BT CoLyThuyet - Cuối Kỳ

8/11/2019 BT CoLyThuyet - Cu i K

1/173

GVHD: PGS. TS. TRNG TCH THIN

BI TP CHC

Tp Hai: NG LC HC


2/173

Chng I: NG LC HC CHT iM

BI TP 1.2: trang 31 (SGK)

Mtxe gong c khilngl 700 kg angchyxungdc

dctheo ngray thngv nghing vimtngang mtgc150. gi cho xe chy u, ta dng dy cp song song vimtdc. Vn tcchyucaxe l 1,6 m/s. Xc nh lccngcady cp lc xe chyuv lc n bhm dngli

trong 4 giy. Hscnchuynngtngcng l f = 0,015v lc cnta coi rngxe chychmdnu.


3/173

150

y

x

N

P

T

v

cF

hnh 1.2

Dng tin 2 ng lc hc im:

4

1

. 1kk

cT Na P Fm F


4/173

Trng hp 1:

Tin 2 tr thnh:

Chiu phng trnh (2) ln trc y.

Ta c:

Chiu (2) ln trc x:

Xe chuynngthngunn 0a

0 2cTP N F

0 cos

.cos . .cos

P

N P m

N

g

0 .sin

.sin . .sin . . .cos

c

c

P F

T P F m g f m g

T

. . . .coscF f N f m g


5/173

Trng hp 2:

. . sin .cos

700.9,81 sin15 0,15.cos15

1677,8

T m g f

N

Xe chuynngthngchmdnu a const

1 0.v a t v

Ta c:

1

0

0 ; 4

1, 6

v m s t s

v m s

20 1, 6 0, 4 04

va m s

t


6/173

Chiu (1) ln trc x,y:

. .sin

0 .cos

cm a P F T

NP

. . . .cosc f N f m g F

sin .cos

. .sin

.c

a g f

T m a P F

m

700 0, 4 9,81 sin15 0, 015.cos15

1957,8 N


7/173

Bi tp1:Cho: . Bqua lccncakhng kh v khilngl xo. Xc nhqui lutchuynngcavtA.

0 0, , ,k m x

k

A

O

x

0x

0

P

hnh 1

t


8/173

Bi sa

Xc nhlunggin l xo khi hcn bng(gin tnh)

Chn trc c gctivtr cn bngcavtA, cphngthngngv chiudnghngxung.

Ox

sF

P

0 1sjx FF P .sF P m g

M: .s tF k

.st

F m g

k k

Chn lc l xo b gin thm mt lng :0t s 0x

0 0

0

0

0

v m s x

x t x

hnh 1.1


9/173

Kho st chuyn ng ca vt A ti mt v tr bt k:

p dng tin 2 ng lc hc:

Chiu (2) ln trcOx:

2

1

. 2

k Skm a F P F

. . .. . .

. . 0

s

t

m x P F m g k m x m g k x k x

m x k x

2

. 0 3x x

Vi:k

m


10/173

Dng nghim tng qut:

Vyvtdao ngiuho:

0.sinx A t

A v c tm t cc iu kin ban u:0

00 0

0

0

.sin0 :

. .cos 0

. .cos 0

x x At

A t

A

xs

t

0 0cos 02

0A x

0.sin2

x x t


11/173

Bi 1.5 trang 31

Mt t chhngckhi lngl6 tn chy xung mtchic ph vi tc l21,6 km/gi. Tlc xung ph n

lc dng hn xe phi chy thm mt qung ng l10m, cho rng khi y t chuyn ng chm dn u. Tnhlc cngmi dy cp (chai dy cp) buc giph, coirng dy cp lun lun cng.

a v

2T

10s m

P

hnh 1.5


12/173

Bi sa

Kho st chuyn ng ca xe:

S dng tin 2 ng lc hc:

hnh 1.5.1

msFP

1a 1v1

N

x

y

1 1 1 1msm a P F N Chiu (1) ln :,x y


13/173

Kho st s cn bng ca ph:

1

1

1 2

0 3

ms

P

F

N

m a

13 N P

Ta c: 2 20 1

1

2

0 36 2 .10

v v a s

a

2

1 1,8 0a m s

Thay 21,8 2 :a m s vo

1 1 6000.1,8

10800

mstF m a

N


14/173

x

y

Q

1N

2.T N msF

hnh 1.5.2

*2 0 5 jx mst F T F

*

10800 54002 2

mstFT N


15/173

Chng II: NG LC HC C H.

Bi tp 2. Cho 1 thanh thng, mnh, ngchttitdinuAB. Khilngcaton thanh l m, chiudi . Thanh ABcttrcz tiimgcO v hptrcz mtgc nhhnhv. Cho bit: Hy xc nhmoment qun tnh cathanh AB ivitrcx, y,z v tm O(thanh AB nmtrong mtphngOyz).

, , , , . m OA a OB b

A

Bz

O

K

u

du

u

kd z

Hnh II.2b

x

kyd y

A

Bz

O

Hnh II.2a

x

y


16/173

Bi sa

Khi lngring cathanh:

2 2 2 2 21

. . . .sin sin

b b

k k

k a a

J m d du u u du z z

,m

kg m

Kho st mt cht im K trn thanh: Chiudi:

Khi lng im K:

du

.km du

.sin

.cos

k

ky

kx

d u

d u

d u

z

Moment qun tnh caton thanh ivitrcz:

DnghtrcOxyz sao cho thanh nmtrong Oyz.DngtrcOu c phngtrng AB v c chiunhhnh

v.


17/173

32 2 3 3

2 2 2

2 2 2

sin . sin3 3

sin3

sin

3

bm u mJ b a

a

m a b a ab b

ma ab b

z

2 2 2y1

2 2

. . . . cos

cos

b

k ky

k a

b

a

J m d du u

u du

Moment qun tnh caton thanh ivitrcy:

3


18/173

32 2 3 3

y

2 2 2

2 2 2

cos . cos3 3

cos3

cos3

bm u mJ b a

a

ma b a ab b

ma ab b

2 2 2

x

1

33 3 2 2

2 2

. . .

3 ( )( )3 3

( )3

b b

k kx

k a a

b

a

J m d du u u du

ub a a b a ab b

m

a ab b

Moment qun tnh caton thanh ivitrcx:


19/173

2 2O 12 3

x y mJ J J J a ab bz

Moment qun tnh caton thanh ivitm O:

Vy: JO= Jx.


