BT CoLyThuyet - Cuối Kỳ
Transcript of BT CoLyThuyet - Cuối Kỳ
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GVHD: PGS. TS. TRNG TCH THIN
BI TP CHC
Tp Hai: NG LC HC
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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:
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2 2O 12 3
x y mJ J J J a ab bz
Moment qun tnh caton thanh ivitm O:
Vy: JO= Jx.
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a b
A
B
b
Hnh II.3
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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
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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
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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
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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:
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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
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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.
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A
O
P
r
M
Hnh II.4
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Bi gii
qtR
qtOM
A
mstF
O
0vN
P
r
M 0
0a
0s
Hnh II.4.1
x
y
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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.
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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
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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
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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
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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
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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
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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:
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.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 +
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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
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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
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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
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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
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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
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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
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a chuyn ng ln khng trt trn mt nghing c nh
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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
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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:
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Theo nguyn lyDAlembert, tas bsung vo a hai thnh
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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:
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Thay vo:
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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
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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
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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
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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
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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
Cho h mt di chuyn kh d c bit:
1 20 , 0q x q
Tng cng kh d ca cc ti:
. . .cos .A f P r
2 . . .cosA
Q Q f P r
0x
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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:
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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
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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
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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
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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
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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:
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A P
M
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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:
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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
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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):
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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
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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
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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:
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.cos 4AN P
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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
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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
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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
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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:
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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
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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:
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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
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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:
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.
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:
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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
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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
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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
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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
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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:
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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:
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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
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Bi tp 10. Dng nguyn l di chuyn khdxc nhcc thnh phn phn lc sau:
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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)
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.A AA A H H x
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+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:
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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
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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:
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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
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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).
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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
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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 :
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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
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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
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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
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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
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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
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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
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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
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Chuyn ng tng i: ln khng trt trn mtnghing ca lngtruA (tm quay tcthil imI).
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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:
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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:
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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
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Cho h mt di chuyn kh d c bit:
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)!
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2. BA m g h
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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
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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
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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
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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
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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
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Bi sa
) Ph t h h
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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
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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.
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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:
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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
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0c cx const x m
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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
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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
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0A N Vi:
. cA P P h
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0mg HC HC
Hnh II.12.3
P
C
0yO
0C
0O H
0
ch
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Gia tc gc ca a:
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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
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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
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A 0
0C
Hnh II.13
0: , 2 , mCho
Bi tp 14.
Ch h h h h (H h II 14) M
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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
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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
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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.
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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
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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
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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:
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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
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2
1 1 2 2: . .O
aTa c v r r
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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
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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
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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
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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
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212 1
1
2
PA P r const
g
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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)
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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)
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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
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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
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Mx
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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:
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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:
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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 :
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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
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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
Tng cng kh d ca cc ti:
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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)
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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:
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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
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2 21 2 2 2 21 3
. . .cos . . . sin .2 2 2
P PP Q r r r
g g g
Vay :
M
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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
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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.
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Bi sa
a. Dofh = +2.v ta cn dng 2 thng sc lp 1 va2
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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
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Hnh II.16.1
1
3R
3r
2O
3O
A
21 Dav
E
av
C
av
B D
EC
Xcnh 3:
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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)
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Vntc tm O3:3
3 3.O
av PO
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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
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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
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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
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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
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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
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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
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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
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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