cơ sở vật lí tập 2

Hat nhti vay chi co hai pinking danh cho no, va ta co thg tro hai phtiong ay bang cac da'u cOng va Cung nhti vay, mOt vat ran hay c6 thg quay xung quanh mOt truc co dinh chi co thg quay

MUC LUC

CHUONG 11 SU QUAY

11 - 3 Cac dai lacing gOc coi nhv cac vec td : N6i them 9

11 - 10 Cong COng suat va dinh li ye cong - dOng nang 29

CHUONG 12 SU LAN MOMEN QUAY VA MO MEN XUNG LUNG

12 - 5 Dinh lust thU hai cua Newton du6i clang gOc 65

12 - 7 MO men Ong luring cua mOt vat ran quay quanh mOt trvc c6 dinh 68

12 - 9 Su ban loan rn6 men dOng long MOt s6 thi du 71

12 - 10 Tien dOng cita con quay hoi chuyen (TV chon) 80

12 - 11 MOmen dOng Wong cludc luring tit h6a (Tv chon) 81

Tieu luan 2 CO hoc dia cac dOng tac quay trong vu dao 96 CHUONG 13 CAN BANG VA DAN HOT

13 - 1 Can bang

13 - 2 Cac diet' kien can cita can bang

13 - 3 Trong tam

13 - 4 Vai thi du ye can bang tinh

13 - 5 Cac cau trilc bat dinh

13 - 6 Dan hoi

On tap va Tom tat

Cau hoi

Bai tap va Bai toan

Bai toan.bO sung

CHUONG 14 DAC) DONG

14 - 1 Dao dOng

14 - 2 Chuyen dOng dieu h6a dOn gian

14 - 3 Chuyen 'Prig dieu h6a dOn gian : Dinh lust ye lac

14 - 4 Chuyen &Ong dieu h6a dOn gian : Khao sat ve nang lUdng

14 - 5 MOt dao dOng tit diet.' h6a g6c don gian

14 - 6 Cac con lac

14 - 7 Chuyen dOng dieu haa don gian va chuyen dOng trOn deu

14 - 8 Chuyen dOng dieu haa don gian tat clan (tv chon)

14 - 9 Dao thing curing bitc va sv cong httong (tV chon)

On tap va TOm tat

Cau hoi

Bai tap va Bai toan

Bai toan b6 sung

CHUONG 15 SU HAP DAN

15 - 1 The gi6i va lac hap dan

15 - 2 Dinh lust hap clan oh Newton

15 - 3 Su hap dan va nguyen Ii chong chap

■ •■

15 - 9 ye tinh : quy dao va nang IttOng (41 chon) 201

15 - 10 Nghien cOu can ke hOn ye ski hap clan (tO chon) 206

CHLIGNG 16 CHAT LUU

16 - 12 SO chay cna chat lou "thud (tO chon) 254

17 - 8 Nang Wong va c6ng sugt trong mOt song Ian truyen (to chon) 292

CHUONG 18 SONG - II

Tieu luan 5 Am hoc ciia phong Ma nhac : Khoa hoc hay nghe thuat 362

SC; QUAY I 11

Trong mon judo, mot van Ong vien yeti hdn vei nho hdn, ma nctm yang vat li có thi thdng inOt van Ong vien khOe hdn, to hdn, ma khong hien bier vat hi Ditt nay duoc chang minh trong dOn*g tac ca ban "hip throw" , trong do, van dOng vien xoay doi Hui quanh hong caa minh va - neu Ong tac thanh cong - quat nga doi thu tren san Neu khong sa dung thich clang vat II hoc thi, cle (pat nga, phai có sac manh rat lon

va co the de dang that bai Nhung uu the thu duoc nhO vat 11 hoc, la gi ?

Nghia la, "quang qua Ming", (xem bai town mdu 11-12)

3

11-1 110AT BONG CUA VAN BONG VIEN TRISQT BANG

DOng tac duyen clang cua mOt ngudi truot bang nghg thuat cd thd dung minh hoa, mOt each thil vi ye phuong dign them mi hai dang chuygn (Ring thuan khigt, hay khong pha trOn

1-11N11 11-1 a) Nghe si trot bang n6i tieng Yamaguchi, trong chuygn dOng tinh tien thuan tily (a) va trong chuygn dOng quay thuan tny (b) Trong trtiong hOp thit nhat, chuygn dOng dtiOc thitc hien theo mOt phaing c6 dinh, trong

trueing hop thU hai - quanh mOt truc c6 dinh

Hinh 11-la trinh bay mOt nglig si truot bang ludt tren mat bang, theo mOt dudng thang, vbi t6c do khOng d6i Chuygn dOng cua co ta la chuygn &Ong tinh tin thuan thy Hinh 11-1.b trinh bay su quay cde co ta vdi mOt t6c (10 khong d6i, quanh mOt true thAng thing, trong mot chuygn dOng quay thuan thy Trong chuong nay, chhng ta se nghien cdu loci chuygn dOng thd hai nay

Sit tinh tin la chuygn Ong theo mOt during thang, la chuygn Ong ma cho dgn day chting ta da nghien cdu chu ygu Su quay la chuygn dOng cue cac banh xe, banh rang, Ong

co, kim d6ng 116, roto cite dOng co phan laic va canh quat may bay len thang DO la chuygn dOng cua electron quay va nguyen ttt quay, cua con bao, va cua cac hanh tinh quay, cua cac sao, cac thien ha DO la chuygn (Ring cua dign vien nhao 10n, van dOng vien nhay

va cua cac nha du hanh vu tru tren qug dao Chuygn Ong quay cO c1 moi noi, quanh ta

11-2 CAC BIEN SO QUAY

Trong chueng nay, thong ta nghien eau su quay &la- mOt vat ran hay c6 the quanh mOt true c6 dinh Dieu han chg thti nhat cd nghia la chting '1:a se khOng xet sit quay oda

nhfing vat nhu Mat Trbi, vi Mat Trbi la mOt qua cau khi, chti khong phai mOt vat ran Dieu han chg thtl hai cua chung ta loai ra ngoai nhting vat gi6ng nhu mOt qua ki dang lan xu6ng theo ban ki Qua cau ay dang chuygn Ong lan, la si quay quanh mOt true

chuye'n dOng

Hinh 11-2 trinh bay mOt vat ran hinh clang bat ki, trong chuye'n dOng quay thuan tay,

quanh mOt true c6 dinh, goi la true quay MOi diem eita vat chuyen dOng tren mOt doting trop, co tam d tren true quay, va mOi diem chuyen Ong duoc cling mOt gdc, sau mOt khoang

thei gian nhat dinh Chung ta d6i chigu dieu nay vOi mOt vat chuyen dOng tinh tien thuan

tiny theo mOt phuong c6 dinh Trong si tinh tign thuan ttly mOi diem dia vat chuyen Ong

theo mOt dueng thang va m6i diem chuygn citing throe ding mOt khocing cach sau mOt

khoang thei gian nha't dinh Dudi day, se thuong xuyen dung ski so sanh gala chuyen &Ong thang va chuygn &Ong quay (chuyen Ong theo gdc)

Sau day, chung ta se so sanh cac dai Wong gdc Wong &tang, vUi cac dai luong dai,

ye vi tri, dO cloi, van t6c va gia t6c

VI TRI THEO GOC

Hinh 11-2 cung trinh bay mOt cluang mOc, c6 dinh tren

vat, vuOng gdc voi true quay, va quay cling vcii vat, chting

ta c6 the mo to chuygn dOng &la vat bang each chi rO vi

tri theo goc eira clueing nay true la gdc ma diking nay lam

vei mOt true co dinh 'Prong h 11-3, vi tri gdc 0 dtioc do dOi

voi true x, va 0 dtioc cho bai :

0 =

-r (tinh bang radian), (11-1)

trong do, s la dO dai cua cung, tren bat ki dubng troll

nao, cat boi true x va duting m6c, va r la ban kinh dtiOng

trOn do

at

Gcic dinh nghia theo each nay nen do bang radian (rad), HiNH 11-2 MOt vet ran hinh clang b , chuyen Ong quay than toy quanh hen la Wang \Tong (vg) hay clO Radian, bhng ti s6 hai do dai, trtic z cua he toa dO Vi tri cua du-bng

la mOt s6 thuan toy, va do do, khong co thti nguyen Vi dO mo'c doi vc3i vOt ran al), 5,, nhung no

dai cfra mot &tong tem ban kinh r la 2.rr, nen trong ea vOng phai vuOng ,goc vOi trnc quay NO la c6 dinh d6i vdi vat va quay ding vdi vat troll co 2n radian

Sau mai \Tong quay tron ven cua &Ong

m6c quanh truc quay, cluing ta khong cho 0

tro Iai bang khong Ngu dtiong m6c quay duoc

hai yang, thi B = zlit rad

'Prong chuy6n clang tinh tign theo mat

&tang thang, thong ta bigt chicle moi thti can

bigt ve vat chuygn clang ngu chUng ta bigt

x(t), vi tri cua no, nhu mat ham cUa thi gian

Cung thg, trong chuygn clang quay thuan tity,

chimg ta bigt duoc moi thti can bigt ye su

quay cita mat vat, ngu chting ta bigt 0(t), vi

tri theo gcic &la &tang m6c, nhu mat ham

cua thoi gian

HiNH 11-3 Vat ran quay ctia

h 11.2, trong tilt dien thang, nhin tit tren xu6ng Mat phang cua ti6t dien vuOng goc voi truc quay Trong vi tri nay, &tong m6c lam vdi truc x mOt goc B

DO DOI GOC

Ngu vat ran tren hinh 11-3 quay quanh truc quay nhu tren hinh 11-4, ma lam cho vi tri theo gcic cua &tong m6c thay d6i tit 01 dgn 02, thi vat thuc hiOn mat do dtii goc cho bai :

AO = 02 - Dinh nghia nay ye do cloi g6c khong nhiing

dung duoc cho vat ran, xet ye than ba, ma can

dung &toe cho moi hqt ciia v4t do

Ngu mat vat chuy6n clang tinh tign theo mat

true x, thi do dcti Ox ciia no co th6 duong hoac

am, thy theo vat chuygn clang theo chieu x tang,

hoac giam Cung vay, do clai goc 6.0 curia mat

vat quay cung co th6 cluong hoac am, thy theo

vat quay theo chieu 0 tang (tren cac hinh 11.3

vä 11.4 la nguoc chieu kim clang ho) hoac

giam (theo chieu kim clang h6)

(11-4)

HINH 11-4 Duong m6c cua vat ran tren hinh 11.2 va 11.3, es vi tri gOc 01, vao thei diem ti va vi tri goc 02 vao thdi dim t2 Luong 4B (= 62 - 01) la dO ddi goc trong khoang thdi gian (= tz - ti) Tren hinh, khOng ve vat

VAN TOC GOC Gia sit rang (xem h 11-4) vat quay dm too vi tri gdc 01 vao thai didm t1 va 6 vi tri gdc 02, vao thdi didm t2 Chting ta dinh nghia van toc goc trung binh cua vat là

trong do AB la dO cloi gdc trong khoang thoi gian At (chit w la chit Hy lap o-me-ga)

V6n t6c goc (arc th6i) a), ma cirang ta se ndi den nhieu hon ca, la gioi han cua ti s6

trong p.t (11-5), khi At den tOi khong Nhu vay :