20/173


21/173

a b

A

B

b

Hnh II.3


22/173

Bi sa

O

a

0

BC

b

AC BC

x

y

0

AC

0BC

x

0AC

x

As

ACx

BCx

AP

N

BP

/ 3a3

b

Hnh II.3.1


23/173

Dng htrc Oxy nhhnh v.

Gi:

Khi tm ca vt A, vt Bvaton h: CA, CB, C

on ng chuynngca vt A khi B chm mtphng cnh: sA

Ta x ca cc khi tm: 0 0; ; ;A B

A BC CC C

x x x x

Ta c:

0 0

2= ; = ;3 3

= ; =3 3

A B

A B

C C

C A C A

ax x b

a bx s x s a


24/173

Ban u ton h ng yn.

0

0

0

= 0

= 0= 0

A

A

A

C

C

C

mxs

vmy

s

0

0

0

= 0= 0= 0

B

B

B

C

C

C

mx sv

mys

Kho st chuyn ng ton h(2 vt).

Hngoi lc tc ng ln h:

; ;A BP P N


25/173

Dng nh l chuyn ng khi tm.

3

=1. = = ; 1

e

C j A BjMa F P P N

Chiu (1) ln trc x:

2. = 0 = 0=

C C

C

mM x x s

x const

0=C Cx x

Theo nh ngha khi tm h, ta c:


26/173

2

=1

. . .= = A B

k kA C B Ck

C

A B

m x m x m xx

M m m

0 0

:

. .. .= = 0A B A B

A BC CA C B C

A B A B

do

m x m xm x m x

m m m m

= 0Cx

0= =C C

const xx

0 0.. .=

.A BA B

A BC CC B C

A B A

A

B

mm x m x

m mm

x x

m

m


27/173

00. . = . .B

AA

BCA B A C CC B

m m mx mxx x

3. . = .

2.

33 3AA B BA A

abssm m m m

aba

= 0BAA B

ms b a

m m

V sA< 0 nn vt A chuyn ng v pha bn tri.


28/173


29/173

A

O

P

r

M

Hnh II.4


30/173

Bi gii

qtR

qtOM

A

mstF

O

0vN

P

r

M 0

0a

0s

Hnh II.4.1

x

y


31/173

a. Phntchchuynng.

Ca vnh: vnh lnkhng trt, nhanh dn, ln trn mt

nghing cnh.y l1 dng chuyn ng song phngvi tm vn tc tc thi Pl im tip xc A.

Ca tm O: chuyn ng thng theo phng ca mtnghing, nhanh dn, hng ln.

Quanh nghc.

Dovnh lnkhngtrt nn tac cc quanhsau y:

0 0 00 0 s v ar r r

Vi lgc quay, lvn tc gc, lgia tc gc ca vnh.


32/173

b.Tnhng nng h

2 2

0

1 1. .

2 2

h

OT m v J

2 2

0

.

.

O

Pm

g

PJ mr r g

v r

Vi do vat lavanh

2 2 2 2 2 21 1. . . . . .2 2

h P P P

T r r r g g g


33/173

Tnh tng cng cc ti

mstA A P A N A F A M

0, (vN NVi vuong goc vi be mat tiep xucvav iem A conh

: A

)

0mstA F

.A M M

0. . .sin .A P Ph Pr 0 0.sin . .sin h s rV

.A P.r.sin M -

0s

O

Hnh II.4.2

0h


34/173

pdngnhl bin thinng nng:

1 0

h h

T T

0h

T

A

ban au heng yen= 0

2 2. . . .sin .P r Prg M

ohm2 vtheothi gian t:

2. 2. . . .sin .P

r Prg

M


35/173

2

2. .

P.r.sing

Pr

M -

0

. .sin. .

2. .

Pra r g

Pr

M

iukinca vnh lnln:M

0 . .sin 0

. .sin

Pr

Pr

chieu achon laung

M

M


36/173

c.Sdng nguyn lyDAlembert.

Tc ng thm ln vnh 2 thnh phn cbn ca hlcqun tnh t ti O.

Vectorchnhcah lcquntnh.

.qt O

R ma

. .sin. . .2 .

qt O

qt O gg

R a

Pr

R ma r

P

P

M

. .sin

2qt

PrR

r

M


37/173

Moment chnh ca hlc qun tnh i vi tm O.

.qt

O O

M J

2

2

. .sin

. . . .2 .

qt

O

qt

O O

M

Pr

r gJ rg

P

M P

M

1

2

qt

OM P.r.sin M -

Khostscnbngcavnh:


38/173

.sin 0 1

cos 0 2

jx qt mst

jy

F P R F

F N P

Giih(1), (2) ta thuc:

.cos 3N P

.sin

. .sin.sin

2

mst qtF P R

PrP

r

M

42

mst

P.r.sinF

r

M +


39/173

d.iukin vnh lnkhngtrt:

. 5

mst msgh t

F F f N

Thay (3), (4)vo2 v ca (5) tanhnc:

. .sin

2 . .costPr

f Pr

M

. .sin . .cos2

tPr f P

r M


40/173

Bi tp 5.Cho 1 chgm 2 vtckhi lng M1vM2clin ktvachu ti nhhnh v. Hsma st trt tnh gia 2 vt

l ft,bqua ma st gia vtckhi lng M2vi sn cnh. Ban u khi chachu tc dng ca hlc F hcnbng. Tm iu kin ca gi tr lc F hai vt cngchuyn ng tnh tin thng theo phngngang nhnhau

(khng trt i vi nhau).

F

tf 1M

2M

Hnh II.5


41/173

Gi ln ltl gia tc ca C1, C2, C.1 2

, ,C C Ca a a

Bi gii

Gi C1, C2, C ln ltlkhi tm ca vt 1, vt 2 vaca

ton h.

V vt 1 khng b trt i vi vt 2 nn:

1 2C C Ca a a a

Kho st chuynngcatonh.

