Giáo trình vật lý đại cương phần cơ học ths nguyễn hoàng tuyến, ths nguyễn thị ngọc vân

Nhuhg, tiiy theo dang chuyen dong cua chat diem, ta nen chon mot he toa dp phu hup de sd didn ta toan hoc cho chuyen dpng cua chat diem la rd rang va thuan tien nhat.. PHUONG TRINH CHUYE

Nguyen Thi Ngoc Van Nguyen Hoang Tuyen

■■4 .

Trtfcfng Dai hoc Dan lap Ccfu Long muon khing dinh la trung tam dao tao nguon nhan li/c co trinh do cao cho khu vifc dong bang song Cdu Long de phuc vu cho- cbng cuoc cong nghiep hoa, hien dai hoa dat nifdc thi phai nang cao chat litong dao tao toan dien Mot trong nhtfng yeu to trong yeu la doi ngu thay co, he thong cac giao trinh va trang thiet bi day hoc.Gung vdi cac nganh cac cap trong toan quoc dang day nhanh tien do xay dting va cung co vi the trong xu the hoi nhap, triidng Dai hoc Dan lap Ciki Long dan tiing biidc hoan thier de difo'ng dau vdi nhtfng thach thu’c do, gop phan difa nen giao due dai hoc Viet Nam diing vffng, ngang tarn khu vile va the gidi.

Ke hoach xay difng ve mot Bo giao trinh sir dung cho cac giang vien lam tai lieu co ban de giang day, cho sinh vien nghien cdu tham khao hoc tap la nhu cau cap bach cua nha trifdng

An pham n&y la gido trinh Vat Ly Dai Ciiong - Phan Co Hoc cua Khoa Khoa Hoc Co Ban, Trird’ng Dai hoc Dan lap Cdu Long, do ThS Nguydn Hoang Tuyen va ThS Nguydn Thi Ngoc Van bien scan

Lanh dao triTdng Dai hoc Dan lap Cufu Long tran trong nhtfng cdhg hien, dong gop cua Quy thay co mdi giang cung nho cac thay co can bo - giang vien co huh cua nha trOdng

ThS Nguyen Cao Dat

trong nha trirdng

Tran trong gidi thieu gido trinh nay den Quy thay co vd cac em sinh vien trildng Dai hoc Dan lap Cdu Long

Vinh Long, thang 09 nam 2009

Q HIEU TRLfdNG

Chitong 1

I nhUng khai NIEM Md DAU

1 Chat diem va chat diem

• Chat diem : Mot vat co kick thiidc nhd khong dang ke so

vdi nhttng khodng each, nhdng kick tkiidc ma ta dang khdo sat got la chat diem Chat diem con goi la hat.

Vi du : khi xet vien dan bay trong khong khi, trai dat quay quanh mat trefi, ta co the xem vien dan, trai dat la nhiing chat diem Khi do, cac vat the nay dUOc khao sat nhu" cac diem (hinh hoc) trong chuyen dong cua no NhUng can lifu y rang, mot vat dUOc xem la chat diem hay khong, khong phai do kich thudc rieng cua nd xac dinh, ma do ty so' gitfa kich thub'c cua nd

va do dai dac trimg cho chuyen dong cua nd xac dinh Ta biet rMng, bdn kinh cua trai dat (R = 6,4.106 m) rat nhd so vdi khoang each trung binh gitfa nd va mat trd'i (r = 1,5 x 1011 m)

Vi vay, khi xet trai dat trong chuyen dong ttf quay quanh true cua nd, ta khong the xem trai dat la chat diem dtfdc

Khi khao sat chuyen ding mot vat, neu ta khong quan tarn den stf quay cua nd va stf dich chuyen gitfa cac bo phan cua nd dbi vdi nhau, ta cd the didn ta vat do bang md hinh chat diem.Chat diem chuyen dong thtfdng dtfqc goi tat la dong diem

• He chat diem : Mot tap hap gom nhieu chdt diem, ma cac

chat diem trong he co tiiang tdc vdi nhau goi la he chdt diem

He chat diem con dtfqc goi la he hat

Vi du: tap hop cac phan ttf khi trong mot binh chtfa la mot

he chat diem, tap hop cac hanh tinh trong he mat trdi la mot

he chat diem

DONG HOC CHAT DIEM• •

2 Chuyen dpng cof hoc va hp quy chieu

• Chuyen dong cot hoc : Su" chuyen dcfi vi tri cua mot vat

doi vdi cac vat the khac trong khong gian theo thcri gian goi la chuyen dong co hoc Chuyen dong co hoc goi tdt la chuyen dong

• He quy chieu : Mot vat dope chon lam moc g&n vdi mot

he toa do va mot gdc thcri gian cung vdi mot dong hd hop thanh mot he quy chieu

Chon he quy chieu khac nhau, ta se nhin thay chuyen dong cua cung mot vat didn ra don gian hay phdc tap Khi nghien ciiu chuyen dong mot vat, ta nen chon he quy chieu nao de chuyen dong cua vat ay dien ra don gian nhat

Nham bieu dien toan hoc cho mot chuyen dong, ta thucfng chon mot he toa do gan vdi he quy chieu Co the chon cac he toa do thudng dung trong giai tich : he true toa do Descartes vuong gdc, he true toa do doc cdc, he true toa do tru, he true toa dp cau, Nhuhg, tiiy theo dang chuyen dong cua chat diem,

ta nen chon mot he toa dp phu hup de sd didn ta toan hoc cho chuyen dpng cua chat diem la rd rang va thuan tien nhat