AO

= lim T = clO

Ngu cluing ta bigt 0(t), cluing ta co thd tim duoc van t6c goc a), bang cach lay dao ham

Dinh nghia cAla van t6c goc khong nining dung duoc cho vat ran quay, xet ye Wan 130, ma

con dung duoc cho 772.6i hat cud v4t do Don vi van t6c gdc thong tinging la radian tren

giay, (rad/s) hoac yang tren giay (vg/s)

Ngu mOt hat chuydn dOng tinh tign theo mOt true x thi van tdc dai v co thd duong hoac

am, tity theo hat chuydn Ong theo chieu x tang, hoac x giam Cling vay, van t6c gdc co da

mOt vat ran quay cung co thd ducing hoac am, tuy theo vat quay theo chieu 0 tang (nguoc

chieu kim citing h6) hoac B giam (theo chieu kim deing h6) DO Ion dia van t6c goc goi la

t6c d6 goc, cung duoc ki hidu la co

GIA TOC GOC

Ngu van t6c gdc cua mOt vat quay khOng phai la hang s6, thi vat co gia t6c gdc Goi co2

va co i Ian luot la van t6c gcic d thbi diem t2 va t1 Gia t6c goc trung binh cfm vat quay

duoc dinh nghia la

-

CD2 1 Act)

trong do Acv la do bign thien cila van t6c gdc trong khoang thbi gian At Gia t6c gOc

(tic th6i) a ma cluing ta se ndi de'n nhieu hon ca, la giai han cim dai Wong nay, khi cho

At tign den tOi khong Nhu vay

(11-6)

Dinh nghia ye gia t6c gdc khong nhang dung &foe cho vat ran quay, xet ye toan bi), ma

con dung duoc cho ca moi hat caa v4t do Thong tinging, don vi gia t6c gdc la radian tren

giay binh phuong (rad/s2) hoac \Tong tren giay binh pinging (vg/s2)

Bai Wan mat! 11-1

Vi tri goc cua mOt duting mdc, tren mOt banh xe dang quay duoc cho boi :

= t3 - 27t + 4,

trong dO t &foe do bang giay, va 0, bang radian :

a) Tim co(t) va a(t)

7

Giai Dd co cluoc co(t), eh-tang ta lay dao ham cda 0(t) theo t :

b) Vao ldc nao thi co = 0 ?

Giai Cho co(t) = 0, ta &roc

c) M6 ta chuydn Ong cda banh xe, khi t 0

Giai Dd tra loi, chiing ta xet cac bidu thdc cda 0(t), co(t) va a(t)

Ltic t = 0, ddOng m6c cua banh xe cs 0 = 4 rad, va banh xe dang quay vdi van t6c goc

- 27 rad/s (tic la, theo chiseu kim clsOng 116 vdi van t6c goc 27 rad/s) va vbi gia t6c gdc bang

khong

Vdi 0 < t < 3s banh xe tidp tac quay theo chieu kim dong h6, nhung vdi van t6c gcic giam dan, vi bay gib no co gia t6c gOc deicing (ngtioc chieu kim d6ng h6) (tim co(t) va a(t), vdi t = 2s, chang han)

Dic t = 3s, banh xe cldng lai ch6c lat (co = 0) va td trade den gib no van quay theo chieu kim d6ng 116 (during m6c bay giO a 0 = -50 rad)

Vdi t > 3s, gia t6c g6c cua banh xe tip tac tang Van t6c goc cda no, ma bay gib cling ngdoc chieu kim d6ng ho, se tang nhanh vi dau cua w va a phd hop vdi nhau

Bai town mfiu 11-2

MOt con quay quay tren san vdi gia t6c gOc

z

True-

z

(a)

HINH 11-5 (a) MOt cat dia hat quay quanh mOt truc thang ding trung slot

truc cita may (b) Van t6c goc cua dia 05 the biL dien bang vectd co,

tren truc quay, va chi xuong dUOi, nhu ta thgy tren h (c) Chung ta thi6t lap chieu &la vectd van t6c goc la chi& !Wong xu6ng, bang cach dung quy tac ban tay phai Khi cac ngon tay cita ban tay phai chay tron quanh dia va chi &tong ma dia quay, thi ng6n tay cat choai ra, chi chi& ciaa w

Va.y : C = 5 rad/s Do do:

trong do, C' dtioc tinh, nhO chi' Yr rang 0 = 2 rad litc t = 0

Chung ta co thg ma ta vi tri, van t6c va gia t6c cua met hat rieng le bang cac vecto Ngu hat bi buOc phai o tren met &tong thang, thi ta khOng that can dgn tiu thg cua vecto Hat nhti vay chi co hai pinking danh cho no, va ta co thg tro hai phtiong ay bang cac da'u cOng va

Cung nhti vay, mOt vat ran hay c6 thg quay xung quanh mOt truc co dinh chi co thg quay theo chieu hoac ngtioc chigu kim dong h6 quanh true dd, va chting ta lai co thg phan biet hai chieu ay, bang cac dau Ong va trit Nay ra cau hed : "Lieu ta co thg xit H dO dili gdc, van t6c goc va gia tgc g6c cita mOt vat ran quay, nhti cac vecto ?" Cau tra l i (voi sit

de dat ma ta se lam rO, dttai

day) la : "Co"

Hay xet van t6c gdc Hinh

11-5a trinh bay mOt dia hat

quay quanh mOt true c6 dinh

Dia co t6c de quay co khong dOi

(= 33 1/3 vg/ph) va co mOt chigu

quay c6 dinh (theo chigu kim

d6ng h6, nau nhin tit tren

xuang) Chung ta quy tide bigu

din van toc gcic cua no bang

mOt vecto httang doc theo chigu

true quay, nhti trail hinh 11-5b

Chung ta chon dO dai cfia vecto

nay theo met thang do thuan

tien nao do, chang han, veil 10

vg/ph bang 1cm

9

(c0

(b)

Chung ta thigt lap met chidu cho vecto co, bang cach dung quy tac ban tay phai, nhu

ta thay tren h 11-5c Khum ban tay phai cho cong theo dia quay, dd cac ngdn tay cda ban

chi chiseu quay Khi do, ngdn cai cda ban choai ra se chi chieu cda vecto van t6c goc Ngu dia quay theo chieu nguuc 14i, thi quy tac ban tay phai se not cho ban bidt rang, vecto van t6c goc bay gib hoeing theo chidu ngticic 141

Khong de gi lam quen \Teri viec bidu din cac dal Wong gcic bang cac vecto Theo ban

nang, chfing ta cho dal rang cd met vat nao do chuydn creng theo chieu dm met vecto 0

day, lai khong phai thg That vay, met vat (vat ran) lai quay quanh phuong cua vecto

'Prong linh vile sii quay thuan ttly, thi met vecto xac dinh met true quay, chit khong xac dinh chieu chuydn Ong cUa met vat nao dci Tuy nhien, vecto cling xac dinh chuydn deng Hon nita, no tuan theo moi quy tac vd phep tinh vecto, da not clan trong chuong 3 Gia t6c

goc a la mot vector' khac, va ne cling tuan theo cac quy Vac ay

Bay gib clan su de dat da nhac ci tren Cac do dai goc (trit phi cluing rat nho) khOng the

coi dirge la vecto Tai sao lai khong ? Tat nhien là chting ta co thd cho chting cd chidu Ian

do 16n, citing nhu cluing ta da lam dgi vOi vecto van t6c goc trong hinh 11-5 Thy nhien do indi la (theo each ndi cda nha toan hoc) didu kign can, nhung chua du Dd co thd bidu din

duct bang vecto, met dai Wong can phai tuan theo quy tac Ong vecto, ma met trong cac quy the do not rang, ndu ban ceng hai vecto, thi thd tu theo de ban Ong chting se khOng

coy nghia gi De cloi gee khong vuut qua duuc phep this nay

De' thay diet' nay, hay dat met quydn sach det tren san, nhu hinh 11-6a Bay gib cho

quydn sach thuc hien hai phep quay 90° lien tigp, 16n dau, quanh true (na.'m ngang) x va

sau do quanh true (thang citing) y, dung quy the ban tay phai cid xac dinh chidu cluong

trong moi trubng hop

Bay gib, voi quydn sach ci cling vi tri ban dal' (h 11-6b), hay thuc hign hai de dbi gOc

ay theo thti tti ngdoc lai (ttic la, tan dau quanh true y, va sau do, quanh true x) Nhu hinh

ye cho thay, quydn sach kat thilc u hai hudng rat khac nhau

Nhu vay, hai Ong tac giong nhau lai sink ra kat qua khac nhau, tity theo thd tu ma anh thuc hign chting \Tay, phep Ong cac de deli khong co tinh giao hodn, tUc la de deli gee

HiNH 11-6 (a) Tit met vi tri ban dgu d ben trai, cho guy& sach hai phep quay 90° lien tiep, phep this nhgt quanh true (nA'm ngang) x, rei quanh true (thang ditng)y (b) Quyen sach vAn nhgn dtldc hai phep quay ay, nhung theo the to ngtIdc 'al Neu de dai goc thing là mot dai Wong vectd thi thU tit oh cac do dai do

se khOng có anh hong Ta thgy ro van cI6 IA cac dO (Jai goc (len) kitting phai là dal Wing vectd, du ta co the' gan cho chOng CA

dO len IA'n chieu

Phuang trinh só Cong thitc dai

khong phai la dai Wong vecto VOi mOt chut kinh nghiem, ban co thd chtIng minh rang cac

vi tri cu6i ding cita cu6n sach cang gan nhau hon, ndu ban dung nhang do cloi nhO hon

90° cang nhieu Trong truong hop gidi han cita nhiing di) clOi vo ding nhO (nhti dO trong

p.t 11-6), cac do doi g6c c6 the coi nhu dai luong vectd

Trong su tinh tin thuan ttiy, chuydn dOng vdi gia toc dai kh6ng doi (thi du, chuydn

dOng cda mOt vat roi) la mot true ng hop rieng quan trong Trong bang 2-2, chting t6i trinh bay mOt boat phuong trinh cua chuydn dOng trong su quay thudn ttly, trubng hop gia

t6c goc khong doi cung quan trong, va mOt boat song song cac p.t cung ap dung dugc cho Bang 11-1

PHUONG TRINH CAC CHUYEN DONG VLSI GIA TOC DAI

VA GIA TOC GOC KHONG DOI

Bien sa Ong n- 4t C6ng thlic g6c PhUdng trinh s6

1 2

= wot + - 2 at (11- 10) co2 = wo + 2a0 (11-11)

trudng hop nay Chung tOi se khong thigt lap cluing, ma chi don gian vigt chting ra tii cac

p.t dai to ong ging, ma the cac dai Wong gcic vao cac dai Wong dai Bang 11-1 trinh bay ca hai boat p.t Ay Dd don gian, chting toi d'a cho xo = 0 va 00 = 0 trong cac p.t ay VOi cac

diseu kien clau ay mOt di) doi dai i x (= x - xo) thi bang x va mOt dO cloi gcic O0 (= 0 - 00)

Lang 0

Bai ton mfiu 11-3

M6t dia mai (h 11-7) co gia t6c g6c khOng d6i a = 0,35

rad/s2 N6 bat Tau chuydn thing tif nghi (ttic la, coo = 0), voi

mOt dtaing m6c nam ngang thy y, tit vi tri g6c 00 = 0

a) lioi dO doi g6c 0 cita dtiong m6c (va do d6, cita dia) luc

t = 18s

Giai Ta dung p.t (11-10), tren bang 11-1 va duoc :

HiNH 11-7 Bai town mAu 11-3 va

tren dia) IA dam ngang

11

1

= (0)(18s) + (.0(0,35 rad/s2)(18s)2 =

= 56,7 rad 57 rad = 3200° 9,0 vg

b) Van t6c gdc caa dia, lac t = 18s la bao nhieu ?

