D W @ Q 8 U S

    v i 5 B @ G @ D D U i 5 B D W D i G D 6

    D U 8 S

    W v D 6 G i W @ Q 8 U i G @ i U W D 6 1 @ 8 @ U B 5 Q @ B i 6 @ 6 6 D 6 W i Q G 5 W i @ 6 1 @ Q @ U 5 D W 5 Q S@ 8 5 W D

    U D G 5 D S@ i 6 5 8 5 D

    0

    b

    2 E

    i i 2 E C

    i v D 5 Q G D

    2

    b

    B

    b

  • 8/8/2019 Mecnica Fsica-Classica

    35/208

    s i Q 6 5 G @ U @ B i 6 i 5 B 5 8 @ A D D U 8 S

    W v D D U i 5 B D j 6 @ G i W @ Q 8 U i G @ i U W D 6

    5 B

    b

    b 1 q

    @ 6 @ 3 D

    1

    D

    4 @

    b 5 B

    T b 1 q S@ @ U 5 W D G D T

    b 1

    D

    4 @

    b 5 B

    b 1 q 6 Si S@ i 6 6 S

    @ 6 @ 1

    D

    4 @ b 5 B

    b 1 q 6 Si S@ i 6 6 S

    @ 6 @ E

    b 1

    0

    )

    1

    " ( " $ " )5

    1

    "

    2 " $ "

    5 B i 6 u v @ @ B W @ U 8 D 6 W i Q G 5 W i @ 6 D D U 8 S

    W v D 6 v 3 @ 5 8 D D v B W D B i G @ i U W D 6 W @ Q j

    8 U D 5 6 B i @ j 6 @ Q v B D Si U 5 8 D 5 B 5 8 D G D 5 1 @ 1 D G 5 6 8 xD Q W 5 D G D D U 8 S

    W v D D i W @ Q 8 U i

    G @ i U W D 6 @ U 8 @ Q W @ D i 5 Q 8 @ U D i I

    Q

    6 @ Q G i

    @

    U @ 6 @ W 8 5 D B @ Q 8 @ D 6

    G 5 6 8 xD Q W 5 D 6 G @ D U i 5 B D W D i B SD 5 B D @ D D 6 8 D B @ Q 8 i B SD 5 B i G i W @ Q 8 U i G @ i U W D 6

    @

    6 D i i Q 8 i 6 G @ U @ 8 i U Q i 1 D U 8 S

    W v D B i @ j 6 @ @ Q 8 U @ G v D 6 W 5 U W v Q @ U x@ Q W 5 D 6

    W i Q W x@ Q 8 U 5 W D 6 G @ U D 5 i 6

    @

    1 Si U 5 8 D S@ 8 D Q @ Q 8 @ D @ 6 8 D 6 G v D 6 W 5 U W v Q @ U x@ Q W 5 D 6 1

    f D v U D b 1 @ 6 8 SD U @ U @ 6 @ Q 8 D G D D Si U 5 8 D G @ v B D D U 8 S

    W v D W i Q Q D G D @D U @ 5 D i G i

    rmin

    rmaxD

    A

    C

    B

    O

    D

    5 v U D b 1

    !

    # 7 ! # !

    % 7 % % # # A # #

    ! 9 % # !

    !! # !

    @ 6 D W i

    1 s i Q 6 5 G @ U @ B i 6 i B i 5 B @ Q 8 i G D D U 8 S

    W v D G @ i 5 6 G @ 8 @ U

    D W D Q W D G i i i Q 8 i 1 i @ j 6 @ 6 @ v 5 G D B @ Q 8 @ Q i 6 @ Q 8 5 G i G D 6 @ 8 D D 6 6 D Q G i 8 D Q j

    @ Q 8 @ @D W 5 U W v Q @ U x@ Q W 5 D 5 Q 8 @ U 5 i U Q i i Q 8 i $ @ Q i D B @ Q 8 @ 8 D Q @ Q 8 @ @D W 5 U W v Q @ U x@ Q W 5 D