3 Khong gian va thdri gian (theo cd hoc cd dien)

• Khong gian trong ccf hoc dupe quan niem khong phu thupc vao thdi gian vzl vat the chuydn dpng trong dd Khong gian cd tinh dong nhat va ding hdd’ng, do do thda man cdc tinh chat cua khong gian Euclid, trong do cac tien de Euclid va cac dinh

ly cua hinh hoc Euclid ddefe sd dung Khong gian ndi tren ddtfc goi la khong gian tuyet ddi

• Then gian cung ddp’c quan niem khong phu thupc vao khong gian, vao vat the chuyen dpng, trdi deu td qua khd, qua hien tai den tUtfng lai, ddi vdi moi he quy chieu Thbi gian ndi tren cung dupe goi la thefi gian tuyet do’i

M f

(C) 0

II PHUONG TRINH CHUYEN DONG - QUY DAO

Viec xac dinh vi tri cua mot chat diem chuyen dong theo thdi gian gpi la xac dinh chuyen dong cua nd Day la bai toan

co ban cua dong hoc chat diem

Do cd nhieu phunng phap xac dinh vi tri cua chat diem, nen

cd nhieu each xac dinh chuyen dong cua chat diem

1 PhtfoTng phap vector

Ta cd the xac dinh vi tri cua dong diem M d moi thdi diem b^ng vecto f diIng tef diem gdc O tdi diem M (O dtfpc chon tuy

y va dufng yen)

OM = r , r difpc goi la vecto ban kinh (hay vectcr tia) cua M.Khi M chuyen dong, r se thay doi theo thbi gian ca ve do Idn Idn phifcmg, chieu Do do, r la ham vectcr phu thuoc dd'i so

vd hifdng t (thdi gian)

Neu xac lap ttfbng minh diipc bieu thde (1), ta cd the dung dilpc vectcr r tai tiing thdi diem t Dau mut cua r d tiTng thdi diem t ay chinh la vi tri cua M trong luc nd chuyen dong.Nhif vay, bieu thde (1) xac dinh quy luat chuyen dong cua M dtidi dang vectcr, nen (1) dupe goi la phiiang trinh chuyen dong

dang vecto.

Phitong phdp nay thifcrng diftfc dung trong ly thuyet, cho phep ta nghien cdu, bieu di§n chuyen dong mot each co dong, tong quat

Quy dao cua dong diem M Id ditdng no vach ra trong

khong gian trong qua trinh chuyen dong.

Quy tich cac diem ngon cua vectcr f khi t bien doi chinh la

quy dao cua M

2 Phtfdng phap toa dp

Tif diem O, ta dung mot he true toa do vuong goc Oxyz Khi

ay vi tri cua M dupe xac dinh bdi cac toa do (x,y,z) cua no

ZA

<4

thi

x = x(t)

(C) M

Do dai dai so cua cung OM ro rang xac dinh hoan toan vi tri cua M Ky hieu : s = OM, s goi la hoanh do cong cua M

Phifcfng phap toa do thildng dtfac dung trong thifc hanh tinh toan vi cu the va thuan tien

Muon tim quy dao cua M, ta can tim bieu thdc lien he gitfa cac toa do cua M bang each khd t trong cac phifcfng trinh chuyen dong (2)

Vi du, khd t trong (2) ta diftfc 2 phifcfng trinh co dang :

'fx(x,y) = 0f2(y,z) = 0

• Neu chuyen dong cua M trong mat phang Oxy

z = z(t) = 0 Phifcfng trinh chuyen dong phang co dang :

x = x(t)

y = y(t)chuyen dong tren difcfng thang Ox

= z(t) = 0 Phifcfng trinh chuyen dong thang co

Day la cac phifcfng trinh cua 2 mat tru cd cac difcfng sinh tifefng dng song song vd'i cac true z va true x Giao tuyen cua 2 mSt nay chinh la quy dao cua dong diem M trong khong gian

3 Phifcfng phap tif nhien

Neu quy dao cua M da biet, ta cd the lay mot diem O bat

ky tren quy dao (C) lam goc va chon mot chieu difcfng tren do

• Neu M

y = y(t) = 0 , z

dang :

s

III VAN TOC

1 Van toe trung binh - Van toe ttfc thdri

Vd'i phifOng phap tif nhien, trong khoang thdri gian At gia suf dong diem M dich chuyen di/qc cung As

v (goi tdt la van toe) duttc dinh nghia :

ds

v = —dt

• v bang dao ham hoanh do cong s cua M theo thdi gian t

• v dac trung cho chieu va toe do nhanh cham cua M

° v > 0 : M chuyen dong theo chieu difOng da chon, va ngtierc lai

° Iv| : bieu thi toe do nhanh cham cua chuyen dong cua M

2 Vector van toe

Vdi phifcrng phap vecto, trong khoang thdi gian At gia sd M

cd ban kinh vecto f bien doi mot lifqng Ar

hay : (5)