Giai Bay gib ta lai dung p.t (11-9) caa bang 11-1, dg duoc

Bai town mfiu 11-4

Voi cai dia mai trong bai toan mau 11-3, ta gid sit rang gia t6c gdc van the (a = 0,35

rad/s2), nhung bay gib gia sit rang dia khOng bat dau chuygn dOng tit nghi ma da có met van t6c goc ban dau co., bang -4,6 rad/s tic la, gia t6c gcic ban dau co tac dung lam dia quay cham

a) V'ao thai digm t nao dia mai se tam thoi ding lai ?

Giai Giai p.t (11-9) co = coo + at d6i voi t, ta duoc

0 - (-4,6 rad/s)

0,35 rad/s2 b) Vao the i digm nao, dia mai da quay da dd de dbi gdc dia no la nam yang, theo chieu quay duong ? (Khi do, de dbi goc caa vach m6c se la 0 = 5 vg)

Giai Ban dau, dia quay theo chieu am, veil c.o = -4,6 rad/s, nhiing gia t6c gcic a caa

no lai duong Si trai dau ban dau gift:a van t6c goc va gia t6c goc co nghia la, dia quay cha.m dan, trong sU quay theo chieu am, dUng lai, va sau do, quay nguoc lai, theo chieu

&tong Sau khi dubng m6c tro tai vi tri dau caa no voi 0 = 0, dia can piled quay them nam yang naa, dd dat de cloi goc ma ta can Day la dieu "can hgt stic chit y" ndu chang ta dung p.t (11-10) :

0 = coo +2at2

The cac gia tri da bigt va dat 0 = 5 vg = 10.E rad, cho ta :

107r rad = (-4,6 rad/s)t + ( 1 ) (0,35 rad/s2)t2

Chu y rang, vai t tinh ra giay, thi cac don vi trong p.t nay deu pith 'Kip BO cac don vi (cho thuan tien) va xdp thu tie, ta duck :

Gidi p.t bac hai nay theo t, va loai nghiem am, ta chicle :

CACH GIAI BAI TOAN

Chie'n thmat 1 : Nhung dap so bat nga

Chd vOi va quang di mot nghiem cua p.t bac hai, vi coi no nhu vo nghia Nhigu khi, nhu trong bai toan mau 11-4, met nghiem bi loai lai coy nghia vat 11

Hai nghiem dm p.t (11-14) la t =

32s va t = -5,6s Chung ta chon nghiem

thti nhat (dtiong) va khong bigt rang

co le nghiem thti hai cling co y nghia

Nhung no la gi ? Thdi gian am trong

bai toan nay co nghia don thuan la mot 60

thdi digm tract lite t = 0, ttic la try& 40 _./ 7 31,4 rd d ( ;5 v i

ltic chung ta Mt Tau quan tam den e 7.0 i i r ar - 7

/ - 32s

I

Hinh 11-8 la dO thi cac vi tri gcic -h2 , I- - 1 - 20 io -f 20 c 30

4 0 ! b

toan mau 11-4, coi nhtt mot ham dia i _i_ l i

I thdi gian, del vdi ca thdi gian dtiong

lan thdi gian am D6 la do thi cua p.t

11-10 (

/

0 =coo + — at2 \ , vol th e hang

2

trong bai toan mai 11-4 Thoi diem am (ate la thdi diem truck 16c

t = 0) da dUdc dila vao Hai nghiem cua p.t 11-14 dUdc ghi bang

s6 co = -4,6 rad/s -yd a = +0,35 rad/s2 cac diem d va e

Diem a -ling voi t = 0, 0 thdi digm do

chting ta da tut y lay vi tri gOc dia vach nide bang khOng Laic do, dia dang quay theo chigu

0 giam va tigp tuc quay nhtt the cho do'n ltic thing lai, 0 digm b, ltic t = 13s Sau da, no

quay tro lai, vdi &tong m6c tro lai vi tri 0 = 0, tai digm c, va di tigp them nam yang (= 31,4 rad), den digm d Digm nay (t = 32s) la nghiem ma eh-tang ta da chap nhan lam dap

s6 dia bai toan dia ta

Thy nhien, hay chi.' y rang dudng m6c da 0 citing vi tri g6c ay ltic t = -5,6s, truce lac

"chinh thtic bat dau" dm bai toan Nghiem nay (digm e) cling co gia tri nhu nghiem 0 digm

d Dieu quan trong lion, khi tai hoi xem nghiem am nay cci thg coy nghia vat li gi, la ban

se biet duce them mot chtit vg sit quay cila dia

Bai toan mau 11-5

Trong mot Ian nghien ctiu ye may bay truc thang, ban xac Binh &floc rang van t6c cua rot° thay d6i tit 320 vg/ph dgn 225 vg/ph, trong 1,5ph, khi rot° quay chain dan dg thing lai

a) Gia t6c g6c trung binh cua canh quat rOto trong khodng thdi gian nay la bao nhieu ? Tit p.t 11-7

b) Vol gia the trung binh nay, sau bao nhieu lau cac canh quat se diing lai, kg tit hic

cluing co van t6c goc ban dau 320 vg/ph

Giai Giai p.t 11-9 (co = coo + at) d6i yeti t, ta doge :

11-5 CAC BIN SO DA' VA BIN SO GOC

'Prong muc 4-7, chimg ta da nghien cdu chuygn Ong trim deu, trong cid met hat di vai van t6c dai khong d6i, theo met &king troll va quanh mot true quay Khi met vat ran, chang han cai yang quay ngda g6 (h 11-9) quay quanh met true quay, m6i hat tren vat chuygn Ong tren during trOn cda no quanh true do Vi vat la vat ran, nen moi hat dia vat

deu quay met \Tong, trong ding met thoi gian, tie la chung co cung met t6c de g6c co

Thy nhien, hat tang ct xa true, thi chu vi during trail cua ne tang lan, va, nhti the, toc

do dal cda no tang cao Ban co thg nhan thay dieu do tren vOng quay ngua go Ban quay

veil ding met t6c de goc co, khong kg ban each tam digm bao xa, nhung t6c de dai cda ban

tang met each dang kg, ngu ban chuygn dgn mep ngoai cda \rang quay

Chung ta nhieu khi can lien he cac bign s6 dai s, v va a dia met digm rieng biet, tren met vat quay, vdi cac hien s6 6, co va a cda vat d0 Hai nhom bign s6 d0 lien he vdi nhau qua r, khodng each vuong goc tit digm ay den true quay Khoang each vuong gcic la khodng each gida digm ay va true quay, do theo met dtiOng vuong gee yeti true Ban kinh r cda

thong tron vach bdi digm do quanh true quay cho ta khodng each vuong gcic

Vi tri

Neu met during mot tren vat ran quay (lucre met gdc 0, thi met digm dia vat chuygn Ong tren met cung tran, qua met khoang s, cho boi p.t 11-1

s = 6 r (gcic do bang radian) (11-15)

Day IA he thilc do dai - gdc dau den cda ta Gdc 0 phai do bhng radian vi ban than p.t

11-15 là dinh nghia s6 do goc bang radian

HINH 11-9 MO ngtIoi cUesi ngtla tren yang quay ngua g6 deu quay quanh truc giCta, vai

ding mOt t6c dO gOc w

Nhung ngttoi nao 6 xa truc hdn thi chuyen dOng v.:3i toe dO dai 16n hon

T6c cto

Dao ham p.t 11-15 theo thei gian, giii r khOng d6i, (Ian den :

ds dO

dt dt

HINH 11-10 Vat ran quay tren

h 11-2, nhin theo mat cat ngang M6'i diem cila vat (thi du, P) chuyen Ong tren mOt &tong tron quanh truc quay (a) Van t6c v cua m6i diem tier) tuydn v6i clueing tr6n tren d6 no chuyen dOng (b) Gia t6c dai dia diem ay (trong trttong hOp tong quat) co hai thanh phan : mOt thanh phan tip tuyen at va mOt thanh phan xuyen tam ar

y

(b)

15

Nhung ds/dt la t6c do, dai (do lon cda van t6c dai) cda diem ta dang xet va do/dt lu tot

de g6c dia vat quay, thanh this

Met lan nita, t6c dO gcic w phai &toe tinh theo radian tren giay Phuong trinh 11-16 cho

ta thay rang, vi moi digm trong vat ran deu cd ding t6c do gOc w, nen ban kinh r lan hon

se cho t6c de dai Ion hon Hinh 11-10a cho thay van t6c dai bao gib cung tiep tuygn yeti quy dao trim cda didm ay, vi ding met nguyen nhan nhu trong chuygn dOng troll (muc 4-7)

phan do cua gia tot dai thi tigp tuygn \TIM dubng di dm digm dang xet Ta goi phan nay la

thanh phari tiep tuyen at cda gia tot dai cda digm ay, va vigt :

at = a r (goc do bang radian) • (11-18) Ngoai ra, nhu p.t 4-22 cho thay, mOt hat (hoac met digm) chuygn dOng tren met dubng

tren co mOt thimh philn xuyen tam ar cda gia tot dai, ar = v2/r, gay ra cac thay ddi ve

phuong cda van t6c dai i";-.)'Thg v, tii p.t 11-16 vao, chung ta co thg vigt thanh phan nay,

thanh :

v2

r = co2 r (gOc do bang radian)

Nhu vay, ta thay tren h 11-10b, gia tot dai cua met diem tren met vat ran quay, ndi chung co hai thanh phan Thanh phan xuyen tam ar (cho bbi p.t 11-19) bao gib cung c6 mat, chdng nao ma van tot gOc cda vat con chua bang khOng Thanh phan tigp tuygn at (cho loch p.t 11-18) thi cO mat chdng nao ma gia tot gcic con chua bang khong

Bai toan mAu 11-6

Hinh 11-11 trinh bay met may quay li tam, dg bat cac nha du hanh vu tru tap chiu

dung cac gia t6c lOn Ban kinh r da dubng trim vach ra boi mOt nha du hanh la 15m

a) May li tam phai quay vdi van tot goc khong d6i bao nhieu, ngu nha du hanh phdi chiu dung met gia tot dai, bang 11 lan gia toc roi to do ? Day phong chdng là gia tot cdc dai ma mOt phi ding lai may bay tiem kith duoc ren luyen tot co the chiu ddng &foe -trong met khoang thbi gian ngan - ma khong bi hoa mat

Giai Vi van t6c gdc khong d6i, nen gia tot gcic a (= dw/dt) bang khOng va thanh phan

tigp tuygn (xem phuong trinh 11-18) cda gia tot dal cung thg Chi con lai thanh phan xuyen tam Tit p.t 11-19 (ar = w2.r) ta cluoc :

ar

v = cox = (2,68 rad/s)(15m) = 40 m/s (Dap s6)

c) Ngu may li tam tang toc deu dam ti nghi dean van t6c gcic trong cau (a), mat 120s,

thi gia t6c tip tuygn &la nha du hanh vu tru la bao nhieu ?

