HU'aNG DAN QUAN LI SU'DUNG
KHAM PHA DirCfNG CYCLOID VA HYPOCYCLOID V 6 l SIT TRO GIUP CUA PHAN MEM CABRI
Dudng cycloid la dudng cong lien tuc dugc ve bdi mdt diem ndm ttdn dudng ttdn khi dudng ttdn lan dpc theo mpt dudng thang ttong cung mpt mat phang. Cung gidng nhu dudng cycloid, mdt dudng ttdn Im ben ttong mdt dudng frdn cd dinh, quy tich cua mdt diem fren dudng ttdn lan dugc gpi la dudng hypocycloid.
1. Dudng cong cycloid a. Ca sa todn cua duong cong cycloid
Gia su rang dudng ttdn cd ban kinh a vd lan tten true Ox, bat dau tir vi tri md tam ciia hinh ttdn ddt tten tryc duong cua true Oy. Duong cong Id quy tich cua diem P fren dudng ttdn tning vdi gde tpa do khi tam C cua dudng frdn C nam fren true Oy. Gde 9 frong hinh ve la gde tao bdi ban kinh CP khi dudng frdn lan tdi vi tri radi. Ndu x, y la tpa dp ciia P thi qud trinh lan cua dudng ttdn keo theo OB = arcBP - aO, do dd
x = OB - OA = OB - PQ - a9 - asinO = a(e - sinG).
y = BC - QC = a - acosO ^ a(l - cos6).
Vi vgy phuang trinh tham sd cua dudng cycloid la
f x - a ( e - s i n e ) /QJ-, l y = a ( l - c o s e ) ^ -*
Rd rdng frong hinh ve, y la ham sd cua x nhung rat khd cd the tim mdt cdng thiic don gian de cd ham sd nay. Dudng cycloid Id rapt trong cac dudng cong raa phuang trinh tham sd don gidn vd de dang ldm viec ban phuang trinh tpa dp Dd-cac.
Tir phucmg frinh (1) chiing t a c o
, _ dy _ asined6 _ sm9
^ ~ dx ~ a(l - cose)de ~ 1 - cosB
Nhdn xet rang dao ham y ' khdng xdc dinh tai 6 = 0,±27:,±4ji,... Nhiing gia tri nay cua 6 tuang iing vdi cdc diera tai dd dudng cycloid tiep xiic vdi Ox, cac diem nay dugc gpi Id diem liii. Cac tiep tuyen ddi vdi ducmg cycloid thi vudng gde tai diem liii.
b.Ve duang cong cycloid tren phdn mem Cabri
Ta dyng dd thi ham so cho bdi phuang trinh tham sd
X - i(t) tj-gjj phan mem y = g(t)
Cabri nhu sau
• Cdch tgo ra gid tri ciia tham so t
. Hifn hf tryc vudng gde Oxy.
Nguyen Ng^c Giang . Lay diera A cd hoanh dp dm tren tryc hodnh.
. Lay diera B Id diem ddi xiing ciia diem A qua goc toa d d O .
. Ndi AB.
. Lay diera T tren doan AB.
. Ggi hoanh dp cua diem T i a t Khi T chuyen ddng thi gia trj tham sd t thay ddi.
• Tinh cdc gid tri x =f(t) vd y^g(t)
Su dung niit lenh "Mdy tinh..." de tinh cdc gid trj x = f(t) va y = g(t). Keo tha gia tri X, y chuyen ra vimg lam viec.
• Dung diem H co hodnh dox^f(t) tren true hodnh
. Sir dyng mit lenh "Chuyen sd do" —»• True O x ^ gid tti X = f(t) ta dupc diem mdi H fren true hodnh, dat ten la H (hodnh dd cua H chinh Id gia tri X = f(tj).
• Dung diem K co hodnh dgy - g(t) tren true tung
. Sir dyng mit lfnh "Chuyen sd do" —> True Oy —> gia tri y
= g(t) ta dugc diem mdi K tten tryc hodnh, dat ten Id K (tung dp ciia K chinh la gid tri y = g(t)).
• DungSemM(x;y):\
Ngdy nhdn hdi 15/12/2012: Nea-
X = f(t) y = g(t) . Dudng vudng goc vdi Ox tai H va dudng vudng goc vdi Oy tai K cat nhau tai M.
. Ndi MH, M K . '
• An di cdc doi tugng khong cdn thiit
• Ve do thi
. Tao vet cho diem M chuyen ddng diem T ta thu dupc vet ciia diem M. Mudn cd ngay do thi ta sir dung mit TiDkP CHI THIET BI GIAO DUC-SO 91-03/2013 • 2 9
HUONG DAN QUAN LI SUDUNG
lfnh Quy tich click vdo dilm M sau dd click vao diem T.