(6)v

Do Idn cua v : |v| =

Hitdng cua v daoc xac dinh bdi cac cosin chi hiring cua no:

cos(i, v) =

dxdt

dy dt

dz dt

• v bang dao ham cua ban kfnh f theo thcti gian t

phitong tiep tuyen vdi quydao (C) tai tifng diem

• v co : s chieu la chieu chuyen dong

do Idn |v| = |v|

IV GIA TOC

1 Vector gia toe

Gia suf tai thbi diem t, M co vector van toe V]_ Sau khoang thbi gian At, tai thefi diem tx, M co vectcr van toe v2, ttfc

vx

Iv|’

2+

Vectcr van toe v dbpc dinh nghia :

, Af

v = hm —

At—>0 At

drV" dt

vv cos(j,v) =-i-;

(7)hay :

2

Do Idn cua a : |a| =

^Hitdng cua a diTOc xac dinh bcfi cac cosin chi hifdng cua nd:

+

dt

dvx dt

dvz dt

2+

dva“ dt

trong khoang thdi gian At = t2 - tj., vectcf van toe cua M bien thien mot ItfOng Av = V2 - v1

Vectcf gia toe a dtfOc dinh nghia :

Av

a = nm —

At->0 At

''dVydtSuy ra : ax j, ’ a

Bang giai tfch vecto, ta cd the chufng minh rang:

Vecto gia toe a luon ntim trong mat phdng mat ti&p vcfi quy dao (C) vd luon httdng ve b$ lorn cua (C) tai titng diem.

2 Hinh chieu cua a tren cac true toa dp

Tir bieu thufe : v = vxi + vyj + vzk, dao ham 2 ve theo t, ta diTOc :

Chuyen dong

Thang

4 Gia toe tiep tuyen - Gia toe phap tuyen

Bang giai tich vecto, ta co the phan tich a gom 2 thanh phan : at va an

= v2

a = 0

Vv2

a <90°

d(v2) dtTir bieu thtfc (9), ta xet cac trilbng ho'p :

• Neu v = const : chuyen dong deu, thi :

a = 0 (trong chuyen dong thang)

(v, a) = 90° (trong chuyen dong cong)

v tang : chuyen dong nhanh dan, thi :

cos(i,a) = ^; cos(j,a) = -^-; cos(k,a) = ^-

3 Dau hi$u oho biet tinh chat nhanh cham ciia chuyen dpng

Ta thay rang sif bien doi cua v2 theo t cho thay sif thay doi gia tri cua v theo t Ta co :

d(v2) dv - = 2v — = 2 va

dt dt

Nhanh dan

a

a = an

a = 90o,at =0

>an

trong do :

♦ at tiep tuyen vdi quy dao, goi la gia toe tiep tuyen

phap tuyen vdi quy dao, gpi la gia toe phap tuyen

Chdong nhanh dan

v a'Va

v-• Y ngliia vat ly cua at va an :

Ttr cac cong thtfc (11), ta thay :

• at dac trifng cho sir bien doi do Idn cua vecto v cua dong diem theo thdi gian at cang Idn thi toe do nhanh cham cua chuyen dong bien doi cang nhieu, va ngtfqc lai

• an dac trimg cho sq bien doi phitong cua vecto v cua dong diem theo thdi gian Dieu nay co nghia la an cang 16n thi vectcf v co phifcfng bien doi cang nhanh, an cang nhd thi vectcr

v c6 phifcmg bien doi cang cham

That vay :

'v2f

MCI

\\a

a > 90°,at 1'4' v

1 Chuyen dpng thang deu

Trong he dern vi SI, v tmh bang —, a tmh bang s

V CAC CHUYEN DONG THANG DON GIAN

Xet dong diem M chuyen dong thang tren true Ox, phiTOng trinh chuyen dong cua M :

an =0

• Ung vdi mot gia tri v xac dinh, an cang Idn neu p cang nhd, nhiing p cang nhd, quy dao cang cong thi phitcfng cua v bien doi cang nhanh, va ngifOc lai

• Ung vdi mot gia tri p xac dinh, an cang Idn neu v cang

Idn, nhimg v cang Idn thi phifcrng cua v bien doi cang nhanh,

va ngittfc lai

Dac biet, trong trifdng hdp quy dao cua dong diem la thang,

p = co Khi do :

dv =

o

Tai t = 0, M cd hoanh do x0 Tai t bat ky, M cd hoanh do

x Ta tlch phan 2 ve phtfcrng trinh tren va cho cac can tilcfng dng, ta cd :

•X

dx =xo

dv

dt "aChon gdc thdi gian t = 0, luc M cd van toe v0 Tai thd’i diem t bat ky, M cd van toe v Tich phan 2 ve phiTOng trinh

vi phan (phan ^bien so) tren vdi cac can tiicrng ufng, ta dtfdc :

I

Vay ; v = at + v0

dxMat khac : — = v

dt

1 2(at + v0)dt -> x - x0 = — at + vot

2

Vay : x = vt + x0

(12) la phiTOng trinh chuyen dong thing deu

2 Chuyen dpng thang bien doi deu

0 t = 0 vq

xo

Do la chuyen dong vdi quy dao thing va van tdc bien doi deu theo thhi gian

Khi do v la ham bac nhat theo t Cd 2 trirbng hop :

• v tang deu theo t : chuyen dong nhanh dan deu

• v giam deu theo t : chuyen dong cham dan deu

Suy ra : a = const, an = 0 (p = co), a = at

Vay : (14)