Giai Vi gia t6c gdc khong d6i trong thdi gian tang t6c cita may li tam, nen p.t 11-9 ap

Mac dam gia t6c xuyen tam cu6i cling ar la ion (va clang bao &Ong), gia t6c dal trong

thdi gian tang t6c lad khong

—

17

CACH GIAI BAI TOAN

Chien thu4t 2 : Cac don vi cho cac bin s6 gcic

Trong p.t 11-1 = slr) chting ta da tti rang buec minh là dung radian cho moi phep

do gac Nghia la chung ta da phai bidu din dO disi gdc bang radian, van t6c goc bang rad/s hoac rad/ph, va gia t6c gcic bang rad/s2 hoac rad/ph2 Ngoai 10 dec nhat cita guy tac nay la

cac p.t chi chtla cac bign s6 g6c, thi du cac p.t gcic Het ke tren bang 11-1 0 day, ban co

thg tity y &mg bat ki don vi nao ban thich, cho cac bign s6 gcic Nghia là ban co thg dung

radian, de, hoac \Tong, chitng nao ban can sit dung cac p.t

Trong cac p.t phai dung s6 do bang radian, ban khong can theo d6i cac bign doi dai s6

dm don vi "radian" (rad), nhu ban phai lam cho cac don vi khac(1) Ban co thd them vao hay ba no di, tuy tiing tracing hop Trong bai Wan mau 11-6a, don vi do da dtioc them vao cho dap s6, can trong bai than mau 11-6b; va 11-6c, no lai bi ba di

11-6 BONG NANG QUAY

Luai ctta cita met cai cua dia quay nhanh cha.c chan la co clang nang Chung ta lam the nao bidu dien duoc clang nang ay ? Chung ta khOng thg truc tip dung tong thitc quen

1

thuoc K = —

2 mu- , vi no chi ap dung cho cac hat, va chting ta khOng bigt Fay gi lam v Thay vao dO,_ chting ta se coi dia ctia (va bat ki vat ran quay nao khac) nhu met tap hop nhieu hat co van tot khac nhau, sau do chung ta co thd citing clang nang cita moi hat cid lai, dd tim duoc dong nang cua than be vat Bang cach do, chting ta thu duoc clang nang

am mot vat quay :

K = 2 — m , + 2 — m v2 + — m 2 2 2 3 3 V2

trong do mi la kh6i Wong cita hat thu i va vi la van tac cua no T6ng phai lay cho moi

hat tao thanh vat ay

Van de la trong p.t 11-20, vi khOng co ding met gia tri d6i yeti moi hat Chung ta giai

quygt van de nay bang each the v tit p.t 11-16 (v = co.r), va duet :

K = 2 m.tr- = 2 m (co • = — (Zinr") 2

trong do, w co ding met gia tri, \Tad moi hat

Dai Wong trong dau ngoac a ye phai cua p.t (11-21) cho ta bigt, kh6i Wong cua vat quay duoc phan b6 the nao quanh truc quay Ta goi dai Wong ay la quan tinh quay(2) cita co" thg d6i veil true ay Nhti \ray :

I = Imfi (quan tinh quay) (11-22)

(1) Vi (rad)", viii n bAt ki, An gill thit nguyen 1 (N.D)

(2) ThUang duoc gni la 'Women quan tinh

tam thanh lai lOn hdri rat nhieu, so v6i thanh (c)

Thg vao p.t 11-21, ta duoc bigu thtic phai tim :

K = 1 Ico2 (gdc do bang radian) (11-23)

Vi chtIng ta der dung he thiic v = co.r khi rut ra p.t (11-23) nen w phai dugc bigu din bang radian/Oon vi SI cita I la kilogam nhan met binh phtiong (kg.m2)

Phuong trinh (11-23) cho ta citing nang cua mOt c6 the chuygn Ong quay thuan tuy,

1

is p.t gdc tuong &tong cong thiic : K = 2 kh.t cho ta dong nang cua c6 th6 chuygn

1 dOng tinh tign thuan thy Trong m6i truong hop, deu co thila s6-

2 Ch6 nao ma M (cd thd

goi la quern tinh tinh tien) xuat hien trong cong attic nay thi I (quern tinh quay) lai co mat,

trong tong thiic kia Sau niter, m6i p.t chtla mOt thila s6 la binh phuong cua t6c do, tInh tign, hoac quay, thy loai chuydn dOng DOng nang tinh tign va dOng nang quay khong phai

la hai dang nang lugng khac nhau Ca hai deu la dOng nang dugc bigu din mOt each thich hop cho moi loai chuydn Ong

Quan tinh quay dia mOt vat quay khong nhfing phu thuOc kh6i luting dm no, ma con phu thuOc kh6i luting dd dtioc phan b6 thg nao d6i vizi true quay ay Hinh 11-12 goi cho

ta mOt each co sue thuyel phut, d6 cam nhan quart tinh quay Hinh 11-12a trinh bay hinh dang ben ngoai cfm hai thanh chat deo, ben ngoai trong rat giong nhau Ca hai ding dai chUng lm, kith thudc va trong luting cua hai thanh deu bang nhau, va ca hai deu can bang quanh trung digm cua chting Ngu ban cam vao chinh giita mai thanh, va dua ra, dua vao rat nhanh, bang mOt chuygn (Ong tinh tidn, thi ban con chua thg phan biet dugc chting Thy nhien, ban se nhan thay mOt

su khac nhau that dang ngac nhien,

ngu bang each xoay di xoay lai c6

tay, ban bat hai thanh quay nhanh

trd di trot lai (nhii mOt cay gay) Mot

thanh thi quay hohn toan d6 clang ;

thanh kia, thi khong ; tren hinh

11-12b va 11-12c cho thay rang,

thanh "d6 quay" cd hai qua nang ben

trong, tap trung b gin tam, va

thanh "khci quay" thi chi:Mg laid hai

dau Mac dau trong lacing cua hai

thanh bang nhau, quan tinh quay

quanh mOt true chinh gala lai hoan

toan khac nhau, vi su phan b6 kh6i

Wong cita chOng khac nhau

19

11-7 TINH TOAN QUAN TINH QUAY

Ndu mOt vat ran tao thanh tut cac hat r6i rac, thi ta co thg tinh dli6c quan tinh quay cila nd theo 11-12 Ndu vat la lien tuc, thi ta co thd thay tong trong p.t 11-22 bang mOt tich phan, va dinh nghia quan tinh quay tro thanh :

I = f r2dm (quan tinh quay lien tuc)

'Prong cac bai toan mau ghi trong muc nay, cluing ta tinh I cho ca hai loaf vat Ndi

chung quail tinh quay Mt ki cua c6 thd nao dgi voi mOt truc quay deli phu thuOc (1) vao hinh dang cua vat, (2) vao khodng each vuOng gOc tut truc quay t6i kh6i tam cua vat, va (3), vao ski dinh hue ng dm vat d6i vdi true

Bang 11-2

VAI QUAN TINH QUAY

Cal vOng, quanh truc gida,

Vanh truc (ling tru) quanh truc

Hinh tru dac (hoac cal dia) quanh true gida

Hinh tru dac (hoac cai dia) quanh during kinh gida

di qua tam va vuang gOc vdi thanh

Thanh mong quanh mot true

di qua dau thanh va vuOng gne vdi thanh

Qua eau dac, quanh mOt dUong kinh bat

ki

Qua cat., rOng, m6ng quanh mOt dual-1g kinh bat

Tam quanh mOt true vuOng g6c, qua tam

12 m(a2

b2)

(j)

Bang 11-2 cho to quan tinh quay caa vai vat ran thOng thugng, quanh cac true khac

nhau Hay nghien cliu can than bang nay, dd coy nigm, su phan b6 kh6i Wong có anh

huang the" nao clan quail tinh quay

Dinh li ve true song song(*)

Mu ban biat quan tinh quay cila mOt vat quanh bat ki mOt truc nao di qua kh6i tam

cua no, thi ban co thg tim dugc quan tinh quay &la no quanh Mt ki truc nao khac song

song vai truc ay, bang dinh 1i v6 tryic song song, do la

= Ikh.t Mh2 (dinh li true song song) (11-25)

'Prong 016, M la kh6i Wong cila vat va h la khoang each vuong gdc gala hai true (song

song) Dung 1cui, thi dinh li dugc phat bidu nhu sau :

Quan tinh quay caa mat vat quanh mat truc bat ki, thi bang quan tinh quay (=

Mh2), ma no co dai vai truc do, neil toan ba kh6i luang caa no tap trung yew khai tam

caa no, Ong voi quan tinh quay (= ikh.t) caa no dai vai true song song, di qua khai

tam cact no

(*) Sach ViOt va sach Phap : dinh li Huyghens hoac Steiner (ND)

21

x

Chung minh dinh li true song song

Goi 0 la kh6i tam cita met vat, hinh dang bat ki, cd tigt dien trinh bay tren hinh 11-13

Dat g6c toa do el 0 Xet met true di qua- 0, vuong goc voi mat phang cita hinh va met truc

khac, di qua diem P va song song voi true thti nhat Goi toa de cita P la a va b

Goi dm la met kh6i luong nguyen t6, co toa de x va y Quan tinh quay cita vat, quanh truc di qua P, theo p.t 11-24, la

I = f r2 dm = f vx - a)2 + (y - b)2] dm

Ta co thg sap xep lai thanh :

I = f (x2 + y2) dm - 2a f xdm - 2b f ydm + f (a2 + b2) dm (11-26) Theo dinh nghia kh6i tam (p.t

9-9), hai tich phan u giiia p.t 11-26

cho ta toa de cita kh6i tam vat (nhan

yea met hang s6) Vi (x2 + y2) bang

R2, trong do, R la khoang cach tit

0 dgn dm, nen tich phan nhat

don thuan la ikh.t, quart tinh quay

dm vat quanh met true di qua kh6i

tam Xet h 11-13, ta thy rang s6

hang cu6i cling trong p.t 11-26 la

Mh2, trong do, M la kh6i Wong toan

phan Mat vay, p.t 11-26 rut lai

thanh p.t 11-25, la he thtic ma ta

phai chi:mg minh

H1NH 11-13 Tiet dien mOt vat tin, vdi kheii tam 6 0 Dinh li truc song song (p.t 11-25) lien he quart tinh quay dia vat quanh mOt true

di, qua 0, v6i quan tinh quay quanh mat nix song song di qua mOt diem, diem P chang hall, d cach kh6i tam mOt khoang h CA hai truc

(feu vuong gOc v6i mat phAng dia hinh

Bai town mfiu 11-7

Hinh 11-14 trinh bay met vat ran, tao beti hai hat, kh6i Wong m, gam yeti nhau bang met thanh, dO dai L, co kh6i Wong khong dang kg

a) Quan tinh quay dia vat quanh met truc di qua tam va vuong gcic thanh (xem h.11-14a) la bao nhieu ?