Ap dyng phuong phdp ndy cho ve dudng cong cycloid cd phuang trinh tham so
j X = a(t - sin t) l y = a(l - c o s t )
ta dugc do thi hen phan
dudng cong hypocycloid. N l u mpt dudng frdn ldn ben frong mpt dudng ttdn co djnh, quy tich cua mpt diem trdn dudng ttdn lan dugc gpi Id dudng hypocycloid. Neu mpt dudng ttdn lan ben ngodi mpt dudng frdn CO djnh, quy tich cua mpt
raera Cabri nhu hinh ve Dudng cong cycloid nhu da nghien cdu d fren dugc Galileo phdt hifn ra lan dau tien vao dau the ki 16. Tuy Galileo phat hifn ra dau tien nhtmg dng khdrig khdra phd them dugc gi vd nhiing tinh chat ciia nd. Ong ben nhd cac ban be nghien ciiu ttong do cd Merserme d Paris. Merserme thdng tui cho Descartes va nhiing ngudi khac vdo nam 1638. Nhiing ket qua dupc tim thay xuat hifn khdng lau sau do. Descartes da tim thay sy kien thidt cho tiep tuyen. Nam 1644, hgc ttd ciia Galileo Id ToriceUi cdng bd phdt hifn cua dng ve dien tich gidi han cua mdt cung. Dp dai ciia mpt cung dugc nha kien true su ngudi Anh Christopher Wren phdt hien nam 1658.
2. Duomg cong hypocycloid a. Ca sa todn eua duang cong hypocycloid
Bdy gid ta se kham phd
diem fren dudng trdn lan dugc gpi la epicycloid.
Chiing ta chi ra cdch bidu dien tham sd dudng hypocycloid. Cho dudng frdn cd dinh cd ban kinh a va mdt dudng frdn lan ban kinh b vdi b < a. Gia thiet tdm ciia dudng frdn cd dinh tning vdi gde tga dp (hinh ye) va dudng ttdn lan be hon bat dau chuyen ddng tai vi tri tiep xiic vdi dudng ttdn cd dinh tai A theo true duong ciia Ox. Vdi 9 ^ dugc chi frong hinh ve, qua trinh lan cua dudng ttdn nhd keo theo cung AB va cung BP Id bang nhau : a9 = b p Tpa do cua PId
X - (a - b) cos9 + bcos9 (p - 9) x - ( a - b ) s i n 0 + bcose((i)-9) Nhtmg p - 6 = ^ - 1 ^ nen phuang trinh tham sd ciia dudng hypocycloid la
x = ( a - b)cose+ bcos a - b „ a - b (3) y = (a - b)sin9 - bsin 9
b Dp ddi cua cung dgc theo dudng frdn cd djnh giiia cdc diem liii ke tiep cua dudng hypocycloid la 27cb. Neu 2jia la mpt bpi nguyen ciia 2jrb thi
•§-ld mpt sd nguyen n thi dudng hypocycloid cd n diem lui vd diem P quay lai diem A sau
khi dudng ttdn nhp lan n ldn frong dudng ttdn cd djnh.
Phucmg trinh fliam s6 cda dudng hypocycloid cd 4 dilm lui cd the viet dudi dang rat dem gidn nhd su dung cdc d|ic tinh lugng gidc. Neu a = 4b thi phuong trinh (3) ttd thdnh X = 3bcos9 + bcos39, y = 3bsine - bsinSe
Nhung
cos39 = 4cos^9 - 3 cos9 sin39 = 3sine - 4sin'e Vi vgy phuang trinh tham so Id: X = 4bcos^9 = acos^9, y
= 4bsine = asin'9(4) Chuyen phuong trinh ndy sang tpa dp Dd-cdc tuong img Id
x' + y* = a' .
Duong hypocycloid vdi 4 diem liti thudng dugc gpi Id dudng asttoid.
b. Ve duang cong astroid tren phdn mem Cabri
Ta ye dudng asttoid tten phdn mem Cabri nhu sau:
. Dyng dogn thang cd dp dai a.
. Dyng dudng trdn ( 0 ; a).
. Hifn hf tryc vudng goc Oxy.
. Dyng diem A ttdn Ox.
. Dyng diem B ddi ximg vdi A qua O vd ndi AB.
. Dyng dilm T(t ; 0) tren AB.
. DyngdlniM(acos^asitft).
. Tgo vdt cho dilm M, chuyen dpng diera T ta thu dugc vet cua diem M la ducmg asttoid.
Xem tiep trang 33
3 0 • TAP CHI THIFT BI GIAO DUC - SO 91 - 03/2013
HUONG DAN QUAN LI SU DUNG Cac nhdm lan lugt bao
cao ket qud thi nghifm qua SDTD, GV tdng kit thdng qua SDTD da thiet kl sin Ien man chilu.