M

A0

Vi tri cua M dd dang diftfc xac dinh bang goc liftfng giac AOM = 0,9 goi la goc dinh vi

(16) la phu’Ong trinh chuyen dong dang goc dinh vi Tif viec xac dinh vi tri cua M thong qua goc dinh vi 0, ta thay rang cac dai lifcmg dong hoc cua M cung co lien quan vdi 0

x^, V! la hoanh do va van toe cua M tai thbi diem t-^

x2,V2 la hoanh do va van toe cua M tai thdi diem t2

VI CHUYEN DONG TRON

1 Khao sat dpng hoc

Xet dong diem M co quy dao la vong tron (O, R) Chon mot diem A co dinh tren quy dao tron lam goc va chieu duOng tren quy dao nay

• Van toe goc trung binh co ditoc dinh nghia :

Hay:

CO A

M

a Van toe goc

Trong khoang thcri gian At, gia suf g6c dinh vi 0 thay doi mot lilcmg A0

_ A0

co = —At

■ co bang dao ham goc dinh vi 6 theo t

■ co dac trifng cho chieu va toe do quay nhanh cham cua M tren quy dao (C) cua no

* co > 0 : M quay theo chieu diftmg da chon, va ngiitfc lai

* |co| : bieu thi toe do quay nhanh cham cua M

• Vecto vdn t^c goc w

Theo dinh nghia, co co :

■ Phtftfng la phiftfng true quy dao tron

■ Chieu la chieu tien cua cai van nut chai, neu xoay no theo chieu quay cua M

I R

^coR

b) Gia toe goc

Trong khoang thefi gian At, gia suf van toe goc cua M bien doi mot litong Aco

• Gia toe goc trung binh p dtftfc dinh nghia: P = 77

• Gia toe goc tifc thdi p (gpi tSt la gia toe goc) difpc dinh

nghia :

* Neu p TT ® : M quay nhanh dan (Hinh 1)

* Neu P TJr : M quay cham dan (Hinh 2)

Trong he don vi SI, co tinh bang —, p tmh bang

2 Chuyen dpng tron deu

Do la chuyen dong vdi quy dao tron va van toe goc khong doi, co = const, p = 0.

(20)(21)(22)(23)

Chon goc thcfi gian t = 0, luc M co goc dinh vi 0q Tai thdi

diem t bat ky, goc dinh vi cua M la 0 TifOng tif nha trong chuyen dong thing deu, ta d£ d£mg tim dnoc phnong trinh chuyen dong trdn deu nhif sau :

0 = cot + 0Q

3 Chuyen d<?ng trdn bien doi deu

Do la chuyen dong vdi quy dao trdn va van toe goc bien doi deu theo thdi gian

-> co la ham bac nhat theo t -> p = const

Chon goc thdi gian t = 0, luc M co goc dinh vi 0O ,van toe goc co0 Tai thdi diem t bat ky, goc dinh vi cua M la 0, van tdc goc la co Tiforng tif nhif chuyen dong thing bien doi deu, ta

co the tim difOc cac edng thufe sau :

CO = pt + CO q ,

0 = i pt2 + co0t + 0O ,2

co2 — co2 = 2p (0 — 0q ),co| - co2 = 2p(02 -0i),(21) Id phifong trinh chuyen dong trdn bien doi deu

Trong (23): ©i,^ la goc dinh vi va van tdc goc cua M tai thdi diem ti

02,co2 la goc dinh vi va van tdc goc cua M tai thdi diem t2

Chu y

1 Cac dai hrqng sau day dtfoc dung de dac tnfng

■ cho chuyen dong thing : x, v, a

■ cho chuyen dong trdn : 0,co,p

dva" dt

dx

v = —dt

a

JI -"/

at = PaR

at = p.R

2 Ta d§ dang thay rang :

• Mol lien he gitfa x,v trong chuyen dong thang deu tiTOng tii nhii moi lien he gitfa 0,co trong chuyen dong tron deu

• Moi lien he gitfa x, v, a trong chuyen dong thing bien doi deu tiTOng tif nhu” moi lien he gitfa 9,(o,P trong chuyen dong tron bien doi deu

3 Ta co the chdng minh dito’c cac cong thdc sau day :

n dco

<-> p = —

dt

a)

©A

BAI TAP CHUONG 1

1 Diem M chuyen dong theo phtfctag trinh (dang tif nhien) :

s = 5 + 3t3 , (s : m, t: s)

a) Xac dinh doan difdng M di ditto trong 3 giay dau

b) Tinh v, at, an cua M tai thdi diem t = 2 s, neu ban kinh cong tai do la p = 18 m

Sau khi bat dau chuyen dong (tif t = 0), diem M chuyen dong nhanh hay cham dan?

2 Diem A chuyen dong tren quy dao theo phittng trinh :

s = it3 - 2,5t2 + 6t , (s:m,t:s).