H ngoi lc tcng lnh:

21 , , ,FP NP Hnh II.5.1

F

tf 1

M

2M

1c

2c

c

N

a1

P

2Px

y

O


42/173


43/173


44/173

iu kin vt (1) khng trt trn vt (2):

1. 8mst msgh tF F f N

Thay (6), (7) vo (8):

1

11 2

. . .tM

F f M gM M

1 2 .tF f M M g

Bi tp 6 Cho mt a trn c ng cht c bn knh R


45/173

Bi tp 6. Cho mtatrn,c, ngchtc bn knh Rv khilng , bylnkhng trttrn mtnghing vivntcban ucatm A a chn dcl . Bitmtnghing cnhnghing mtgc i

viphngngang v chiudi mtnghing l . Chobit: bn knh R,

a) Hy phn tch chuynngcaav tm A a. Tmmiquan hnghcgiachuynngcaav

tm A a.b) Tnh ngnngcho av tngcng tc ngln a.c) Tnh vntcv gia tccatm A a. Cho nhnxt hai

ktquny.d) Tm iukinvgi tr ac thlnln cht

dc.e) Xc nhcc thnh phnphnlctiimtipxc I.f) Tm iukincagc nghing alnkhng trt

trn mtnghing.

m

0

A

v

0

Av

0, , , , , .A tm v f f


46/173

g. Cho :g1. phn tch li chuyn ng ca a. Xc nh

vachn cc ta suy rng.

g2. xc nh cc lc suy rng tngngg3. thit lp hphngtrnh vi phn chuyn ngcho h.

3ttg f

hdof

Hnh II.6

,

,A As v

I

A

R


47/173

a chuyn ng ln khng trt trn mt nghing c nh


48/173

Do :

Quan hng hc giachuynngcaav chuynngcatm A akhi alnkhng trt:

a chuyn ng lnkhng trt trn mtnghing cnh.y l trng hp c bit ca chuyn ng song phngvi tm vn tc tc thilim tip xc I.

AAv

R

Phn tch chuyn ng ca khi tm A a: tm A achuyn ng thng vi qu ol ng thngsong song vi mtnghing cnh v cch mtnghing y mt khong bng bn knh a.

A

; A A A Av a

AA

s

R AA

a

R


49/173


50/173

c. Vn tc ca tm A:

Dng nh l bin thin ng nng:

. .sin ;A AA A P P h P s cao hnglncng m

1 0h hT T A

22 03 3 sin4 4

const

A A AP Pv v P sg g

20 4 .sin3A A Av v g s o hm 2 vtheo thi gian t:


51/173

Theo nguyn lyDAlembert, tas bsung vo a hai thnh


52/173

g y y gphn cbn ca hlc qun tnh:

thias trong trng thi cn bng.

qt

qtA

R

M

.qt A AR m a

(do tnh ln nn bo(-))2

. . .sin ;3

qt A AR m a P

.qtA A AM J

2 2

2

1 1 1.sin . .sin

2 3 3

qt AA

aP PM R P Rg

gR

Rg

Vit cc phng trnh cn bng:


53/173

Thay vo:


54/173

Thay,vo:

1.sin . .cos

3 tP f P

tan 3 tf

g.g1. Do nn a va lnva trt trn dc.y

cng l trng hp c bit ca chuyn ng songphng vi tm vn tc tc thi khng phi lim tipxc I.

3 ttg f

Bc t do ca h: 2hdof

Chn 2 ta suy rng: . (hnh 3.1),x

x


55/173

Hnh II.6.2

I

A

msF P

N

g2.

Xc nh lc suy rng tng ng vi ta suyrng :1 xQ Q

1q x

Cho h mt di chuyn kh d c bit:

1 20, 0qq x

(a ch trt m khng ln v )0


56/173

Tng cng kh d ca cc ti:

;

. .

.sin . .

.sin .

0

cos

sin .cos

ms

ms

A A P A F

P h F x

P x f

A

N x

P f P x

N

A x

N

P f

(a ch trt m khng ln v )0

hx

A Hnh II.6.3

0x


57/173

Lc suy rng:

Hnh II.6.4

0

I

A

msF

P

0x

N

0 :N v khi chiu lnphng trtphng ca lc

nng N vung gcso vi phng trt.

1 sin c

sin co

o

s .

sx A f Px

Q Q

f M g

Xc nh lc suy rng tng ng vi ta suy2Q Q


58/173

Lc suy rng:

(a quay quanh tm A c nh hay a trt khng ln)

Xc nh lc suy rng tng ng vi ta suyrng

2Q Q

2q

. . .

ms

ms I

A P A NA A F

F s f N r


1 20 , 0q x q


. . .cos .A f P r

2 . . .cosA

Q Q f P r

0x


59/173

g3. Dng phng trnh Lagrange 2:

2 . . . .cosQ Q f M g r

Hnh II.6.5

, 1, 2ii i

d T TQ i

dt q q

0

.Is r

A

msF

P

0x

N

Xc nh ng nng h:


60/173

Xc nh ng nng h:

2 2

2 2 2

1 1.

2 2

1 1 1. .

2 2 2

h

A AT M v J

M x M r

10

T T

q x

(khng c x ch c o hm ca x)

1

.d T d T

M x

dt q dt x

2

0T T

q

(khng c )

21d T d T


61/173

2

2

1. .

2

d T d T M r

dt q dt

Do :

2

. sin cos .

1. . . .cos .

2

r

M M

M Mr

x f g

f g

sin cos2 cos .

x f g

f g

r


62/173

Bi tp 7. Cho chnhhnh v. Bitbn knh r, P, =const, Q, rng rc l vnh trn ngcht. Dy mm, nh,

khng gin, khng trt trn rng rc, lun cng. Ban uhng yn.a) Hy phn tch chuyn ng ca cc vt rn trong h.

Thit lp quan hng hc gia cc vt.b) Xc nh ng nngcho ton hv tng cng ca cc

ti tc ng ln h.c) Xc nh gia tc ca vt A v gia tc gc ca rng rc

B.d) Tnh lc cngdy ni vt A.

e) Tm iu kin ca moment nhnh dy ni vt A bchng. Xc nh li gia tc ca A vagia tc gc carng rc B.

M

M

M


63/173

Hnh II.7

a) Phn tch chuyn ng:

Vt A: chuyn ng tnh tin thng ng, nhanh dn, cchiuhng xung.

, ,B B B

, ,A A As v a

M

A

P

Q

B

r

Rng rc B: chuyn ng quay nhanh dn, cng chiu


64/173

kim ng hquanh trcvung gc vimtphnghnh vviqua tm B cnh (tm B cnh).

Thit lp quan hnghcgia cc vt:.

.

.