GiAl Theo p.t 11-22, ta co :

1

1

I = .r12 = (m)L ) 2 + (m) (-2- L) 2 = mL2 1 (Dap s6) b) Quan tinh quay dia vat quanh met true di qua met dau thanh va song song veli true thti nhat, nhti tren h.11-14b, la bao nhieu ?

Giai Ta co thg dung dinh li true song song, dia p.t 11-25 Chung ta vita tinh dticic /kh.t

trong phan (a), va khoang cach h giita hai true song song, thi bang nita de dai cua thanh

True quay

C/77

Ta có thg nghi6m lai kel qua nay, bang each tinh tr‘ic tigp, theo p.t 11-22

I = Infi = (m)(0) (m)(1-)2 = mL2 (Dap s6)

Bai town mfiu 11-8

Hinh 11-15 trinh bay mot thanh ming, d6ng

tinh, kh6i Wong M va do dai L

a) Quan tinh quay cila no quanh mot true

vuong goc vai thanh, va di qua khoi tam no, la

bao nhieu ?

Giai Ta chin nguyen to kh6i Wong la mot

khuc dx cita thanh Tam dia kink nay a vi tri

x Kh6i Wong dia mot don vi dO dai cita thanh

la do do, kh6i Wong dm cua nguyen to dx

là :

(al

(6) HiNH 11-14 Bai toan mAtt 11-7 MOt vat ran, g6m hai hat, kh6i lUdng m, n6i voi nhau bang mOt thanh,

kh6i lUdng khOng clang ice

Digu nay phi hop vai kgt qua cho trong bang 11-2(e)

b) Quan tinh quay dia thanh quanh mot true vuong goo vOi thanh, di qua mot da"u thanh,

Bai town mAu 11-9

Met phan tii axit clohidric gom met nguyen ta hidre có kh6i Wong mH bang 1,01 u va

met nguyen tit do, co kh6i Wong mci bang 35,0u Tam cua hai nguyen ti o each nhau met

khoang d = 1,27.10-10m = 127pm (h.11-16) Quan tinh quay cua phan tit quanh met truc

vuOng goc vdi doing n6i hai nguyen va di qua khoi tam cua phan t*, là bao nhieu ?

Goi x la khoang cach tit kh6i tam cua phan tit toi nguyen ti clo Khi do, theo h.11-16 va p.t 9-3, chang ta thay :

-max +mH (d - x)

ma ±mH

Hay la mei = rn H(ci x)

Phuong trinh nay cho ta :

m H mCI + n2,H

Theo p.t 11-22, quail tinh quay quanh met truc di qua

khol tam la

Cac don vi nay Maa thuan tign khi phai dung cham vai quail tinh quay cua phan tii Neu

ta dung angstrom lam don vi de dai (1A = 10-1°m), thi dap s6 thanh I = 1,58uA2, met don vi con thuan tien hon nua

Bai town mai., 11-10

VOi tong nghg hien dai, co thd chg tao met banh da co kha nang tick trit nang Wong old cho chay met cai OtO Ban clau nang ltiong dugc trit dudi dang Ong nang quay, khi banh

db dugc met cai may lam cho quay Sau 00, nang lugng tich trii dligc chuygn clan cho Oto, bang met hg truygn Ong, lac cho chay oto sit rang met banh da nhti vay dugc chg tao

dudi dang met hinh tru dac, co kh6i ltigng M = 75 kg va cd ban kinh R = 25cm Mu banh

quay voi 85000 vg/ph, thi no trO dugc met Ong nang quay la bao nhieu ?

Giai Quan tinh quay &Ia banh da hinh tru, theo bang 11-2(c) la :

(a) (b

Lacing nang Wong nay, ngu &toe sit dung \Teri hieu suat dtt kidn, se lam cho mat ato nho

chay chang 200 dam

11-8 MOMEN QUAY

Qua nam cita canh cita &toe dat hgt sac xa ban le cap., vi mat li do xac dang Ndu ban mu6n mcs mat canh caa nang, chac chan ban phai tac dung mat laic ; tuy nhien, chi the' thOi, thi data du Ban plied dat laic vao ch6 nao, va day theo hue:trig nao cfmg quan trong Ndu ban OA ltic cita ban gan ban le hon qua nam, hoac dadi mat g6c bat ki khac 90°, d6i vol mat phang caa cita, thi ban phai dung mat lac ldn hen, so veil khi ban dat lite vao qua nam ya vuOng g6c vdi mat phang cua cita

Hinh 11-17a trinh bay tit dien caa mat vat c6 thd quay dttoc quanh mat true di qua

0 va vuOng g6c vdi tia dien Mat laic F dtioc dat vao diem P ; vi tri caa no doi vdi 0 dude

xac dinh boi vectcs vi tri 77> Cac vectcs F viz i.>ram vbi nhau mat g6c ¢ (de' don gian, chang

ta chi xet nhfing laic khong co thanh phan song song veil true quay)

HiNH 11-17 (a) Mot lac F tac dung tai diem P vac) mot c6 the co the quay dU6c quanh rn0t truc di qua 0, vuOng

g6c v6i mat phang cua tilt dien ye tren hinh MOmen do lite nay dung IA Fr sin0 _÷

(b) MOmen c6 the vitt IA F, r, trong dO F, IA thanh phan tiep tuy6n cua F (c )4 MOmen (cua lvc F)

cling co the vi6t IA Fr1, trong dO r1 IA canh tay dOn cita F

De' xac dinh xem F tac clang thg nao vao sti quay caa vat quanh true quay, ta phan tich

F ra lam hai thanh phan (h.11-17b) Mat thanh phan, goi la Meath phan xuyen tam Fr

httOng theo F.> Thanh phan nay kitting gay ra su quay, vi no tee dung theo mat daring keo

dai qua 0 (Ndu ban it& mat canh cita song song vdi mat cita, thi ban khong lam canh cita

quay doge)

Thanh phan kia caa F, goi la thanh phan tiep tuyin Ft thi vuong g6c co do ldn

Ft = FsinO Thanh phan nay mdi gay ra su quay (Ndu ban day mat canh cita, vuong goc

vdi mat canh cita, thi ban lam quay canh cita)

Kha nang lam cho vat quay caa Ft khong nhang phu thuac ctibng do dia no, ma can

phu thuac Ft, va do d6 laic F, dtioc dat each 0 bao xa De bao ham ca hai ydu t6 ay, chting

ta dinh nghia mat dai Wong goi la momen quay (hay cita laic F) Z, bang tich caa hai thita

so, va vigt :

* 'Prong thuat ngit tieng Viet ; "mOmen cua mot Iktc, hay "mOmen We"

25

lam cho vat quay

Trail h.11-17b, chUng ta co thd coi r, (10 Ion cua r, la canh tay dim dia thanh phan loc Ft

Mdmen quay (tit tigng Latinh torque, nghia la xodn) co thd d6ng nhat met cach khong chat the lam vdi tac dung lam quay hoac tac dung xoan &la laic F Khi ban tac dung mOt the vac) mOt vat thi du mOt cai tuavit hoac mOt cal cle Eng - dg lam

quay vat do, thi ban da tac dung mOt momen Don vi S/ cita momen la niutcin nhan

met (N.m)* Mdmen r la duong ngu no co xu hudng lam cho vat quay ngooc chieu kim Tang ho, theo chieu lam tang 0, nhu trail h.11-17 NO la am, ngu no co xu hating

lam vat quay theo chieu kim d6ng h6

Dinh nghia cua momen quay trong pt 11-28 co thg vigt dat9i dang tich vecto

Mdmen quay rla mOt vecto htiong vuong goc yeti mat phang chtla 17>va r' DO ldn cita

r duoc cho bcii p.t 11-28 , 11-29 va 11-30, con chieu cua no &toe cho boi quy tac ban tay phai Chung ta se tro lai p.t 11-31 trong chuong 12

11-9 DINH LUAT THI1 HAI CUA NEWTON CHO Sid QUAY

Hinh 11-18 trinh bay met tracing hop don gian ye

so quay quanh mOt true c6 dinh C6 the quay g6m co

mOt hat Hang Id kh6i Wong m gan olau mOt thanh

khong kh6i Wong, do dai r Mot luc F, nhu hinh ye,

tac dung va lam cho hat chuygn dOng trail mOt duong

tram Hat co thanh phan tip tuygn cita gia t6c at, tuan

theo dinh luat thti hai ella Newton :

Dai Wong trong dau ngoac u ye phdi cua p.t 11-32

la quan tinh quay dm hat quanh true quay (xem p.t

11-22) Do do p.t 11-32 rut lai thanh :

• = la (gcic do bhng radian) (11-33)

• Niuton nhan met con la thin vi tong nhien, momen va tong la hai dai Wong hoan toan khac nhau, va kh6ng dii0c

lAn 10n voi nhau Gang thUang thldc bleu thi bang jun (1J = 1N.m), nhung mOmen thi khOng

Thy thong ta da suy ra p.t 11-33 cho truing hop dac bidt cua mot hat rieng le quay quanh mot true c6 dinh, no van dung cho moi vat ran bat ki quay quanh mot true c6 dinh,

vi bat ki vat nao nhu vay cung c6 thd coi nhu tap hop cua nhieu hat rieng le

Bai toan rnAu 11-11

Hinh 11-19a trinh bay mot cai dia &Ong tinh c6 kh6i Wong M bang 2,5 kg va co ban kinh R bang 20cm, lap tren mot cai true na.'m ngang c6 dinh MOt vat nang kh6i Wong m bang 1,2 kg treo vao mot soi day khong trong Wong quail quanh mep dia Tim gia t6c cua vat nang khi roi (gia thigt rang no roi), gia t6c g6c cua dia va sac cang cua day Day khOng tricot va khOng co ma sat a true

Giai Hinh 11-19b trinh bay gian &I vat - tut do dia vat nang Vat nang duoc gia t6c

twang xuang, thanh this trong Wong cua no (= mg) phai ion hon stic cang T cua day Theo

dinh lust Newton thti hai, ta co :

T - mg = ma

Hinh 11-19c trinh bay gian do vat - ti do cua cai

dia Momen quay tac dung vao dia la - TR, la s6 am vi

no quay theo chidu kim deing h6 Theo bang 11-2, cluing

1

ta bigt rang mOmen quail tinh I cua dia la

-2 MR' (Hai

laic khac cung tax dung vao dia, la trong Wong Mg dm

no va laic phap tuygn N do gia do tac dung vao dia Ca

hai luc d'eu tac dung vao true cua dia, tuy nhien, chting

khong tac dung mOmen quay nao vao dia) Ap dung dinh

lust thtl hai cua Newton dual dang g6c r = la va

p.t' 11-18 cho dia, ta duoc :

Khi thay a bang aIR chung ta da thaa nhan - vi day khong tricot - rang gia toe dai cua

vat nang bang gia t6c dai (tiap tuygn) cua mep dia

Ph6i hop hai p.t 11-35 Ara 11-36, ta duoc :

Nhu ta dot loan, gia t6c cua vat roi phai nho hon g va slit gang cua day (= 6,0 N) phai nha hon trong Wong cua vat treo vac) day (= mg = 11,8N) Ta ding thay rang gia t6c cua

vat va slit tang Gila day phu thuOc vao khai Wong oda dia nhung khong phu thuOc ban kinh cua no De' kim lai, chung ta chti y rang cong thtic da suy duoc a tren tien doan rang

a = - g va T = 0, trong trueing hop dia khong co kh6i Wong (M = 0) Day chinh là didu

ta pita"' chb doi : vat nang roi don thuan nhu mat c6 thd ttl do, va keo theo soi day

Gia tog gcic cua dia duoc suy tit p.t 11-18 :

a —48 m/s2

a — R

0, 20m

Bai town rnAu 11-12

Dd quat nga mOt dal this 80kbang dOng tac judo co ban quang qua hong, ban du dinh keo a° cua anh ta bang mOt hie F va mOt canh tay don di = 0,30m, tit mOt diem tua (truc quay) tren hong phai dia ban, va ban quay trOn anh ta quanh truc cid vcii gia t6c goc bang -12 rad/s2 nghia la gia tag theo chieu kim d6ng ho (h 11-20) Gia sit rang quan tinh quay

I cua anh ta dai vai didm tua 1a 15 kg.m2

a) Cueing do cua F phai là bao nhieu, nen dau tien, ban keo gap anh ta vd phia truoc

dd dat kh6i tam cua anh ta len hong ban (h 11-20a) ?