3Xac dinh hoanh do cong va van tdc tai thcfi diem dau (t = 0)

b) Tim thdi diem ma v = 0 va hoanh do cong cua A tai thdi diem do

c) Xac dinh chieu chuyen dong cua A tren quy dao

3 Tren doan difcfng thang dai 12,5 km tau hoa giam van tdc tif

60 km/h xudng con 40 km/h Xem tau chuyen dong cham dan deu Xac dinh gia tdc a va khoang thdi gian tau chay tren quang dtfdng do

4 Puli co ban kinh R quay quanh true n^m ngang (qua diem

O — hinh ve )

B

R

o—

(x,y : m; t: s)

(x, y : m; t: s)

(c, d la hang so')

a) Xac dinh quy dao cua M

b) Tinh tri so' cua v va a cua M tai cac thdi diem t} = 2 s

b) Tim tri so' cua v va a

c) Neu ngoai hai phiicfng trinh chuyen dong tren, N con co them phitong trinh chuyen dong : z = 5t,z : m Cho biet quy dao cua N trong trtfdng hop nay

7 Mot chat diem chuyen dong tren mot dirdng tron ban kfnh

R theo phitong trinh :

s = ct2 + dt,

Vat nang A chuyen dong theo phitong trinh :

y = — yt2, y = const2

a) Tim bieu thufc van tb'c v va cdc gia toe a, at, an diem B nam tren vanh puli

b) Tim bieu thufe van to'c goc co va gia to'c goc 0 cua diem

B nay

5 Diem M chuyen dong theo phifcfng trinh :

x = 0,2t2 + 10

y = 0,15t2 -5

va a theo t.

d = m/s ,

b)

c)

b) Neu y nghia vat ly cua lefi giai d tren neu :

a) khi cau dang dtfng yen

Tmh van toe v, cac gia toe at,an

T>^

g

11 Mot vat diftfc tha red tif mot khi cau dang bay d do cao

300 m Hoi sau bao lau vat mi tdi mat dat, neu luc tha vat thi:

Ap dung bang so

t = 0,5 s

8 Mot chat diem M chuyen dong theo mot quy dao phang vdi gia toe at = c va gia toe an = bt4, trong do c,b la hang so, t

la thdi gian tinh tir luc bat dau chuyen dong Hay tim :

a) Phnong trinh chuyen dong cua M tren quy dao (dang tif nhien)

Bieu thde ban kinh cong p cua quy dao theo hoanh do cong s

Bieu thde gia toe a cua M theo hoanh do cong s

9 Mot chat diem C chuyen dong cham dan deu theo dirdng tron ban kinh R sao cho tai moi thdi diem at va an deu bang nhau ve do Idn Tai t = 0, C co van toe v0 Tim bieu thde van toe v, gia toe a cua diem C theo quang dirdng S m& no dich chuyen diTOc

10 Hai vat difOc nem thang ddng len cao tif ciing mot diem vdi cung van toe dau v0 = 24,5 m / s, vat nay sau vat kia khoang thdi gian t = 0,5 s

a) Hoi hai vat gap nhau sau khi nem vat thii hai bao lau va

d do cao nao ?

R = 2m, c = 3m/s2,

14 Mot vat chuyen dong thang bien ddi deu, di het quang dirdng AB trong 6 s Van toe cua vat khi qua A bang 5 m / s, khi di qua B bang 15 m/s Tim chieu dai quang dudng AB

15 ThUdc A cd chieu dai 25 cm treo tren tufrng bang mot soi day Trong titdng, d phia dudi thud'c co mot 16 sang nhd B Hoi canh dudi cua thudc phai d cao hem 16 sang mot khoang h bang bao nhieu de khi dot sdi day cho thifdc red, thi/dc se che khuat

16 sang trong khoang thdi gian 0,1 s ?

b) khi cau dang ha thang xuong vdi van toe 5 m/s

c) khi cau dang bay thang len vdi van toe 5 m / s

12 Td mat dat, ngudi ta nem mot vien da M vdi van toe dau v0 theo phUcrng hop vdi mat dat gdc a Hay xac dinh :

a) Phtfcrng trinh chuyen dong vd quy dao cua M

b) Do cao Idn nhat ma M cd the dat diftfc

c) Tam xa cua M (khoang each td vi tri nem M den vi tri

nd roi cham dat)

d) Khoang thdi gian M chuyen dong

Cho gia to'c trong triThng g khong do'i, bo qua life can cua khong khi len vien da

13 Mot sqi day vat qua mot rdng roc co dinh B ,mot dau day

buoc vao mot chiec xe A dat tren duhng ndm ngang, dau kia dcfqc keo vdi van toe v Tim van toe V ciia xe tren dud’ng vao luc doan day buoc vao xe cd gdc nghieng a so vdi mat dudng

B

!

B L-i

16 Ngufri ta nem dong thcfi hai vat len cao theo phtfcfng thang ddng Vat A tif mat dat vdi van tdc v0 = 20 m/s , vat B tif do cao H so vdi m^t dat vdi van toe v0 = 10 m / s

a) Hoi sau bao lau hai vat se gap nhau? Neu H = 5 m ?b) Xac dinh gia tri cua H de hai vat gap nhau sau khi vat

B da qua do cao cifc dai va trade khi vat B chuyen dong qua vi tri xuat phat cua nd Cho biet gia toe trong trifdng

g = 10 m / s2 Bo qua life can cua khong khi len vat

17 Mot chile xe dang chay vdi van tdc 90 km/h, tai xe bdng phat hien d phia trade each xe 30 m cd mot ngi/di di gitfa dildng, nen voi phanh xe vdi gia tdc 5 m/s2 Hoi xe con chay tiep trong thdi gian bao lau va cd khdi dam vao ngadi do khong?