    @ 8 @ U 5 i U Q i i Q 8 i % 1 4 @ 3 D

    i 8 @ B i u v @ D D U 8 S

    W v D G @ B i U D D @ @ W 8 v D U @ 6 8 @

    q

  • 8/8/2019 Mecnica Fsica-Classica

    36/208

    B i 5 B @ Q 8 i 1

    S@ i @ U S

    i G i U D G 5 D 1 ' i U i v 8 U i D G i i U D 5 i @ W 8 i U B v G D W i Q 6 8 D Q 8 @ j

    B @ Q 8 @ G @ G 5 U @ W W D i Q i B @ 6 B i 6 @ Q 8 5 G i 1 4 @ 3 D i @ U S

    i G i D Q v D U i 8 @ B i u v @ @ @

    G @ B i U D D @ U W i U U @ U v B xD Q v i b i U @ @ B i D 5 U G @ D

    1

    U @ U @ 6 @ Q 8 D i

    U D 5 i @ W 8 i U Q v B 5 Q 6 8 D Q 8 @ @

    i U D 5 i @ W 8 i U Q i 5 Q 6 8 D Q 8 @ C 1 4 @ i 6 @ U S

    i G i 6

    @ 6 D i W i B @ Q 6 v U SD @ 5 6 5 1 @ 1

    6 @ Q G i @ G i 5 6 Q Sv B @ U i 6 5 Q 8 @ 5 U i 6

    D Si U 5 8 D S@ @ W D G D i 5 6 D i B G @ v B W @ U 8 i @ U S

    i G i G @ 8 @ B i 5 v D D i B @ Q i U

    B Sv 8 5 i W i B v B G @ @

    D D U 8 S

    W v D i 8 D @D i 6 5 W D i 5 Q 5 W 5 D 1 4 @ @

    Q D i

    6 D i W i B @ Q 6 v U SD @ 5 6 i U D 5 i @ W 8 i U @ U W i U U @ U SD 8 i G i i @ 6 D W i @ Q 8 U @

    @

    @ D

    Si U 5 8 D Q v Q W D @ W D U SD 1

    6 Si U 5 8 D 6 @ W D G D 6 6 D i U D U D 6 Q i @ Q 8 D Q 8 i Q i W D 6 i G i 6 i 8 @ Q W 5 D 5 6

    '

    @ -

    8 i G D 6 D 6 Si U 5 8 D 6 Q 5 8 D 6 5 B 5 8 D G D 6 6 D i @ W D G D 6 1

    0 1 ' " 2 " 5 $ x

    2 5 ' ! 2 4 $ " " )5 1 "

    s i Q 6 5 G @ U @ B i 6 v B D D U 8 S

    W v D 6 v 3 @ 5 8 D D v B W D B i G @ i U W D 6 W @ Q 8 U D 5 6 u v @ G @ 6 W U @ @

    v B D Si U 5 8 D G @ Q 5 G D @ D v Q W D i 1 6 v D @ Q @ U 5 D B @ W xD Q 5 W D S@ G D G D i U

    2

    0

    b

    C

    b

    C

    B

    b 1 q {

    D 6 G @ @ Q G @ G i 8 @ B i D 8 U D S@ 6 G @ i i

    G

    G

    G

    G

    G

    G

    G

    B

    4 v 6 8 5 8 v 5 Q G i @ B b 1 q { i 8 @ B i 6

    2

    b

    "

    $ %

    '

    $

    )

    % 1 2 4 5 7 9 A C

    E F F I Q S U W X Y ` U I

    ' b b e g

    X iU q

    b

    W

    $

    4

    )

    C

    S s W

    '

    ` Q W F U

    b u

    U

    g u b w

    ' x

    ` F `

    g

    I

    ' b

    Q ` X `

    4

    $

    C 7

    # ( 4 C b

    1 % b$ i6 2 5 C % B 2 % 4 1 ! # $

    x

    # $

    B % $ j $ B C B C r C () 2 P B m n ! r

    z | ~ | ~

    | | ~

    |

    I Q W F

    g

    ` S F ` W s W ` X U I

    b u

    `

    e

    ` U I W

    %

    x

    "