4. Kit lu^n
GV su dyng SDTD trong day hpc se cung cap cho HS cd cai nhin tdng quat vl vdn dl dang hgc t£^. Thdng qua SDTD do cdc em ty thilt kd cd the danh gid dugc iniic dp ty hoc tap, mure dp hieu biet va nam b^t van de d mdc dp nao, GV cd die nhanh chdng dieu chinh cho phu hgp.
Ngoai vifc diilt ke SDTD cho cac bai 1hi^ hanh, GV cdn cd the thilt ke cac Mi luyfn t ^ . GV cd the hirdng ddn HS 1^
SDTD eho bai mdi, ghi chep kien ihiic tren ldp, dn tap khi thi cir, ^ ke hoach ca n h ^ minh hga cac y tirdng cua ca nhan. Sii
dung SDTD ttong day hgc that sy la ddi mdi phuong p h ^ day hgc, gdp phan nang cao hifu qud dgy hgc hda hgc ndi rieng va cac mdn hgc khac, gdp phan nang cao nang lyc nhan thiic cuaHS.
Tdi lifu tham khao 1. Tran Dinh Chau, Dgng Thi Thu Thiiy Dgy lot - hgc tot cdc mdn hgc bdng bdn do tu duy, NXB Giao dye Vift Nam, 2011.
2. Nguyen Xudn Trudng, tdng chil bien, Sdch gido khoa Hoa hgc II, NXB Gido due 2007.
3. PGS.TS. Nguyin Thj Sim (Chu bien), TS. Le Van Nam, Phuang phdp dgy hgc hoa hgc, NXB Khoa hoc va ky thuat, 2009.
4. Tony Buzan, Sa do tu duy, NXB Tong hgp TP Hd Chi Minh, 2008.
Summary IMindMap huge role in teaching chemistry, especially the exercises. IMindMap in teaching chemistry to help lectures become more intuitive, more scientific and logic. Instmctional practices to help students IMindMap to maximize the ability to think, to create, and stimulate the imagination, inspire students to bring positive results in teaching chemistry. In this article we introduce you how to build and use IMindMap from advanced exercise class taught organic chemistry 11.
KHAM PHA DirdNG CYCLOID..,
3. Mpt s6 tinh chdt cua dvdng cycloid \k astroid
Vi dv 1 ^
Chi ra rang difn tich xdc dinh bdi mdt cung ciia dudng cycloid bang ba ldn difn tich cua dudng ttdn lan.
That vdy, mpt cung dugc xdc dinh khi dudng ttdn chuyen dpng dung mpt vdng trdn xoay. Vi vgy sit dung tich phdn tinh dif n tich vdi tham sd 6 la tham sd bifn lay tich phdn
A=Jydx = J y 2 | i 9 = Ja(l-cos6) aO - cosewe = J a^a - cose)^de
= a^ J a - 2cose + cos^ e)de
2Jt 1 =
= a ^ J O + cos^e>de = a^J"d6 + 1 ° ° a^ J - (1 + cos 2e)de ^ 3jta^.
Vi du 2
Xet dudng thang tiep xuc vdi dudng asttoid tgi diem P ttong gde phan tu thii nhat.
Chi ra rang dogn thang dugc tgo bdi khi tiep tuyen nay cat bdi cdc true tga dp cd dp ddi khdng ddi, khdng phy thupc vao vi tri cua P.
That vdy, tu phuang trinh x
= acos^G, y = asin^9, hf so gde cua tilp tuyen Id
y' = -i- = = - tan6 dx -3asinecos^ede Nen phuomg trinh ciia tilp tuyin Id y - asin^G = -tan9(x- acos^O).
Chiing ta tun giao vdi ttyc Ox bdng each cho y = 0 va tim X x = acos'6 + asm^ecos9 = acos9.
Tuong ty, giao vdi ti-yc Oy Id y = asinG. Vi vay dogn tiidng dugc tgo bdi khi tiep tuyen cat
(liep trang 30) bdi cac tryc tga do cd dp ddi la V^cos^ + a^sin^ ^ a la hang sd.
Tren day la mdt sd khdm phd xoay quanh cdc dudng cycloid vd hypocycloid. Bdi viet ndy can trao ddi gi thera?
Mong dugc su chia se ciia cdc ban.
Tai lifu t h a m khao 1. George F. Simmons, Gidi tich mgt bien so, Gido trinh trudng Dai hpc Thiiy lpi.
2. Phgm Thanh Phuong, Dgy vd hgc todn vai phdn mem Cabri, tdpl Hinh hoc phang, NXBGD, 2006.
S u m m a r y This article will explore to the cycloid, hypocycloid curves and some their properties by the aid of Cabri interactive software.
TAP CHI THIET BI GIAO DUC-SO 91-03/2013 • 3 3