18 Mot dto bat dau chuyen dong vdi gia tdc 0,5 m/s dung

vao luc tau dien vdOt qua nd vdi van tdc 18 km/h Hoi van tdc cua dto khi nd dudi kip tau dien ? Biet gia tdc cua tau dien la

0,3 m/s2

19 Mot vat chuyen dong bien ddi deu vdi van tdc dau

18 km/h Quang difdng nd di ditoc trong giay thtf nam ke tir luc bat dau la 4,5m Tlnh :

a) Gia tdc cua vat

b) Quang dtfdng vat di daqc sau 10 s

Bo qua site can cua khong khi len vat Cho biet gia toe trong trirdng g = const.

21 Mot may bay bay ngang vd’i van toe Vi d do cao h so vdi mat bien, de tha bom trung mot t&u chien dang chuyen dong tren mat bien vdi van toe v2 trong cung mat piling thing dtfng vdi may bay Hoi may bay phai cat bom khi nd d each tau khoang each 1 theo phudng nam ngang la bao nhieu ? Giai bai toan trong hai trifdng hop : may bay va tau chuyen dong cung chieu va chuyen dong ngircc chieu Bo qua sdc can khong khi

22 Tit mot dinh thap cao H = 25 m, ngudi ta nem mot vien bi

B len phia tren vdi van toe v0 = 15 m/s theo phitong hop vdi mat phing ngang gdc a = 30° Hay xac dinh :

Phirong trinh quy dao cua vien bi B

Khoang each td chan thap den diem ma B cham dat

Do cao Idn nhat B dat dude

Van tdc cua B khi cham dat

23 Mot vdlang sau khi bat dau quay dirqc 1 phut thi cd van tdc gdc 700 vdng/phut Tinh gia tdc gdc cua vdlang va so vdng ma

vd l£ng quay diftfc trong phut ay Biet ring vd l&ng quay nhanh dan deu

24 Mot banh xe quay cham dan deu, sau 1 phut quay van tdc gdc cua nd giam tir 300 vdng/phut xudng con 180 vdng/phut Tim gia tdc gdc cua banh xe va so vdng banh xe da quay dude trong phut ay

cua C va ban klnh cua dia qua C.

25 Mot dia trbn ban kinh R = 10 cm, luc dau dang ddng yen, sau do quay xung quanh true cua no vdi gia toe goc bang 3,14 rad / s2 Sau giay thdr nhat, tinh :

a)

c)

26 Mot ngifdi muon cheo thuyen qua song co dong nifdc chay Neu ngitfi ay cheo theo hudrng tit diem A - ben day bd, den diem B - ben kia bd (AB vudng goc vdi dong song) thi sau khoang thdi gian tj = 10 phut thuyen se tdi bd ben kia tai diem C each B mot khoang s = 120 m ve phla xuoi dong Neu ngtfdi ay cho mui thuyen ludn ludn chech mot goc a so vdi AB

ve phia ngiftfc dong, thi sau khoang thdi gian t2 = 12,5 phut, thuyen se tdi dung diem B Hay tim :

Chieu rong 1 cua con song

Van toe u cua thuyen doi vdi midc

Van toe v cua ndde doi vdi bd song

Chiftfng 2

DONG LUC HOC CHAT DIEM • • •

Ccf sor cua dong lye hoc la cac dinh luat dutfc tong hop tuf cac ket qua cua hang loat thi nghiem va quan sat ve chuyen dong cua cac vat the Cac dinh luat nay lan dau tien diTOc Newton trinh bay mot each hoan chinh va co he thong trong tac pham

“Cac nguyen ly cua Triet hoc tif nhien” cua ong (1686 )

I DINH LUAT I NEWTON

1 Phat bieu dinh lu£t

Moi vat giU nguy&n trang thdi drtng yen ho&c chuyen dong th&ng deu neu tong cac ngoai lite tdc dung l&n vat being khong.

2 Quan tinh

Dinh luat nay neu len mot tinh chat rat tong quat cua moi vat the, do la tinh bao toan trang thai chuyen dong: vat dang dtfng yen, til no khong the thay doi trang thai dilng yen do, vat dang chuyen dong thSng deu, tu* nd cung khong the thay doi van toe cua nd difqc Tinh chat bao toan trang thai chuyen dong, tdc gid nguyen trang thai chuyen dong nhif cu cua cac vat the, goi la qudn tinh De thoat ra khdi trang thai nay, vat phai chiu tac dung cua ngoai life khac khong

Khoi litqng m cua vat Id so do mile do qudn tinh cua nd Vat

cd khdi ItTOng cang Idn thi quan tinh cua nd cang Idn, tu'e la kha nang gid nguyen trang thai chuyen dong cua vat cang cao,

va ngirqc lai

3 He quy chieu quan tinh

Trang thai ddng yen hay chuyen dong ma dinh luat tren de cap chi dung doi vdri nhttng he quy chieu dac biet goi la he quy chieu quan tinh

F = F1+F2 + + Fn

He quy chieu qudn tinh Id he quy chi^u gdn vdi mot vat td

do, do Id vqt khong chiu tdc dung cua Ide ndo hay tong hap lilc ten vqt triet tieu.