Giai Neu kh6i tam cua anh ta a tren truc quay, thi vecto trong luting cita anh ta khOng tao mOmen_3uay nao quanh true do Momen quay dOc nhat tag dung vao anh ta, do do, la

do laic ken F cua ban Theo p.t 11-29 va 11-33, dai vcii mOmen quay theo chieu kim &Ong

Giai Trong trueing hop nay, trong

Wong cua dgi this ban tao ra mOt mOmen

quay duong (ngugc chieu kim dang h6), Khol tam

cua thti

va 11-34, ta co :

1144fes thong lai mOmen cua ban Theo p.t 11-29

Theo phan (a) chting ta bigt rang so

The gia tri nay va cac dtt kin da biet, ta

clugc :

= d2m g = la,

VI *

Cann tay don d1

cua luc kOo

cua ban

F

Tryc cua

hong

Cant) tay don d2

cua trong Itfigng dal thti

F

HINH 11-20 Bai town mAu 11-12 Deng tae judo gang qua

h6ng, that hien dung (a) va sai (b)

Canh tay don d,

cua Arc Mc

cua ban

(0,12 rn)(80 kg)(9,8 mls2)

Kgt qua cho thay rang ban phai keo manh them 50%, neu ban dau ban khong keo gap ngudi d6i thin dd dila kh6i tam caa anh ta len hong ban Mot vo si judo gioi thi bigt bai hoc nay, nhd vat 11 (Trong bai "Nha khoa hoc nghiep du" (Tap chi Scientific American, thang 7-1980) cd phan tich mon judo va aikido)

CONG - BONG NANG

Tren h.11-18, hay tutting Wong rang c6 thd (g6m mot hat rieng 10 kh6i luong mpn

dau snot thanh, cd khgi luting khong clang kg) quay (bloc mot gcic dO Cong do luc F thuc

la p.t gdc Wong dating cua p.t 7-10,

Day la p.t gdc tuong dung cua p.t 7-32, P = F.v, cho ta t6c do ma F thuc Men

cong, tren mot hat chuydn Ong theo mot phuong c6 dinh, viii t6c do v

De" tim cai Wong duong gdc cua climb li cong - dOng nang, ta xuat phat tit p.t 11-38,

ma thay Ia vao mOmen quay V Nhti vay :

Bang 11-3

MOT SO HE THIIC THONG DUONG GIfrA CHUYEN DONG TINH TIEN

VA CHUYEN DONG QUAY

Tinh tign thuan ttly (theo Quay thuan ttly (quanh

Van t6c v = dxldt , Van t6c gdc w = deldt ,

Dinh 11 ding- W = AK Dinh li cong-• W = AK

Tinh tich phan nay, va dung phiiang trinh 11-23 (K = -2- 1 ko2), ta cluac :

W - 2 /co22 - 2 - /co2 = Kf - K = AK 1 (11-40)

Dinh li cOng - dOng nang nay cho ta thay rang ea/1g tirtic hien bai momen quay toan

phan tac dung vao mat c6 thd quay thi bang do bign thien cua clang nang quay cila vat dd

Dd la cai Wong &acing gdc cua dinh 11 cong - clang nang cloi vai chuydn clang tinh tign (p.t 7-21), ma thong ta co thd nhac lai, nhtt sau : cal-1g thvc hien tren mat c6 thd (hoac mat hat) bai laic toan phan tac dung vao vat, thi bang do bign thien clang nang tinh tign cua vat do

Bdng 11-3 tom tat cac p.t ap dung &doe cho sot quay dm mat c6 thd quanh mat true c6 dinh, va cac he thilc Wang duong, ci6i vai chuydn clang tinh tign

Bai toan mAu 11-3

a) Trong each b6 tri trail h.11-19, dia quay duac mat gdc bang bao nhigu, sau 2,5s ? sit rang he bat Tau chuydn clang tit nghi

Theo p.t 11-10, dat coo = 0 va dung gia tri dm a da tinh &toe trong bai than mau

11-11, ta cittac :

0 = odot + 1 at = 0 + 1 ) (-24 rad/s2)(2,5 s)2 = -75 rad (Dap s6)

b) Van t6c gdc cua dia, luc t = 2,5s la bao nhieu ?

Giai 'Pa c6 thd tim dude, vei p.t 11-9 Gat wo = 0 va dung gia tri ciia a tinh &fee trong

bai toan mau 11-11, ta duec :

w = co + at = 0 + (-24 rad/s2)(2,5s) = -60 rad/s (Dap s6)

c) DOng nang cua dia llic t =2,5s la

Mot cach khac de' giai bai toan nay, la

tinh do bign thien Ong nang quay cua

dia, bang each dung dinh 11 cong - dOng

nang Mu6n vay, ta phai tinh cong thuc

hien tren dia bdi momen quay tac dung

vao dia Momen quay tac dung bai stic

cang T cua day la khong d6i, thanh this,

theo p.t 11-38

Of of

W = f rde = r f dB = T(01 - 0i)

trong do, T la stic cang ciia day (= 6,0N

D6i vei momen quay, ta dung -TR, D6 quay dia ban tac dung mOt momen quay va

sinh mOt cong d@ cho dia mOt dOng nang quay

xem bai toan mau 11 - 11) va R (= 0,20m)

la ban kinh cila dia Dicing 02 — 01 chinh la dO ddi g6c ma ta da tinh trong (a) Nhu vay :

W = r(Of - Bt) = - TR(Of - = - (6,0 N)(0,20 m)(- 75 rad) = 90J (Dap s6)

Vi Ki bang khOng (he bat dau chuydn

dOng tit nghi), nen p.t 11-40 cho ta thay

rang dap s6 la bang K

Bai toan mau 11-14

MOt vat ran, Om mOt cai yang mong

(kh6i liking m va ban kinh R = 0,15m) va

hai thanh manh (in& thanh co kh6i Wong

m va dO dai L = 2,0R) duec sap xgp nhu

tren h 11-21 Vat co thd quay &Lige quanh

mOt true nam ngang, trong mat phang

veng train va di qua tam yang train

H1NH 11-21 Bai toan mail 11-14 MOt vat Om mOt cai vOng va hai thanh, có th6 quay quanh mOt true

31

a) Quan tinh quay I dia vat quanh true quay, tinh theo m va R, la bao nhieu ?

Giai Theo bang 11-2 (i) vOng c6 quail tinh quay Ivg = 1 mR2 quanh dtiong kinh cua

- kh.tA

no Theo bang 11-2 (e), thanh A co quan tinh quay / = mL2/12 quanh rnOt true di qua

kh6i tam va song song vdi true quay Dd tim quan tinh quay I A cua thanh A quanh true

quay, ta dung dinh li true song song 11-25

IA = kh.t.A Mil ikt A = 12 + R + -2- = 4,33mR2, trong do, ta da sit dung &It kin L = 2,0R, va trong Ito kh.t A (= R + L/2) là khoang

each vuong g6c giiia tam clidm thanh A WI true quay

Ta cung lam Wang tu, cho thanh B : quan tinh quay cua no /kii.t3 quanh mit true huong theo thanh thi bang khong, va quan tinh quay IB cua no quanh true quay la :

IB = Ikh.tii Mhkh.t_13 = 0 + mR2 = mR'

Khi van eking vien Kurt Thomas, quay quanh xa de bat dau bunt-1g cho minh rdi xuCing, thi clang nang quay ma anh da có, luc

o chiS thrip nhAt, khi quay, chuyen (met phrin 16n) thanh the nang hap clan CO sd chuyen la nhd cang am ma memen quay quanh xa thUc hien tren anh ta, tnc la tong sinh ra bai trong ludng cua anh, trong Wong nay dat vao ki& tam cua anh

Trong do hkh.t.B (= R) la khoang each vuong gcie gitta thanh B va true quay Nhu thd,

quail tinh quay I dia vat ran quanh true quay là

1

/ = Ivg + IA + B = — 2 mR2 + 4,33 mR2 + mR2 = 5,83mR2 5,8 mR2 (Dap s6)

b) Bat dau chuydn Ong tit nghi, vat quay quanh true quay tit vi tri huOng len tr8n, nhu

h 11-21 FHA t6c dO g6c (2) cua no quanh true quay, khi no 16n ngudc ?

Giai Trude khi vat chuydn dOng khoi tam cua nod each true quay mOt, khoang v kht' trong do yom dude cho bai p.t 9-5

m(0) + mR + m(R + L/2)

trong cid, 3m là khoi Wong dm vat Khi vat quay, kh6i tam cua no xu6ng Uri cling mOt

khodng each tOi true quay Nhu vay, kh6i tam da chiu mOt do diii thAng citing

AYich.t = -2R Trong qua trinh di xu6ng, the nang hap clan U cita vat duoc chuygn thanh

dOng nang quay K cua no DO bign thien AU cua thg nang bang tich cua trong Wong vat

3mg) vOi do alai thang ding AYkh.r V4Y

ON TAP VA TOM TAT

Vi tri g6c

DO mo td su quay cua mOt c6 the quanh mOt true c6 Binh goi là true quay, chung ta

gid dinh rang co mOt ditimg m6c Kan voi vat, vuong gdc of i true quay, Ira quay ding vUi

vat Chung ta do vi tri g6c 0 cua dubng ay so vai mOt true c6 di/1h Khi 0 do bang radian,

0 =

-r (do bang radian),

trong dd, s IA do dai cua cung trim, ban kinh r, quet bcii 0 S6 do, bang radian lien he

vdi s6 yang, ya s6 do, bang :

Do tliri g6c

Co the quay quanh mOt true, ma co vi tri gcic bign thien tit 01 dgn 02, thi co 416 dai g6c

trong dd, 00 là s6 clueing, ngu quay nguoc chi6u kim citing h6, va là s6 am, khi quay theo

chi6u kim &Ong 116

Van to'c goc vet to'c do g6c

Ngu mOt vat quay doge mOt goc 00 trong khodng thbi gian At, thi van t6c goc trung

binh 65 cua no là :