A B

A B

A B

s r

v r

a r

b).

h A BT T T

Vt A chuyn ng tnh tin: 212

A APT vg

Vt B chuyn ng quay: 21

2

B B BT J

ng nngca h:


65/173

A P

M


66/173

Gia tc ca A v gia tc gc ca rng rc B:

AA P sr

p dngnhl bin thin ng nng:

1 0

h hT T A

212

A AP Q v P sg r

M

o hm 2 vtheo t:

12 .

2 A A AP Q v a P v

g r

M

. 0A

Pra g

P Q

M

c).

Gia tc gc ca rng rc:


67/173

Gia tc gcca rng rc:

d) Xc nh lc cngdy.p dng nguyn lyDAlembertkho st scn bngca

vt A:

A

B

Pa r

gr P Q r

M

.Aqt A AR m a

A

qt

P PP r rR g Pg P Q P Q

M M A

P

yAT

A

qtR

Hnh II.7.1

Aa

Sau khi b sung vo thi vt A cn bng.A

R


68/173

Phng trnh cn bng:

Chiu ln trc y:

Sau khi bsung vo thi vtA cn bng.qtR

0Aqt A AR T P

0Ajy A qtF T R P

AA qt

A

P

rT P R P P P Q

Q rT PP Q

M

M

e)iu kin dy khngb chng (dy cng):


69/173

e) u dy g b c g (dy c g)

Khi dy b chng:

Gia tc vt A:

Gia tc gc rng rc B:

0 0 .AT Q Q r r

M

M

Vy iu kin dy b chng: .Q rM

Aa g

2.B gQ r

M

Bi tp 8. Cho chng yn thiim ban u nhhnh


70/173

p g yve. Rng rc B l mt a trn, c v ng cht. Hsmast trt tnh vang gia vt A vamt phng ngang cnhl v . Cho bit: Dy mm,nhe, khng gin, lun cng, khng trt trn rng rc.a) Tm iu kin ca gc A trt c trn mt

nghing.b) Cho , dy lun cng.

b1) Phn tch chuyn ng cc vt rn trong h. Tm miquan hvng hc gia cc vt.b2) Tnh ng nngcho ton hvatng cng tc ng lnh.

b3) Xc nh gia tc ca A vagia tc gc ca rng rc Bb4) Tnh lc cngdyb5) Tm iu kin ca dy ni vt A b chng. Xcnh li gia tc ca vt A vagia tc gc ca rng rc B.

, , , , , .t

r P Q const f f M ,

tf f

ttg f

M

Mr


71/173

a) iu kin vt A khng trt (A cn bng v dy chng):

P

Q

B

r

A

Hnh II.8

Kho st s cn bng ca vt A:

T do ha vt A:


72/173

.cos 4AN P


73/173

iu kin vt A khng trt:

Thay (3) v (4) vo 2 vca(5), ta c:

A

. 5mst msgh t AF F f N

.sin . .costP f P

6ttg f iu kin vt A trt:

7t

tg f

b1. Phn tch chuyn ng ca cc vt rn trong h:

b) V nn vt A trt ctrn mtnghing.ttg f


74/173

Hnh II.8.2

M

P

Q

B

r

A

B

B

B

As

AvAa

Vt A tnh tin thng, nhanh dn theo phngca mt


75/173

Rng rc B quay nhanh dn, theo chiu kim ng h

quanh tm B cnh. Quan h ng hc:

nghing v vi chiu hng xung.

; ;A A AB B Bs v a

r r r

b2. ng nng ca h:h A BT T T

2 2

22 2 2

2

1

2 21 1 1

.2 2 2 4

A

A A A

B AB B A

PT m v v

gvQ Q

T J r vg r g

Vi:

21h QT P v


76/173

Tng cng cc ti:

2 2AT P v

g

msA A FA PA M

. .sinA AA P P h P s

. B AA sr

M

M M

cos . ms msd A A A AA F F s f N s f P s

Vi:

sin cos

AA P f sr

M

b3.p dng nh l bin thin ng nng:


77/173

3

1 0

h hT T A

21

sin cos2 2

A A

QP v P f s

g r

M

1 2 sin cos2 2 A A AQP v a P f v

g r M

sin cos

2

A

P fra g

QP

M

iP f

M


78/173

b4. Kho st chuyn ng ca vt A:

sin cos

2

AB

P fa r g

Qr

P r

Ta c:.ms AF f N

Hnh II.8.3P x

y

AN

msF A

AT Aa

Dng nh l chuyn ng khi tm cho vt A:


79/173

Chiu (8) ln hai phng x, y

4

1

. 8eA A j A A ms

j

m a F P N T F

. .sin

90 cos

A A msA

A

m FTa

N

P

P

cos 10AN P

.sin .

sin cossin cos

2

A ms A AT P F m a

P fP r

P f P gQg

P

M

sin cosP P f

M


80/173

sin cos

sin cos

2

A

P P fr

T P fQ

P

b5.iu kin dy cng:

sin cos2

11

2

A

QP f

rT

Q

P

M

0AT

sin cos 0

2

Qf

r

M

.

sin cos2

Q rf

M

iu kin dy chng:


81/173

.

sin cos2

Q rf M

Xc nh li v :Aa B

sin cosAa f g

2

2

.B

gQ r

M

Bi tp 9. Cho chnhhnhv:


82/173

a). H c lun cn bng vi mi loi ti tc ng haykhng? Ti sao?

b). Dng nguyn l di chuyn kh d xc nh cc thnhphn phn lc ti ngm A.

A B

q const

Hnh II.9

Bi sa


83/173

a. (T gii) Tnh bc t do ca h: Dofh

S vt rn: n=1 (thanh thng nm ngang)

Tng rng buc ca cc lin kt:

3lk

R (ngm phng)Do bc t do ca h:

3

3.1 3 0

lk

h

Dof n R

V dofh 0 nn h lun cn bng vi mi loi ti tc ng.

b). Lin ktcmt rng buc c gillin kt n(vd: khp bn l trt lin kt thanh) Lin kt ngm


84/173

Xc nh thnh phn phn lc HA:

Gii phng lin kt n cn tr chuyn ng tnh tinthngtheo phngngang:

d: khp bn l trt, lin kt thanh). Lin kt ngmphng c 3 rng buc s c xem l tng ngvi 3 lin kt n. xc nh cc thnh phn phn

lc ca lin kt ngm ta gii phng ln lt tng linkt nv xem cc thnh phn phn lc xut hin nhllc hot ng bsung.