Trong ta nhien khong co vat th do, vi vay khong cd mot he quy chieu quan tinh dung Tuy nhien, trong ky thuat va ddi song, thufrng anh hifdng quay cua trai dat len chuyen dong cua cac vat khao sat la khong dang ke, nen h? quy chieu quan tinh (gan dung) ditac chon la he quy chieu gan lien vdi mot diem ddng yen tren trai dat

II DINH LU AT II NEWTON

Life diftfc bieu didn bang mot vectcf nen life cd ba yeu to

■ Phifcmg, chieu cua vectcf bieu dien phtfefng, chieu cua life Difbng thang ma doc theo do life tac dung len vat goi la difbng tac dung cua Ihc

■ Do dai cua vectcf (theo ty le tif chon ) bieu dibn tri sb cua life

■ Goh cua vectcf bieu dien diem dat cua life

• Nguyen ly chong chat life

Tac dung dong thdi cua cac life F|, F2, » Fn co phifcfng dong quy len mot vat tifefng difefng vdi tac dung cua tong hop life F bang tong hinh hoc cac life thanh phan F|, F2, , Fn

n

F = ZFk

k=l

_ F

a = — m(1) dirac goi la phitong trinh co ban cua dong life hoc chdt diem.

Trong (1), F tmh bang N, m tmh bang kg, a tmh bang

s

2 Phat bieu dinh luat

Diidi tdc dung cua life F, chdt diem khdi liMng m chuyen dong vdi gia tdc a Gia tdc a cung hiidng vdi life F va co do Ion a ty le thuan vdi do Idn F cua life vd ty le nghich vdi khdi litang m cua vat.

Doi vdi he defn vi SI :

^Nhdn xet

Neu biet trade lac F, ap dung (1) ta se sac dinh daoc gia tdc a, th do ta se tinh daoc van tdc v va ban kinh f (vi tri) cua vat Vay neu biet lac tac dung len mot vat ta cd the xac dinh daoc chuyen dong ndi chung cua vat do Neu biet trade lac

va vi tri dau, van tdc dau (rg, vq - goi la cac dieu kien dau) cua vat ta cd the xac dinh daoc chuyen dong cu the cua vat do

$ Ngaoc lai, neu biet trade gia tdc a (ho&c van tdc v, ban kinh r) cua vat, van dung (1) ta se xac dinh daoc lac F tac dung len nd

Dinh luat II Newton chi nghiem dung ddi vdi he quy chieu qudn tinh.

(4)

III DINH LU AT III NEWTON

Tren day ta mdi chi xet moi lien he gitfa Itfc tac dung lenvat va gia toe ma vat thu diWc Trong thtfc te, moi stf thay doitrang thai chuyen dong deu xay ra do ket qua taong tac gitfa

cac vat Dinh luat III Newton xet den svf tifcfng tac gitfa hai vat, phat bieu nhif sau

chdt diem A tac dung l&n chdt diem B mot luc F thi chdt diem B cung tac dung ten chdt diem A mot lUc F cung

pliitong, ngiiqc chieu vd cung do Idn vdi F

F' = -F

Hai life nay goi la life va phan Inc Chung khong phai la cap

life can bang vi chung co diem dat khac nhau.

Dinh luat III Newton dung doi vdi moi he quy chieu, khong doi hoihe phai la he quy chieu quan tinh.

IV DONG LUONG VA XUNG LUONG

1 TriTcrng hqfp chat diem

Xet chat diem M co khoi lirqng m chuyen dong vdi van toe

v Theo dinh nghia, dong lifotag P cua M la mot vectcr dirqc xac dinh nhif sau

F la life tdc dung len vat

(4) difqc xem 1& mot dang khac cua dinh luat II Newton Tit bieu thtfc (4), suy ra : dP = Fdt

nghia vat ly cua dong luting vd xung hting :

■ Dong liftfng P la dai Inong bao gdm ca khoi liiong m

va van toe v cua chat diem, nen P dac trimg cho chuyen dong cua chat diem ve mat dong life hoc

Cu the, trong hien tiTo’ng va cham, dong Iddng mot vat dac trimg cho kha nang truyen chuyen dong cua vat ay len vat the khac

■ Xung lifcmg Fdt la dai lifOng dSc trimg cho tac dung cua luc F trong khoang thdi gian dt

Neu F = const, (5) trd thanh : AP = FAt, At = t2

Trong he SI, P tinh bang kg^

2 Fdt Hay AP = P2 - ^ =I

Fdt dutfe goi la xung IdOng cua hie F trong khoang thdi gian dt Tich phan hai ve bieu thu'e tren vdi cac can chay tiTOng ting nhu” sau :

P : Pj -> P2 <-> t: t^ -> t2

n jrj n flk=l k=l k=l

Ta xet tang thanh phan trong (6) :

■ = 0, vi cac noi lac luon xuat hien tang cap tnXc ddik=l

(theo dinh luat III Newton), nen tong cac noi life trong he luon bang khong

n

k=1

len h? Cac lac khong dong quy vi dftt len cdc chat diem khde nhau, tong cua chung khong goi la tong hop lac Re cd tdc dung khac vdi tac dung cua tong hop lac

dP = RedtSuy ra :

lite tdc dung ten he trong khodng thai gian do.