AO

w

76,7*

Van t6c goc (tile thbi) co cila vat la :

de

Ca w lan co deu là vecto, ca hai deu cd chieu cho boi quy tac ban tay phai, tren h 11-5

Chung la citiong, d6i v6i sur quay nguoc chieu kim (king h6, va la am, d6i vOi sti quay theo

chieu kim citing ho DO lOn cua van t6c gdc la t6c d6 gOc

Gia tdc goc

Ngu van t6c gdc cua mOt vat bign thien tii co l dgn cot trong khoang thji gian

At = t2 - t1, thi gia tot goc trung binh a cua vat, là

Ca a 16'n a deu là cac vecto

Phircrng trinh Ong hoc, ddi voi gia t6c gdc khong ddi

Gia t6c goc khong cloi (a = const) la mOt trubng hop clac biet, quan trong cua chuydn

Ong quay Cac p.t dOng hoc thich hop, duoc cho tren bang 11-1, la :

Bien so ddi vd bien so" goc

MOt didm tren mOt vat ran quay 6 each true quay mOt khodng vuong goc r thi chuydn dOng tren mOt duOng tron ban kinh r Ngu c6 th6 quay mOt gdc 0 ; thi di6m chuydn

&Ong tren mOt cung trOn cd do dai s, cho boi

s = r (gdc do bang radian), (11-15)

trong dci, 9 tinh ra radian

Van t6c dai v cua didm `Ay thi tigp tuygn vOi citiOng trop, t6c do dai v cua didm (dO lon cua van tot dai) duoc cho be i :

trong dci w la tot dO gdc (do bhng radian tren giay) cua co thd

Gia tot dai a cua didm cd ca thanh phan tiep tuyin Ian thanh phAn xuyen tam Thanh

phAn tigp tuygn là

trong do*, a la do lon cua a t6c gdc (do bang radian tren giay binh phuong) cua co` thd

Thanh phAn xuyen tam dia a la

v2

ar = —

r = cot r (gcic do bang radian)

Hong ming quay va qucin tinh quay

Deng nang K cua met c6 the quay quanh mot true c6 dinh duoc cho bai :

K = 1 ko2 (do bang radian),

trong do, I là quan tinh quay cua vat, duoc dinh nghia la :

I dgi vOi met he chat didm rdi rac, va la :

I f r2dm

(11-19)

d6i vUi met vat, co kh6i Wong phan b6 lien tuc Cac r va rt trong cac bidu thlic nay la

khoang ca.ch vuong gcic, t M true quay, dm moi nguyen t6 khgi Wong trong vat

Dinh li true song song

Dinh li true song song lien he quan tinh quay ciia mot vat quanh met true bat ki vOi

quan tinh quay cua cling vat dd, quanh mot true song song di qua kh6i tam :

6 day, h la khoang cach vuong gdc gifia hai truc

Mornen quay (mOmen lift)

MOrnen quay la tac dung lam quay hoac lam xoan mot vat quanh met truc quay, gay ra

bcfi mot life F Ndu F tac dung vao met didm, xac dinh be i vecto vi tri r doi voi truc, thi

moment (dai Wong vecto) la

DO Ion cua mOmen quay la

= rFt = r1F = rFsin0, (11-28, 11-29, 11-30)

trong dd Ft la thanh phan cua F vuong gdc veri r, va 0 la gdc giila r va F Dai luting

rl la khoang each vuong gdc gala true quay va dueing keo dai cua ding thong di qua vecto

F Duong thong ay goi là duiing tac dung cua F va r1 goi la canh tay don dm F Thong

tu, r la canh tay don cua Ft

Don vi SI cilia momen quay la niutcm nhan met (N.m) Momen quay la duong ngu no cd

xu hudng lam cho vat quay nguoc chieu kim d6ng h6, va la am, nau nd co xu hudng lam vat quay theo chigu kim dong

Dinh 101 thth hai cua Newton dtrai clang gac

Dinh luat quay, Wong tai dinh lust thi hai ciia Newton, la

trong do Et la mOmen quay toan phgn tac dung vao met hat hoac met vat, I la quail tinh quay oda hat hoac cua vat quanh true quay, va a la gia t6c gdc tong hop quanh truc

Cong quay va ndng

Phuong trinh dimg dd tinh ding va tong suat trong chuydn &Ong quay thi ttiong tu vai

cac p.t chi ph6i chuydn Ong tinh tian, va la :

of

ei

35

HINH 11-22 Cali hai 6

(a) HINH 11-23 Cali hai 7

1 Cac dai Wong gcic 9, w va a co thg bigu dien bang de, thay cho radian, trong cac p.t

quay, ciia bang 11-2, &doe kilo/1g ? Hay giai thich

2 Theo nghia nao, ma radian lai la mOt don vi "tit nhien" de' do g6e, con do la" la don

vi "thy y chon", cling cua dai Wong ay ? (XOt dinh nghia cda hai don vi ay)

3 Vecto bidu dien van t6c goc cua met banh TA quay quanh mOt true c6 dinh co nhat thigt phai u tren true ay kitting ?

4 Hay thi nghigm cho met quygn sach quay theo each tren h 11-6, nhting bay gib dung

nhting dO deli goc 180°, thay cho 90° Ban kgt luan thg nao vg vi tri et/6i cling cua quygn sach ? Dieu do co lam thay d6i suy nghi cua ban vg viec nhUng de cloi goc (hiiu han) lieu

ed the coi nhu dai Wong vecto dtioc khOng ?

5 Giu canh tay phai ciia ban chile xueng, long ban tay hueng vao din ban Gill c6 tay ban cho cling (1) Gio canh tay cho tbi ltic no nam ngang, va tr6 phia trttoc (2) Giang ngang canh tay cho clgn luc no httOng sang phai Va (3) sau cid, ha no xueng sat ngtioi ban Long ban tay ban se huaing vg phia trade mat Mu ban lam lai, nhting ciao ngtioc thil tti

cac Ong tac, thi tai sao long ban tay lai kizOng httong ye phia trude ?

6 Gia sit ban gill mOt d6ng tign tren ban, trong khi

ban lan mOt deng tien thti hai quanh no, ma khong cho

mep trout (h 11-22) Khi dgng thil hai tres vg vi tri ban

dau no da quay dtioc mOt gdc bao nhieu ?

7 CO thd nghien coil ski quay &la Mat Trbi bang

each theo d6i cac vet den, nhiing tran bao tit tren Mat

Troi, co mau t6i tren nen sang etia dia Mat Troi Hinh

11-23a trinh bay vi tri ban Tau cua nam vet va h

11-23b, vi tri cua nam vet ay sau met vOng quay dia

Mat TrOi Ban ke't luan the nao, ye cau tao cilia Mat

Troi, tit hai phep quan sat ay ?

8 Tai sao bigu dien a trong bigu thtic

1

0 = co

° t + - 2 at- bang vong tren giay binh phtiong, thi

duoc, con trong bigu thtic at = ar lai khong duec ?

9 MOt c6 the co thg quay cltioc quanh met true c6

dinh Vat co the co gia tee gOc khac khong ngay ca khi

van t6c gdc ctia vat: (cd 10, chi tile th6i) bang 0, hay

khOng ? Cai Wong clueing ye tin.h tign, dm van de nay la gi ? Hay cho nhiing thi du minh hoa ca hai trong thai ye quay va tinh tign

10 Met nguoi chgi gon vung met cay gay On, va bat mot dugng dai tit vi tri phat bong Moi digm dia cay gay cd ding mot van tec gdc w a moi the i digm, chiing nao ma cay gay con chuygn Ong, hay khOng ?

11 Khi chUng ta n6i rang met diem tren xich dao c6 t6c dO gdc 2nrad/ngay, thi chting

ta nghi den he quy chie'u nao ?

12 Chti y dgn ca au tu quay dia Trai Dat, va ca chuygn Ong tren quy dao (ca hai chuygn &Ong deu dung chieu), hay giai thich, met cai cay, ban ngay, hay ban dem, thi chuygn Ong nhanh han ? Cau tra lcsi dia ban dugc cho clgi vgi he quy chigu nao ?

13 Tang Wong cd mot banh da quay quanh true cua no va xet met diem tren mep no

Khi 'Jamb da quay vai van toe gdc khong dei, thi digm ay cd gia tgc xuyen tam khong ? Co

gia toc tiep tuygn khOng ? Khi banh da quay vgi gia toe gdc khong dei, thi diem ay cd gia t6c xuyen tam khong ? Co gia toe tigp tuyen khong ? DO lgn cua ca.c gia t6c d6 thay dei the nao theo thbi gian ?

14 Gia sit rang ngugi ta yeu cau ban xac dinh quang &Wing di dugc bai cai kim, trong met may quay dia clang hoat dOng Ban can bigt thong tin gi ? Bien luan theo each nhin tit mot he quy chieu (a) c6 dinh trong phong, (b) c6 dinh dgi vgi cai dia quay va (c) c6 dinh dgi vgi cai can dm may quay dia

15 He fink giita van t6c gdc cua met cap hai banh rang ghep vgi nhau, c6 ban kinh khac nhau, la gi ?

16 Khi tinh toan quail tinh quay dia met vat, cd the coi kh6i Wong dm nd nhu tap trung vao kh6i tam, dugc khong Neu dugc, thi giai thich tai sac Neu khong, hay cho mot thi du de" phan bac

17 Khi ban thing thang, thi Tian tinh quay dia ban d6i voi true nao (a) là nh6 nhat vb (b) la lgn nhat ? Dgi vgi (a) va (b) ban cd the lam thg nao de' thay del gia tri quan tinh quay dm ban ?

18 Nen hai dia trOn ding trong lugng va ding de day, dugc lam bang kim loai cd kh6i lugng rieng khac nhau, cd dia nao cd quan tinh quay quanh true giita dia chung, lgn hon khOng ?

19 Hay ggi ra met phuong phap dd do bang thuc nghiem quan tinh quay quanh met true nhat dinh dm met c6 the cd hinh dang phtic tap

20 Nam vat ran cd ding kh6i Wong, cd Wel dien trinh bay tren h 11-24 Cac tiet dien

cd chieu cao bang nhau va chieu Ong lgn nhat, bhng nhau Vat nal) cd quan tinh quay lgn nhat quanh true di qua kh6i tam va vuOng gcic vgi tigt dien ? Vat nao cd quan tinh quay nho nhat ?

Lai Vahh Za[7,0hdahg /7%hh / i hi>>h ch4a /1/

HINH 11-24 CALI hoi 20

37

21 Hinh 11-25a trinh bay mOt cai thy& met, c6 mQt nits bang g6 va mQt nita bang

thep, va quay duoc d 0, ch6 Tau bang g6 Mot luc F chicle dat vao dau bang thep tai a Tren

h 11-25b, cai thudc lai quay dtioc es 0', ch6 dau bang thep, va cung luc do lai dat vao dau

bang g6, tai a' Trong hai truing hop, gia the g6c c6 cung gia tri hay kh6ng ? Ndu khong, thi trong trudng hop nao gia t6c goc ldn hon ?