    4

    $

    C

    b

    s

    w

    ` F ` X

    b w g

    Q Q

    X I Q W W

    g

    g

    Q Q

    b

    I Q W

    b u

    g

    `

    ' x

    x x

    S

    b

    s

    b w

    U

    g u b '

    ` F U

    w

    ` `

    g

    Q

    b

    W

    w -

    I

    b

    U I

    w

    U

    g u b w

    S ` S

    s W

    w

    b

    X W S

    g

    Q

    b

    W

    w -

    I

    b e

    ` U Y

    b ' b

    S

    g b

    X

    '

    W ` F

    b b '

    I F

    b

    X U I

    b

    $

    '

    W U ` X

    w

    W

    '

    `

    e

    W

    w

    iU

    b

    F

    7

    X U W X

    w

    b

    Q W F

    g

    Q Q

    X I Q W s

    b w b

    4

    $

    C

    $

    `

    g

    Q Q

    b

    I Q W X W U

    b

    F W

    4

    $

    C

    $

    7

    g b u b '

    I

    e

    `

    w

    ` X iU

    b

    ` X

    w

    ` ` F

    b

    F

    ' g b

    F F I

    g b

    iU qW ` F W Q s

    b w b

    X

    '

    W

    b

    F

    g w b

    F

    5 7

    9

  • 8/8/2019 Mecnica Fsica-Classica

    37/208

    UefUef

    r

    r

    a) b)

    E

    E

    EE

    E

    r r rr r r

    2

    0

    1

    1

    0

    1 0 2

    1 0 2

    I

    g w b

    5 7

    m n

    m n

    b

    C

    `

    5 7

    -

    C

    U W X U

    u g

    Q W F W F `

    g

    I X `

    `

    b

    F s

    b w

    U

    g u b

    F ` Q ` X `

    w

    I

    b "

    ` X q

    b

    W `

    u b

    F

    '

    ` F U

    w

    ` ` Q

    g

    Q

    b

    U I

    w

    U

    g

    X

    e

    `

    w

    ` X U I

    b '

    `

    w b

    I W

    $

    7

    ` S X W U

    b

    F W

    b

    C

    S

    b

    ` X `

    w

    I

    b

    `

    u

    ` ` Q ` X `

    '

    I Q I X

    g

    ' b

    W Q W I Q ` X W

    '

    ` I

    b '

    ` F `

    w

    s W F F

    `

    u

    S F `

    e

    W

    w b g

    Q ` X

    b ' b g

    Q s W

    g

    U W

    b

    s

    b w

    U

    g u b

    U W X I X

    g b b '

    ` F U

    w

    ` `

    w g

    Q

    b

    W

    w -

    I

    b u

    I Q I

    b ' b

    7 $

    ` I

    b

    S X W ` X

    b

    X W S

    '

    ` F `

    w

    U I

    w

    U

    g u b w

    S F ` X

    '

    W

    u

    I Q I

    b ' b

    s `

    u b

    F U I

    w

    U

    g

    X

    e

    `

    w

    ` X U I

    b

    F

    $

    $ &

    `

    $

    $

    %

    7 (

    W U

    b

    F W

    -

    C

    S F `

    b

    ` X `

    w

    I

    b

    `

    '

    I Q I X

    g

    ' b g

    Q s W

    g

    U W

    b

    s

    b w

    U

    g u b

    s W

    '

    `

    '

    ` F U

    w

    ` `

    w g

    Q

    b

    W

    w -

    I

    b

    I X X I

    b

    U W Q

    $ 0 $

    %

    W

    g g

    Q

    b

    W

    w -

    I

    b u

    I Q I

    b ' b

    b u

    g

    ` 2

    $

    2

    $

    7 4

    Q W I Q ` X W

    ' b

    s

    b w

    U

    g u b

    ` Q

    U

    b w b

    U `

    w

    F I U

    b

    F W

    b u

    Q ` X `

    '