AB

q

A

H

x

Hnh II.9.1

Cho h mt di chuyn kh d .x


85/173

Tnh tng cng kh d:

p dng nguyn l di chuyn kh d:

Xc nh phn lc NA:

Gii phng lin kt n cn tr chuyn ng tnh tinthngtheo phngng.

A A

A

A A A q H

H

AH

x

0 0AA H

Cho h mt di chuyn kh d .y

qAN


86/173

p dng nguyn l di chuyn kh d:

Tnh tng cng kh d:

Hnh II.9.2

A B

q

0y

.

A AA

A A N A q N y ql y

N ql y

0 0 0A AA N ql N ql

Xc nh thnh phn phn lc MA:


87/173

Gii phng lin kt n cn tr chuyn ng quayquanh tm A.

Cho h mt di chuyn kh d .

Hnh II.9.3

AB

q

AM

x

. y x

dx

Tnh tng cng kh d:


88/173

AA M AA q Ta c:

A AA M M

0 0

2 2

0

. . . .

. .2 2

A q q dx y q dx x

x qq

2

2

A

qA M


89/173

Bi tp 10. Dng nguyn l di chuyn khdxc nhcc thnh phn phn lc sau:


90/173

cc thnh phn phn lc sau:

Hnh II.10.1

Hnh II.10.2

AB

q P q

2M q

2

Ca)

q

A C

M

Bb)


91/173

.A AA A H H x


92/173

+p dng nguyn l di chuyn kh d:

- Xc nh thnh phn phn lc NB:

A

0 0AA H

+ Cho h mt di chuyn kh d :

Hnh II.10.4

A B

q

P q

MC

B

N

By Cy

Tnh tng cng kh d:


93/173

Xc nh thnh phn phn lc NA:

Gii phng lin kt ncn trchuyn ng tnh tintheo phngng:

2

2

.(2 . ) 2 . .(3 . ) .

(2 2 ).

BB

B

A A N A q A P A M

N q P M

N q

20 2 2 0 BA N q BN q

2


94/173

Hnh II.10.5

A B

q P q

MC

AN

du

u

y

Ay

Cy

Cho h mt di chuyn kh d :

Tng cng kh d:


95/173

AA P AAA MNA q

2

Ay tg

2 . Ay

. .2 . A A A AA N N y N

. y u

22 2 2

0 0 0

. . . . . .2

u

A q y q du q u du q

22 . A

q q


96/173

Th vo:

. Cy

2. . . . CA P P y q q

2. . A M M q

2 2.2 . 2 . . 0 AA N q q M

2 AN q

b).


97/173

Xc nh cc thnh phn phn lc HA: Gii phng lin kt n cn tr chuyn ng tnh tin

theo phng ngang:

Hnh II.10.6

q

A C

M

B

AH x

Cho h mt di chuyn kh d :x

Tng cng kh d: .A AA A H A q H x

0 0AA H


98/173

Gii phng lin kt NA:

Hnh II.10.7

Ay

AC

2

M

B

x

dx

By

AV

Cho h 1 di chuyn kh d :

Tng cng kh d :


99/173

Ta c:

A AA VA A q M

. 0 y x y x

2

0 0 0

2

. . . . . .2

2

xA q y q dx q x x dx q

q

2. . AA M M q

2 . Ay


100/173

Gii phng lin kt VC:

..2

A A A

A

A V V y

V

2 21.2 . 0

2

AA V q q

2 2112

2 4

A

q qV q

qM

B


101/173

Hnh II.10.8

A CB

u

du

Cy

CVBy

2 u

.y u

2 2

. . . . A q y q du u q du

2

2 2 24. . .

uA q q q q


102/173

2

. . .2 2 2

32

A q q q q

q

2. . A M M q

2 . Cy

.2 .

C C C

C

A V V y

V

CA A q A M A V

232 .

CA q V


103/173

2 2

2

3

.22

C

C

q

q q V

2 235

2 2 4

C

q q

V q

Bi tp 11. Cho mt chgmchnh lngtruA tit din tami t t ht kh B Vt A


104/173

gic vung v ng tr trn, ng cht, khng yB. VtA ckhi lng m1 ta khng ma st trn mt phng ngang c

nh. VtB c bn knh r, khi lng m2 lnkhng trt trnmt nghing ca lng truA (hnh chiungca trcng trtrnlB). LngtruA chu tc ng ca lc F nhhnh v. Chobit: m1, m2, F, , r.a) Hy phn tch chuyn ng ca cc vt rn trong h. Xc

nh bc tdo ca hv chn cc ta suy rng cho h.b) Vit biu thc xc nh vn tc tuyt i catm B v tnh

ln ca vector vn tc tuyt i ny.c) Xc nh ng nngcho ton hv cc lc suy rng tng

ng.d) Thit lp hphngtrnh vi phn chuyn ng cho ton h.

Cho bit khnngca mnhcgii c hphngtrnhvi phn ny khng? Nu gii c hy xc nh gia tc ca

lng tru A v gia tc gc ca ng tr B.

x r


105/173

a) Phn tch chuyn ng ca cc vt rn trong h Vt lntruA c chuyn ng tnh tin thng theo phng

ngang.

Hnh II.11.1

A

F

BIv

2P

1P

N I

B

ng trung tuyn


106/173

Chuyn ng tng i: ln khng trt trn mtnghing ca lngtruA (tm quay tcthil imI).


107/173

g g g ( q y )

Dng nh l hp vn tc ca im:

Be

Ba

Brv vv

Tnh ln vector vn tctuyti im B:

Hnh II.11.2

BI

v

I

B

A

av

B

av

(khi tng 2 gc bng thicos gc ny bng - cosgc kia).

*

B B

B B A

B A BI

a a

r

a a

I

ev v v v

v v

vv

Vi:


108/173

c). ng nng h:Ah B TT T

Cng thc lng trong tam gic thng:

2 2 2

2 2

2 cos

2 cos

B A BI A BIa a a

A BI A BI

a a

v v v v v

v v v v

M: ; . . A BIav x v r r

2

2 2 2. 2 .cos . .Bav x r r x

2

2

1 1

1 1. .

2 2 A AaT m v m

xVi:


109/173

2 2

2

2

2

1 1

.2 2

B B

a BT m v J

Ta c: 22. ;BJ m r

2 2 2 2 2

2 2

2 2 2

2

1 12 cos . . .2 2

12 . 2 cos . .