Cdc tritctng hotp dac biet

1/ Neu Re = 0, tu” (8) suy ra :

k=l

Hay : P]^ * P2 + + Pn = constVay neu vecto chinh cua ngoai life tac dung len he bang 0, thi tong dong li/Ong cua he bao toan

HP2/ Chieu (8) len true Ox bat ky : —- = Rj

dtNeu R® = 0 , thi Px = const, A z A 7

hay : Plx + P2x + + Pnx = const,

trong do :Pkx = hchOxPk (k = 1,2, , n)

2Redt

iS dt dtk=i "" k=i

P goi la tong dong liftfng cua he

The vao (6), diftfc ket qua :

dtdPdt

I

Vay, neu hinh chieu cua vectcf chlnh cua ngoai life tac dung

len he tren mot phUOng bang 0, thi theo phUOng do tong dong

luong cua he bao toan

Chu y : Trong cong thtfc (7) va (8) ta thay khong co mat

cua noi lire, nhu vay noi life khong dong gop gi trong tong dong

luong cua he Noi chung, chuyen dong cua he ve toan bo la ket

qua tdc dung cua ngoai lUc ma thoi

V NGUYEN LY TUUNG DOI GALILEO LUC QUAN TINH

1 Nguyen ly tifofng doi Galileo

Bang nhieu thi nghiem co hoc va dUa vao tinh chat dong

nhat, dang hudng cua khong gian, tinh chat dong nhat cua thdi

gian trong moi he quy chieu quan tinh Galileo phat bieu

nguyen ly tUOng doi sau day :

Cdc quy luqt co ban cua co hoc deu di&i ra nhu nhau trong

cdc he quy chitu quan tinh khde nhau.

That vay, neu ta lam thi nghiem co hoc d tren mot con tau

dang chuyen dong thi.ng deu (he quy chieu quan tinh co

v = const) thi cac hien tuqng do xay ra giong het nhu khi con

tau dang ddng yen (he quy chieu quan tinh co v = 0) : tha roi

mot vat, vat se roi theo dudng thing ditng ; de vien bi tren

san, no se nam yen mai ; do node vao mot binh, mat thoang

cua nUcfc se co dang phing nim ngang,

Ta co the hieu nguyen ly nay theo each khac :

Moi he quy chieu quan tinh deu tUong ditong nhau khi xet

cdc hien titqng co hoc.

Ve sau, trong thuyet tuefng doi hep (1905) cua Einstein, ong

da phat bieu hai tien de quan trong noi tieng Mot trong hai tien

de do co noi dung mb rong nguyen ly tUOng doi Galileo nhu sau :

Cdc quy luqt co ban cua vat ly hoc noi chung d^u di&n

nhit nhau trong cdc he quy chieu quan tinh khde nhau.

* He quy chieu co dinh 0 goi la he quy chieu tuyet doi.

* He quy chieu O' chuyen dong doi vdi he quy chieu O goi

la he quy chieu titong doi

De phan biet chuyen dong cua dong diem M doi vdi cac he quy chieu tren, ta cd cac dinh nghia :

• Chuyen dong cua M doi vdi he quy chieu tuyet doi O goi la chuyen dong tuyet doi Khi do, quy dao, van toe, gia toe, cua M goi la quy dao tuyet doi , van tdc tuyet doi , gia tdc tuyet doi,

• Chuyen dong cua M doi vdi he quy chieu tifcfng doi O' goi la chuyen dong tuong doi Khi do, quy dao, van tdc, gia tdc, cua M goi la quy dao tuttng dd'i, van tdc tifqng doi, gia tdc titong doi,

• Chuyen dong cua he O' doi vdi he O goi la chuyen

dong keo theo Tirctag ting van tdc, gia tdc, cua O'

goi la van tdc keo theo, gia tdc keo theo,

b) Dinh ly cqng van tdc

Xet chuyen dong cua

chat diem M ddi vdi hai

he quy chieu :

* He quy chieu tuyet

ddi Oxyz

* He quy chieu tiTdng

ddi O'x'y'z' chuyen dong

tinh tien ddi vdi h§ O,

• ^tyd: v^n t®c c^a m

• vtgd : van toe tiWng doi cua M

• vktheo : van toe keo theo cua O'

(11) la noi dung dinh ly cong van toe.

c) Dinh ly epng gia toe

Dao ham hai ve (11) theo t, ta lai diftfc :

d _

dT

Cac ky hieu :

• atyd : gia toe tuyet doi cua M

• atgd : gia toe tirong doi cua M

• aktheo : gia toe keo theo cua O'

(12) la noi dung dinh ly cong gia toe

A Chu y : cac cong thtfc (11), (12) chi dhoc dimg khi he quy chieu O' chuyen dong tinh tien doi vdi he quy chieu O Trong tnfdng hop he O' chuyen dong bat ky, cac dinh ly cong van tdc

va cong gia tdc kha phufe tap

(14)

(15)

3 Lite quart tlnh

Goi m la khoi lut/ng cua chat diem M dang xet, va :

• ^ktheo = ^0 toe cua he quy chieu O')

Cong thde (12) co the viet lai :

a = a' + a0Nhan hai ve bieu thde tren cho m :

* atgd - a

* a a^ (a trong cong thde dinh luat II Newton, a = )

khong qudn tinh (he quy chieu co gia toe)

trong he quy chi$u khdng qudn tinh.