    I

    e

    `

    w

    ` X ` F

    ' b

    F I

    g b

    iU q

    b

    W

    "

    "

    7 5

    W

    '

    `

    w

    I

    b

    Q W F

    b

    Q

    -

    ` Q

    U W X F I

    '

    `

    w b w

    W U

    b

    F W ` Q

    g

    ` W

    "

    "

    % 7

    "

    7 (

    ` F ` U

    b

    F W

    b

    s

    b w

    U

    g u b

    s W

    '

    ` s

    b

    F F

    b w

    s W

    w

    g b u

    g

    `

    w

    s W X W

    '

    W ` F s

    b

    iU W

    7 @

    ` Q W F S s W

    w

    b

    X W S

    '

    W I F U

    b

    F W F W

    b u

    Q ` X `

    '

    I

    e

    `

    w

    ` X ` F

    7

    (

    W U

    b

    F W

    b

    C

    S F `

    b g

    Q ` X

    b w

    Q W F

    u

    ` ` Q ` X `

    b

    ` X `

    w

    I

    b

    S W

    -

    ` Q W F

    g

    Q

    b

    W

    w -

    I

    b u

    I Q I

    b ' b

    U W Q U

    b w b

    U `

    w

    F I U

    b

    F F ` Q `

    u

    Y

    b

    X ` F

    b

    W

    w -

    I

    b

    U I

    w

    U

    g u b w

    $

    $

    S W s W ` X U I

    b u

    `

    e

    ` U I W ` Q

    g

    Q Q

    X I Q W ` Q

    $

    $

    7 $

    I B ` Q W F

    g

    ` ` F

    b

    W

    w -

    I

    b

    U I

    w

    U

    g u b w

    ` ` F

    b

    `

    u

    7 (

    W U

    b

    F W

    -

    C

    S

    b

    W

    w -

    I

    b

    U I

    w

    U

    g u b w

    ` I X F

    b

    `

    u

    s W

    w

    g

    `

    b

    F U

    b w b

    U `

    w

    F I U

    b

    F

    ' b

    W

    w -

    I

    b ' b

    s

    b w

    U

    g u b

    F q

    b

    W

    W

    b u

    Q ` X `

    b u

    `

    w b ' b

    F s

    b w b " D

    "

    S ` Q

    g

    Q Q

    b

    I Q W ` Q

    $

    $

    7

    F G

    I P R T V X Y ` b d f d i q r t r f w x i r Y i d ` w r r V t V w d X ` r

    m n

    x

    V q d

    ` X w d ` f V q

    i d V X b ` k mV d Y r t r f w x i r w d q r t V Y Y ` n ` ` b r b d b d b d Y f d o d f i q r xV f n ` w r

    ` f i r f d Y w xr o d

    z z {

    (

    b

    X

    b

    g w

    ` B

    b

    ` Q W F

    ' g b

    F I Q s W

    w

    b

    X ` F

    e

    W

    w

    iU

    b

    F

    g

    `

    '

    `

    w

    I

    b

    Q

    '

    W s W ` X U I

    b u

    4

    $

    C

    $

    S

    b e

    W

    w

    iU

    b

    w b

    I U

    b

    `

    b e

    W

    w

    iU

    b

    `

    u

    ` U

    w

    I U

    b

    ` X

    w

    `

    ' g b

    F U

    b w

    b

    F

    '

    ` F I X

    b u

    W s W F W

    7

    9 5

  • 8/8/2019 Mecnica Fsica-Classica

    38/208

    Q

    b

    s

    b w

    U

    g u b

    F

    g

    ` I

    b b

    ` X `

    w

    I

    b

    s W ` X U I

    b u

    $

    ` Q

    g

    Q

    b

    ` X `

    w

    I

    b

    s W ` X

    U I

    b u

    `

    e

    ` U I

    b

    4

    $

    C

    %

    5

    $%

    $

    2

    b

    B ` Q W F W ` F

    g '