(a)

fF

(6)

1-1iN1-1 11-25 CSu hoi 21

22 Hay binh Juan tang lieu trong cac diet' khang dinh sau day, ye su truot tuygt (a) Khi truot xu6ng d6c, ngtthi ta can nhiing ban truot (xki) khOng duoc quay d6 dang (b) Khi truot xu6ng d6c cci vat chudng ngai, can nhfing xki quay c16 dang (c) Vi vay, quail tinh quay cila xki dung dg truot xu6ng d6c phai lcsn lion, so vdi xki xu6ng d6c c6 chudng ngai (d) Coi rang gida xki va tuygt c6 ma sat nhO, va kh6i tam cila ngueri truot tuygt d tren kh6i tam &la xki mOt chtit, thi nguoi truot tuygt phai lam the nao cho ca momen quay, dg ngoat, hoac dd than mOt cai ngoat (Trich : "Vat li hoc ve truot tuyel d ch6 ngoat" cita J.I Shonie va D.L Mordick Tap chi "The Physics Teacher" thang 12-1972)

23 Xet mOt cai thudc dung thang thing tren bang (khong ma sat) Neu thudc dg, thi kh6i tam dic no di theo dudng nao ?

BAI TAP VA BAI TOAN

Muc 11-2 CAC BIEN S6 QUAY

1E (a) G6c d tam, tinh ra radian, than mOt cung d0 dai 1,80m tren mot dtidng trim

ban kinh 1,20m, la bao nhieu ? (b) Hay bigu didn gdc nay bang dQ (c) Gdc gida hai ban kinh cita mQt cludng trOn la 0,620 rad DQ dai cita cung bi than la bao nhieu, ngu ban kinh

la 2,40m ?

2E Trong khoang thdi gian t, banh da ctia mOt may phat diOn quay &doe mOt gcic

0 = at + bt3 - ct4 trong do, a, b, c la hang s6 Vigt bigu thtic (a) cita van the gdc va (b)

cita gia the g6c cita banh da

3E Mat Trdi chic chtIng tad each tam he Thien Ha ciia chting ta 2,3 x 104 nam anh

sang, va chuydn dQng quanh tam d6, yeti tot dO 250 km/s, theo mOt du0ng tron (a) Thoi gian dg Mat Trdi di hgt mQt yang quanh Thien Ha, la bao nhieu ? (b) Kg tit ngay hinh thanh, each day chdng 4,5 x 109 nam, Mat Trdi da quay duoc bao nhieu yang ?

4E Vi tri g6c chic mOt digm tren mep cua mOt cai dia dang quay duoc cho bdi

8 = 4,0t - 3,0t2 + t3, trong d6 6 tinh ra radian, ngu t toot do bang giay (a) Van t6c g6c

lhic t = 2,0s va t = 4,0s la bao nhieu ? (b) Gia tot goc trung binh trong khoang thdi gian tit t = 2,0s dgn t = 4,0s la bao nhieu ? (c) Gia t6c g6c the thdi luc bat dau va luc cuoi

khoang thoi gian dO, la bao nhieu ?

HINH 11-26 Bai town 10

0

5E Vi tri g6c cua mot diem tren mot cai banh xe dang quay &roc cho bdi

= 2 + 4t2 + 2t3, trong do, B duoc tinh ra radian va t, ra giay Luc t = 0, thi (a) vi tri

gcic va (b) van t6c g6c, la bao nhieu ? (c) Van t6c gcic luc t = 4,0s la bao nhieu ? (d) Tinh gia t6c g6c luc t = 2,0s (e) Gia t6c gOc co phai la hang s6 khong ?

6P MOt banh xe quay voi gia t6c goc a cho bai a = 4at3 - 6bt2, trong do t la thoi gian,

va a va b la hang s6 Neu banh xe co t6c do gOc ban dau coo, hay viet p.t (a) dm t6c dO

goc va (b) cfla dO doi gcic, theo th6i gian

7P T6c dO goc cUa (a) kim giay, (b) kim phut va (c) kim gib cita d6ng 116, la bao nhieu ? 8P Mot ngubi choi bang chay gioi co the nem qua b6ng chay ye phia dich, vOi van t6c

85 dam/h, bang quay 1800 vg/ph Qua bong chay quay duoc bao nhieu yang tren dieing di Uri dich ? De don gian, to thita nhan rang quy dao dai 60 fat la mot du6ng thang

9P MOt van dOng vien nhao lOn quay tron duoc 2,5 Ong tit cau nhay cao bon mat nu6c

10m Cho rang van t6c ban dau theo phuong thAng dung, la bang kh6ng, hay tinh van t6c gc5c trung binh trong litc nhao lOn

10P MCA banh xe co tam nan hoa va co ban kinh 30cm

NO &toe lap tren mot cai true co dinh va quay voi t6c dO

2,5 vg/s Ban muOn phcing mot mid ten dai 20cm, song song

vdi true va di qua banh xe, ma khong cham vao mot nan

hoa nao Gia sit rang ten va nan hoa deu v6 ding manh :

xem h.11.26

(a) Ten phdi c6 t6c di) t6i thieu la bao nhieu ?

(b) Ngam vao ch6 nao gifta truc va vanh, co thanh van

de gi khong ? Neu co, thi ngam vao ch6 nao tot nhgt ?

Muc 11-4 SV QUAY vOi GIA T6C GOC KHONG D6I

11E MOt cai dia, ban dau co van t6c 120 rad/s, quay cham dan vcii gia t6c g6c khOng

d6i, bang 4,0 rad/s2

(a) Bao nhieu thbi gian da troi qua, tai litc dia diing lai ?

(b) Dia quay duoc mot goc bao nhieu, tru6c khi thing ?

12E Mam caa mot may quay dia dang quay vdi 33-3 vg/ph thi quay cham dan va diing

lai, 30s sau khi tat may

(a) Hay tinh gia t6c gOc (khong d6i), theo vg/ph2 (b) Mam quay duoc bao nhieu \Tong, trong th6i gian ay ?

13E T6c dO gdc cita mot may oto tang tit 1200vg/ph len 3000 vg/ph, trong 12s Gia tic

gcic, tinh ra vg/ph2, neu gia sit la kilo/1g d6i, bang bao nhieu ? (b) Trong thOi gian do, may quay duoc bao nhieu yang ?

14E MOt banh da nang, dang quay quanh truc, thi quay cham lai vi ma sat 6 6 true

Cu6i phut thi nhat, van t6c g6c ciia no bang 0,90 van t6c g6c ban dau (van t6c g6c ban dau la 250 vg/ph) Coi luc ma sat la khong d6i, hay tinh van t6c gcic cita banh, -cu6i phut thti hai

15E Banh da cua mot cai may dang quay voi 25,0 rad/s Khi tat may, banh da quay

cham dan voi gia tic khOng d6i va ditng lai sau 20,0s Hay tinh (a) gia t6c gcic cua banh

da (ra rad/s2) (b) Goc (ra radian) ma banh con quay, cho den litc thing lai va (c) s6 \Tong banh quay duoc, cho dgn ltic &mg

39

16E Bat dau chuydn clang tit nghi, met cai dia quay quanh true dm no vOi gia tee gdc khong ci6i Sau 5,0s no da quay duoc 25 rad (a) Gia t6c gdc cua no trong thai gian do la bao nhieu ? (b) Van t6c gdc trung binh la bao nhieu ? (c) Van t6c gdc ttic thai cua dia cu6i thai gian 5,0s là bao nhieu ? (d) Gia sit rang gia t6c khong thay ct6i, thi 5,0s tip theo dia quay them duoc met gdc bang bao nhieu ?

17E Met rang roc (Wong kinh 8,0 cm co met day dai 5,6m quan quanh mep Bat dau chuy6n clang tit nghi, rang roc nhan duoc gia t6c gdc khong d6i 1,5 rad/s2 (a) Rang roc quay duce met gdc bang bao nhieu, thi soi day duoc than h6t ? (b) Viec do lau bao nhieu ? 18P Met banh xe quay cluoc 90 yang trong 15s, t6c de gdc cua nd vao cu6i thai gian do

la 10vg/s (a) T6c do gdc ciia no, vao dau quang thai gian 15s la bao nhieu, neu gid dinh rang gia t6c gdc khong ct6i ? (b) Bao nhieu thai gian da trei qua, tit lac banh xe bat dau chuydn clang tit nghi, den lee bat dau cua khodng 15s tren ?

19P Met banh xe co gia t6c gdc khOng d6i 3,0 rad/s2 Trong khoang thai gian 4,0s, no

quay cluoc met gOc 120 rad Gid sit rang banh xe Mt dau chuydn clang tit nghi, thi no da

pH" chuydn clang bao lau, truec khi bat dau khoang 4,0s do ?

20P Met banh xe bat dau chuy6n clang tit nghi, quay yeti gia toe gdc khong d6i 2,00 rad/s2 Sau met khodng thai gian 3,00s nao dd, no da quay dugc 90,0 rad (a) Ba.nh xe phai quay bao nhieu lau, trine Mc Mt dau khodng 3,00s do ? (b) Van t6c gdc cua banh xe lae Mt Tau khoang 3,00s ctd, la bao nhieu ?

21P Met banh da quay dtioc 40 \rang, tit ltic bat dam quay cham lai, vai van t6c 1,5 rad/s, cho den khi ditng (a) Gia sit rang gia toe khOng ddi, thi thai gian can d6 ditng la bao nhieu ? (b) Gia toe gcic la bao nhielf ? (c) Nd can bao nhieu thai gian d6 quay ducc 20 yang dau, trong s6 40 yang ay ?

22P Luc t = 0 met banh da co van toe gdc 4,7 rad/s, gia t6c gdc - 0,25 rad/s2 va throng mac a 00 = 0.(a) Duang mac se quay cluoc met gdc cue dai Omax bang bao nhieu, theo chieu

duong ? Den thai didm t nao, thi diiang mac it (b) 0 = —1 0

max va (c) = -10,5 rad (xet

dd quay hdt 60 yang da neu, (c) thai gian can thiet de dat t6c de 10 vg/s va (d) s6 yang quay tit bac nghi cho Ten luc ctia dat toe de gdc 10 vg/s

24P Bat dau chuy6n clang tit luc nghi la Lac t = 0, met banh xe nhan citioc met gia t6c gdc khOng d6i Khi t = 2,0s, thi van t6c gdc cda banh xe la 5,0 rad/s Gia toc ti6p tuc cho Ten khi t = 20s, khi do gia tae dot nget thai Banh xe quay cluoc met gdc bao nhieu trong khodng thai gian tit t = 0 den t = 40s ?

25P Punxa la mot sao notrOn quay nhanh, va phat cac xung vo tuye'n co do Ong IDO

cao, mot xung trong mai yang quay cfia sao Chu ki quay T la thai gian cho met yang quay

Nguai to da co &roc chu ki dd, bang each do tiled gian giita hai xung Hien nay, punxa

mien trung tam tinh van Con Cua (xem h.11-27) co chu ki quay T = 0,033s, va quan sat

cho thay, no dang tang yeti toe dO 1,26.10-5 s/nam (a) ChUng minh rang van toe gdc CO dia

sao lien hO yOi T beg co = 2.r/T, (b) Gia tri cua gia toe gdc, tinh theo rad/s2 la bao nhieu, (c) Neu gia toe gdc cua no khong d6i, thi sau bao nhieu nam nita lcd tit bay gib, pun xa se

ngitng quay ? (d) Punxa duoc sinh ra tit met vu no cua sao sieu mei, vac) nam 1054 sau