2

B

T m x r r x m r

m x r r x

2 2 21 2 2 21

. . . . .cos . .2

hT m m x m r m r x

Xc nh lc suy rng :1 xQ Q


110/173


Hnh II.11.3

A

F

0

rBIv

2P

1P

N I

2C B

0 x

A

av

1 20 ; 0q x q

(a khng quay nn ton hl mt vt duy nht: m = m1+m2)!


111/173


112/173

2. BA m g h


113/173

d) Dng phng trnh Lagrange 2:

2. . .sin . m g r

2 2 . .sinA

Q Q m g r

, 1, 2ii i

d T TQ i

dt q q

1

0

T T

q x

1 2 2. .cosd T d T

m m x m r dt q dt x


114/173

y l hphngtrnh vi phn cp 2 cc kdgii.Cch gii c trnh by chi tit:

1dt q dt x

2

0T Tq

22 22

. .cos 2

d T d T m r x m r

dt q dt

1 2 2

2

2 2 2

. .cos

. .cos 2 . .sin

m m x m r F

m r x m r m g r

t 1 2;X x X


115/173

Nghim:

2

1

1 2 2

2

2

1 2

2

1

2

2

2 2

. .cos.sin 2

. .cos

cos 22 . . .sin .cos

2 .c

12 sin 22

2 co

os

s

F m rg r

X xm m m r

rr F m g r

r m m m

F mc

r

g

m m monst

m m F


116/173

1 2

2

1 2 2

1 2

21 2 2

.sin .cos

2

cos .sin

. .

cos

cos

cos 2

m m F

g

X

m m g F

m m m

m m

c

m

r

r

r

onst

Bi tp 12. Cho mt ac dng hnh qut ng cht, c,dy iu bn knh R gc chn tm l 2 ng yn


117/173

dy iu, bn knh R, gc chn tml 2 ,ng yn

thiim u vi gc nghingl 0(gc hp bi phng

thng ng vatrc i xng ca a) c khi lng m.ata khng ma st vi mt phng ngang cnh.

a) Phn tch chuyn ng ca a vatm O ab) Xc nhv tr ca khi tm C a.

c) Xc nh phngtrnh qu o ca khi tm Ca nu chn htrc ta nhhnh ve:d) Tnh ng nngca a vatng cng cc ti tc

ng ln a.e) Tm vn tc gc ca a v gia tc gc ca a.

xy


118/173

Bi sa

) Ph t h h


119/173

a) Phn tch chuyn ng ca a:

a chuyn ng song phng trong mt phng hnh vvi tm vn tc tc thi khng phi l im tip xc I.

Phn tch chuyn ng ca tm O a:

b). Do a hnh qut c mt trc i xng nn khi tmC ca a s nm trn trc i xng ny.

Tm O a chuyn ng thng vi qu o l ng

thng song song on thng c nh v cchng thng c nh mt on bng bn knh ca a.Do , vn tc v gia tc ca tm O nm trn ngthng ny.

O

0 0 0 0;

v

Dng h trc ta vung gc mi gn lin vi asao cho trc trng trc i xng ca a.

Outu


120/173

Kho st mt din tch vi phn k thuc a nh hnh v:

0ct

O

r

dr

kdA

Kk

uu

d

Hnh II.12.1

Gi: A

l khi lng ring a

l din tch a.


121/173

l khi lng ring a.

Ta c:

2

:A R rad

2 ;

m m

A R

.cosku r

.kdA dr r d

2k k mm dA dr rd R

(vt phng (kg/m2); vt dy (kg/m3)

(uk: ta u ca im K).

(mk: khi lng ca din tch dAk).

p dng cng thc nh ngha ca khi tm:


122/173

2121 cos . .

k k

kc

A

m u

mu r dr d m m R

2

20

3

2

03

2

1cos .

1sin

3

12sin

3

R

c

R

u r dr d R

r

R

R

R


123/173

0c cx const x m


124/173

d). ng nng a: (a l hnh trn c ng cht).

Vy phng trnh chuyn ng ca khi tm C l: 0 ;cx

ng thng ;c y ; .c cv y a y

2 21 1 .

2 2c cT mv J

Ta c: ( l gc hp bi trc u v trc Oy).:

2 2

0 0 .c cJ J m OC J m u

Theo nh ngha:

2 2

0 21

. . .kk A

mJ m OK r dr d r

R

3

0 2

0

Rm

J r dr dR


125/173

0

4

2

0

4

2

4

2

4

R

R

m rR

m R

R

2

0

1

2J mR (ging cng thc hnh qut)

22

2c c

RJ m u


126/173

0A N Vi:

. cA P P h


127/173

0mg HC HC

Hnh II.12.3

P

C

0yO

0C

0O H

0

ch


128/173

Gia tc gc ca a:


129/173

o hm 2 v (2) theo thi gian t:

2 2 2 2 21 1 1sin cos . sin .2

. sin .

222

2c c

c

u u R

g u

2

2 2 2

sin cos .

1sin

2

c c

c

u u g

u R

Bi tp 13. (Chasa).Cho mt thanh thng mnh, ng cht, tit din iu, khi


130/173

g glng m v chiu di 2ta khng ma st trn mt phng

ngang cnh. Ban u thanh ng yn vi gc nghing0.a. Hy phn tch chuyn ng ca thanh AB. Tmphngtrnhqu o caim A, ca khi tm Cv im B.

b. Tnh ng nngca thanh vatng cng cc titc ng ln thanh.c. Xc nh vn tc gc ca thanh vagia tc gccan.

B


131/173

A 0

0C

Hnh II.13

0: , 2 , mCho

Bi tp 14.

Ch h h h h (H h II 14) M


132/173

Cho chnhhnh v(Hnh II.14).

A

B

2r

1r

1O

2

O

2P

1P

M

Hnh II.14

Cho r1, r2, P1, P2, .Dyccc tnh cht sau y:mm, nhe, khng gin, khngtrt trn cc vt v lun

cng. Bqua ma st ti khpbn l O1 v xem nhnh dyAB lun c phng thngng. Rng rc O1 la trnc, ng cht v rng rc O2

lvnh trn ng cht.a. Xc nh bc tdo ca h.