    W

    g b u

    I

    b

    I W

    ' b

    F W

    w -

    I

    b

    F

    g

    `

    b

    s

    b w

    U

    g u b

    s W

    '

    `

    '

    ` F U

    w

    ` `

    w

    ` X

    '

    W

    ` Q U W X

    b b

    F

    g b

    ` X `

    w

    I

    b

    W

    b u

    S

    b

    s

    b w

    I

    w ' b w

    ` s

    w

    ` F ` X

    b

    iU q

    b

    W

    w

    b

    U

    b '

    `

    4

    $

    C

    `

    w

    F

    g

    F

    $

    7 4

    -

    I ` Q W F

    b

    `

    g b

    iU q

    b

    W

    ' b

    W

    w -

    I

    b

    S `

    7 4 5 7 5

    C

    S X

    b

    F ` U iU q

    b

    W

    5 7 7

    "

    $ % ' ) 1 3

    5

    $ 7 % $ ' B C D 3 F C G C 7 B C %

    b

    Q W F

    b

    X

    b u

    I F

    b w b

    `

    7 4 5 7 5

    C

    $

    R

    %

    "

    T

    U W F

    4

    )

    )

    C

    U

    U W Q

    T

    V

    %

    R

    %

    %

    "

    5

    "

    U

    `

    e b

    B `

    w

    W ` F

    g '

    W

    g b

    X I

    b

    I W

    ' b

    F W

    w -

    I

    b

    F s W F F

    ` I F X W s

    w

    W

    - u

    ` Q

    b '

    ` X ` s

    u

    `

    w

    7

    b

    B ` X

    '

    W

    R

    S ` I X

    w

    W

    ' g

    B I X

    '

    W

    b

    F

    g b

    X I

    ' b '

    ` F

    `

    %

    4 5 7

    C

    ` a

    V

    "

    5

    "

    %

    %

    4 5 7

    C

    W

    -

    ` Q W F

    `

    $

    "

    a

    U W F

    4

    )

    )

    C 2 4 5 7 5 C

    E `

    g b

    iU q

    b

    W

    4 5 7 5 C

    `

    b

    `

    g b

    iU q

    b

    W

    '

    `

    g

    Q

    b

    U W X I U

    b

    7 $

    `

    4 5 7

    C

    s W

    '

    ` Q W F

    '

    ` `

    w

    Q I X

    b w

    W

    b u

    W

    w

    Q

    X I Q W s W F F

    `

    u '

    `

    "

    S

    "

    5

    "

    %

    %

    2 W

    g "

    0

    %

    5

    %

    U

    4 5 7 9 C

    U W X

    '

    I iU q

    b

    W

    b

    W

    -

    I

    ' b

    g b u

    I

    b

    I

    b

    Q ` X ` X

    b

    `

    7 4 5 7 9 c C

    "

    0

    "

    4

    $

    C

    %

    5

    $

    %

    $

    %

    5

    %

    2

    E `

    g b

    iU q

    b

    W

    4 5 7 5 C

    `

    b

    `

    g b

    iU q

    b

    W

    '

    `

    g

    Q

    b

    Y I s `

    w -

    W

    u

    ` S s

    b w

    b -

    W

    u b

    S `

    u

    I s F ` W

    g

    U I

    w

    U

    g

    X

    e

    `

    w

    ` X U I

    b

    U W X

    e

    W

    w

    Q `

    a

    F `

    b

    S

    w

    ` F s ` U I

    b

    Q ` X ` S Q

    b

    I W

    w

    g

    `

    S I

    g b u b

    S Q ` X W

    w

    g

    `

    W

    g

    I

    g b u b

    B `

    w

    W

    f

    a

    7

    W

    w -

    I

    b

    Y I s `

    w -

    W

    u

    I U