Chn cc h ta suyrng cho h.

constM

b. Phn tchchuynng choccvt rn trongh. Phn


133/173

tchchuynngphchpca tm O2.Vitbiu thc

tnhvntctuyti choimny.

c.Tnhng nngchotonh.

d.Xcnhcclc suyrng tng ngvicctasuy

rng chn.

e.Thit lphphng trnh vi phnchuynng choh.

Cho bit kh nng c th gii h phng trnh ny

khng?Ti sao?

Bi sa

a M


134/173

a.Dofh= +2

Chn 2 ta suy rng:

q11;q22

A

B

2r

1r

1O

2O

y

2P

1P

1

M

2

Hnh II.14.1

b. Phntchchuynngccvt.


135/173

Rngrc O1:

Chuynng quay quanh tm O1c nh. Rngrc O2:

Chuynng songphngvi tmvn tc tc thi khng

phil imtipxc A. Phntchchuynngphchpca tm O2.

Chuynngko theo:

Tnhtinthngngcngvi dy. Chuynng tng i:

Quay quanh tmvntctcthi Aivi dy.

Vitbiuthctnh: (Hnh II.14.2)M


136/173

2 2 2O O O

a e rv v v

A

B

2r

1r

1O

2O

y

2P

1P

1

B

av

A

av

2

2O

rv

Hnh II.14.2

2

2

O

eO A

e a O A B

v yv v


137/173

2

2 2

2 2

2

1 1 1 1

2

2 2 2 2

. .

.

e a O A B

e a a

O Or rO O A

r O

r

v v v r r

v AO hay v yv v

v r r

Vi :

2 2 2 2

1 1 2 2

2

. .

ve

O O O O

a a e r

ctor

v v v v r r chkhi

c.

ng nng tonh:1 2O OT T T he

1

2 211 1 1

1

2 2O

PJ m r r

g

Ta co:


138/173

2

2 222 2 2. .O

PJ m r r

g

Ta co:

1

1

2 2 211 1 1

1. . .

2 4 O O

PJ r

gVi : T

2 2

2

2 2

2 2

2 2 22 21 1 2 2 2 2

1 1 .2 2

1 1. . . .

2 2

O O

a OT m v J

P Pr r r

g g

2 2 2 2 22 2 21 1 1 2 1 2 2 2. . . . . . . .

2

O P P PT r r r r g g g


139/173

2

1 1 2 2: . .O

aTa c v r r


140/173

1 21 2

1 1

.

.

r rdt dt

r

Tnhtng cngkh d:

2A A A P M

21 2. . OA P s M

22 11.

OO

a

s: r

dt dt

M v

2 1 1.Os r


141/173

2 1 1. .A P r Vay : M

Lc suyrng Q1:

1 2 1

1

.A

Q P r

M

Tnh lc suy rng Q2:

Chnmt dichuynkh d dcbit choh:

1 20 ; 0

2

1 1 2 2: . .O

aTa c v r r


142/173

1 2 21 2 2. . .r r r

dt dt dt

22 OO

a

sv

dt

Ma:

2 2 2.Os r

Tng cngkh d:

2

2

2 2 2 2. . .O

A A P

P s P r

Lc suyrng Q2:

A


143/173

2 2 2

2

.A

Q P r

1 2 1

2 2 2

.

.

P r

Q P r

Vay : Q M

e.

Dng phng trnh Lagrange 2 (i vi h khi mt cn

bng):

, 1, 2ii i

d T TQ i

dt q q


144/173

212 1

1

2

PA P r const

g


145/173

21 2

222

. .

2..

PB r r const

gP

C r constg

at :

1 2 2 1

1 2 2 2

. . .

. . .

A B r D

B C P r E

PM +

1 1 2 2; X

at : X1 2

1 2

. .

. .

A X B X D

B X C X E

Thay cc ktquvo phngtrnh Lagrange 2, ta c:

1 21

. .

.X const

CD BE

A C B

A E BD

(rng rc 1 quay nhanh dn u)


146/173

22 2

. .

.

X consA E BD

A C B

t

(rng rc 2 quay nhanh dn u)

Bi tp 15.

Cho mt c h nh hnh v (Hnh II 15)


147/173

Cho mt chnhhnh v(Hnh II.15).

Hnh II.15

A

C

B

P

Q

Ir

A la trn c, ng chtcbn knh r v trng lngQ AB l thanh thng mnh ng cht tit din u di


148/173

Q. ABlthanh thng, mnh, ng cht, tit din u, di ,trng lng P. Cho r, , P, Q, , a A ln khng

trt trn mt phng ngang cnh.Bqua ma st lnca a v ma st ti khp bn lA.a. Xc nh bc tdo cho hv cc ta suy rng cho

h.

b. Phn tch chuyn ng ca cc vt rn trong h. Phntch chuyn ng phc hp ca khi tm C thuc thanhAB. Vit biu thc tnh vector vn tc tuyt i cho imC ny.

c. Tnh ng nngcho ton h.d. Xc nh cc lc suy rng tng ng vi cc ta

suy rng chn cho h.

constM


149/173

Mx


150/173

Hnh II.15.1

A

C

BP

Q

Ir

M

2

1

AA

av

C

ev

C

av

C

rv

Chuyn ng song phng trong mt phng cha vt vi

atrn A:


151/173

Thanhthng AB:

Chuyn ng song phng trong mt phng cha vt vi

TVTTTl im cha xcnh.

tmvntctcthi (TVTTT)l imtipxc I.

Phn tch chuyn ng phc hp ca khi tm C ca

thanh AB.

Chuynngko theo :tnhtincngvi tm A.

Qu o tm A l ng thng . AA a Av

Chuyn ng tng i: quay quanh tm A

C A

e av vDo:


152/173

Chuyn ng tng i: quay quanh tm A.

Vit biu thc tnh:

c.ng nng ton h:

C C C

a e rv v v

2 2 2

2

22

1 2 2 1 2

2 cos

. . . .cos . .2

C A C A C

a a r a r v v v v v

r r

A ABT T T he

C CArv v AC Do :

2

21 1 .2 2

A A

A a A AT m v J Vi :


153/173

1

2 2

1

; . .

1. .

2 2

A

A a

A A

A

Ta

Qm v rg

QJ m r r

g

co:

2 2 2 2

1 1

2 2

1

. .2 4

3.

4

A Q QT r rg g

Qr

g

2

21 1 .2 2

AB C

AB a C ABT m v J

P


154/173

2

2 2

;

1. .

12 12

AB AB

C AB

Ta Pm

gP

J mg

co:

2 2 2 21 2 2 1 2. . . .cos . .

2 6 2AB P P PT r r

g g g

2 2 2 2

1 2 2 1 2

1 3. . . .cos . .

2 2 6 2

P PT P Q r r

g g g

heVay :

d.

Tnhlc suyrng Q1ngvitasuyrng q11 .

Cho hmt di chuyn khadic bit: 1> 0 ; 2= 0



155/173

Tng cngkh d caccti:

1

A A

M

M .

1

1

A

Q

M

Tnhlc suyrng Q1ngvitasuyrng q2 2.

Choh mt dichuynkh d cbit: 1 = 0 ; 2> 0.

iA A P P h P

Tng cngkh d cacctitc ng:(Hnh II.15.2)


156/173

2 2. sin2

CA A P P h P

2 2

2

.sin2

AQ P

1

2 2.sin

2

Q

Q P

Vay : M

2

0C

2

C

A

2

Ch

P

Hnh II.15.2

e.

Phng trnh Lagrange 2:


157/173

Phng trnh Lagrange 2:

1

1 1

d T TQ

dt

2

2 2

d T TQdt

2

1 2 2

1

1 2. . .cos .

3 2

T PP Q r r

g g

Vi :

2 2

1 2 2 2 2

1

1 2. . . . sin . cos .

3 2

d T PP Q r r

dt g g


158/173

2 21 2 2 2 21 3

. . .cos . . . sin .2 2 2

P PP Q r r r

g g g

Vay :

M


159/173

2 2 2g g g

2

2 1 2 2. .cos . . .sin2 3 2

P Pr P

g g

Bi tp 16. Cho

1 2 31 2 3 1 2 2 1 3 3, , , , , , , , , 2 2 2 .A O O Om m m m J J J R r R r r M M


160/173

a. Xc nh bc tdo cho hv chn ta suy rng cho

h.

b. Phn tch chuyn ng ca cc vt rn trong h. Xc

nh vn tc gc ca rng rc kp 3 vavn tc tuyt i

cavtA.

c. Tnh ng nngcho ton h.

d. Xc nh cc lc suy rng cho h.

e. Vit hphngtrnh vi phn chuyn ng cho h. Gii

hphngtrnh ny.


161/173

Bi sa

a. Dofh = +2.v ta cn dng 2 thng sc lp 1 va2


162/173

mi xc nh cv trca ton h.

Chn 2 ta suy rng q1 1; q22

b.

Phn tch chuyn ng cc vt: Rng rc 1: quay quanh tm O1cnh.

Rngrc 2: quay quanh tm O2c nh.

Rng rc kp 3: chuyn ng song phng.

VtA tnhtinthngng.

1O1

r2R

O

1MB

av

D

2M


163/173

Hnh II.16.1

1

3R

3r

2O

3O

A

21 Dav

E

av

C

av

B D

EC

Xcnh 3:


164/173

Tac:C B

a a

E D

a a

v v

v v

1 1 1

2 2 2

. .. 2 .

B

a

D

a

v r rv R r

Ma:

Xcnh tmvntctcthirngrc 3:(Hnh II.16.2)


165/173

Vntc tm O3:3

3 3.O

av PO


166/173

3 3a

1 1

3 1 21 2

. 3 .1 2

23

C

av r r

Ta co: PC =

1 213 31 2 1 2

232 2

PO PC O C r r r

3 1 22

.3

O

av r Vay :

Do dy khng gin nn:

3

12

OA

a av v


167/173

3

3 1 2

1 2

23

2

3

OA

a av v r

c.

Tnhng nng h:

1 2 3 AT T T T T he

1 1

1 2 2

1 1

1 1.

2 2O OT J J Vi :

2 2

2 2 2

2 2

1 1. .

2 2O OT J J

1 1


168/173

3

3

3

3 3 3

23 2

3 3

2 2 2 2 2

3 1 2 1 2 1 2 1 2

2 2 2 2 2

3 1 3 2 3 1 2

1 1.

2 21 4 1 1

2 . . 4. 4 .2 9 2 9

1 2 2

4 . . . 2. . . .18 9 9

O

a O

O

O O O

T m v J

m r J

m r J m r J J m r

2

2 2 2

1 2 1 2

2 2 2 2 2

1 2 1 2

1 1 4. 2 .

2 2 9

2 2 4. . . . . . .

9 9 9

A A

A a A

A A A

T m v m r

m r m r m r

1 3

2

1

2 2

3

1 4 1 4. .

2 9 9 9O O A

A

J m m r JT r

he


169/173

3

3

22 2 22

1

2

2

3

3

1 4 4 4.

2 9

22

9 9

9

.

O O

O A

A

C

B

J m r J

J m m r

m r

d.

Tnhlc suyrng Q1:

Choh mt dichuynkh d cbit: 1> 0 ; 2= 0

Tnh A : 1 3 AA A A P A P M

3

1 1 1

3 3 3 3

.

. . . O

A

P P h m g s

Vi : M M


170/173

1 3 12

.3

AA gr m m

M

33 1 21 22 23 3OO

a

sv r r

dt dt dt

Ta co:

3

3 3 1

1

2 . . .3

2. . . . . .

3A A A A O A

A P m gr

A P P s m g s m gr

Do o:

3 1 2 1

2 2.

3 3Os r r

1 1 31

2.

3 A

AQ m m gr

M


171/173

Tnhlc suyrng Q2:

Cho hmt di chuyn khdc bit: 1= 0 ; 2>0

Tnh A :

2 3 AA A A P A P M

3

2 2 2

3 3 3 3

.. . . O

AA P P h m g s

Vi : M M

33 1 21 22 2

3 3

OO

a

sv r r

dt dt dt

Ta co:

2 2


172/173

3 1 2 2

2 2.

3 3Os r r

3

3 3 2

2

2. .

3

2. . . . . .3

A A A A O A

A P m gr

A P P s m g s m gr

Do o:

2 2 32

2.

3 A

AQ m m gr

M

2 3 22

.3

AA gr m m

M

e. Vit h phng trnh vi phn cho h bng cch dng

phngtrnh Lagrange 2:

1 2d T T

Q i


173/173

1 2

1

2 . Cd T

dtA

1 2

22

TC

d

dt B

1 2

0T T

2

, 1, 2ii i

dQ i

dt

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