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Bandul Pegas
IIIIIIIIIIIIIIIIIIII V= O
-^ #^ of^ panjang^ awa)^ pegas^ :^ lo y 3 i M Persamaan (^) gerak bandul^?
- Menentukan^ derajat kebebasan
- ketika bandul (^) bergerak tidak (^) hanya - (^) y berubah (^) , (^) ruya juga berubah. Shy derajat^ Kebebasannya^ ada 2 - &
* Persamaan Lagrange dg 2 DK^ (i^ &8)
- (^) Tentukan Koordinat (^) /polar - > (^) r& (^) 6)
- (^) Transformasi koordinat #(d) = = (^) r
- = sin^ A y =^ Co5^ - x =(rsino)^ = (^) Sino + rAsino^ (rIKonstan)^ Ed = Elym (r
- r'se dt = i sin o + r asin t^ de^ -Emp = Mr O d^ dt^ · = (^) mr
- (^) = sin o (^) + 8 Cost i = (rcost) = r (^) CO50 + rdco e · (^26) = limlrtrf
[K(r
2r+ ei) + mgr (050) dt ar = (^) cost + r d cost do^ = d EMro2-k(r-bi)^ =^ MrE-b(r-10) dt i =^ ICO50-rEsin^ t maka
- masukkan (^) energi kinetik I(84) = · &
T = jM( *
- (^) y T = ( m)(aino
- (^) 2 coro + (^) [custo + rain2]) mr (^) = (^) mrt-klr-lot
-- M
Z^ -1~
M(r (sin
= cos"
o) +^ r(05t +^ <in 2^ -) i (^) = (^) r 8 -
L
T = i^ =^ 82 -W^ Ir-lo2) jm/r = (^) + (^) r
Up :^ 12K^ OX^
·^ /K^ (r-10)^ Solusi^ untuk^ bagian^ r
< (^) Plegasfaci (^) V =
- mgr cost V= < K (^) (r-lo) - mgr cost -(86) =^ =^ Solun^ y
- Masukkan he^ fungsi Lagrange 1 =^ T-V^
& = (^) Mr E. = - Mgrsi 1 = [m(r +^ r^ 2)^
- Ek(r-lo)" + mgrcost (^) · - 2) = Mr. shgMr = -gran o · & solusi (^) u ↓ -^ = -sino
bagiano -kr-er+^ l
* Energi Kinetik
Devajat Kebebasannya - &^ Ac^ T = <M , 8 ,^ +^ -M, V
e (^2) , &^ &2^ =^ Konstan
jm/xI +^ y^ )^ +^ 5 M(x= +^ y,2)
- 1 > (^) x ,^ x,x^ I = Em lit (^) + (^) M2[lit + (^) lit+ (^24) , 28 , (05(0, - A2)]
- =^ I, sin O^ Xc :^ l, sin^8 ,^ +^ ha sin^2 T = MitMa)l" + Emelit +^ Malilit,^ E,^ (05(81^ -Oc) I,^ =^ - l, Cos (^) O (^) , Yc = -^ I, (^185 8) ,^ - le (^) CO5 Oz v = x (^) V = (^) M , (^) gY (^) , +^ M29Y
= m , g [ - 2 , 20581) + Mc
9 (
- l, <850 (^1) - lc(05E2) · (^) X = (2, cint1)^ = A sin (^) ti + e (^) ,^ drino^ e dt (^) = - (^) M, gl,^2058 ,^ -^ McGl,^ COS^ O^ ,^
M2g 2228582 = i = in (^) , + (^1) , Osint ,o^ e
Konstan
dt V = - (m , + M2) gl, COSO ,^ - Meg IcCO5 Oz
= 0 + (^1) , (^8) , 10581 = (^) & (^) , (^8) , C058 (^) , masukan he (^) fungsi Lagrange · i , =
1 - 2,^ 20581)^ =.
d 2058 -1, dist
dt^ At t^ L^ =^ T-^ V
E(MitM2) "^ +^ Emelit^ +^ Malilit,^ E,^ (05(0,^
- O2)
- (^0) + 1 , , sin (^) O (^) , = (^1) , (^) , sin (^) O,
+ (m , + M2) 90 , COSO , + Meg IcCO5 Oz
· xc =* )^ (^1) , <in t, +Ic sin to (^) e jika (^) detanya berapa (^) fungsi kecepatannya = (^1) , 8 , (^2058) , +^ 12G2 20582 · (^) i = -1, 1058 , - &c C85 (^) & (2)^ =^2 ,
(^) E
- (^2) , 8 , sin^ A, + l,^ sin^ z
= Ed =e
↑ +?)^ =^ It,^ "^ cos'o,^ +^ 2,^ ",sino^ ,^ )
= (^) , (x2 +^ j?)^ =^ Cit,^ Cost, +^24 , 12 8^ , 22058 ,^ CO5O2+^ &c^ <COs" Ozt esin, +^21 , l it in^ - , sin 2 +^ lesinO = 1 ,^8 , (Co5"t,^ +^ sin201) +^ lit (cost, +^ sin^
-2) +^2 lilze, Es 2 2058 , 20582 +^ <in^8 , <inO2) identifas (^) trigonometri cos[8 (^) , -82) =^ CO58, 28582 + (^) SinO, sin (^) Oz = (^) lit, (^) +l + 2 lilz - (05 (^) (t 1 - E2)
/VI
M V^ =^ O
Et
=) migy^ ,^ -Megy^ ,^
- My one L - Y (^) , = (^) M (^) = 9 - Mzg
- M
= g(M , - Mc - M3) ·I^
M
C-Y V^ =/E) = V
m -
i( m (^) , +^ M2 + (^) M3) + i (^) (M3 -M2) = G (M , - M2 - M3)
Y ,^ =^ Y ,^ L^ &^ C^ =^ Konstant^ is = g) M , - M2 -M3) - y z(M3 - M2)
Y2 =^ ( L^ - Y (^) , ) +^ Y Y3 = (- y (^) , ) +^ (l - Yc) 1M^ ,^
- (^) Mc +^ M3) untuk (^) Y i =^ i, · 26 =^ (TM,^ ee & / ↓ 2i, ;+ (^5) M3 (^) y-2) · i=^ - y Ye^ I^ I ,^ +^ y^ =^ jz- y,^ /^ 24 , - / M i (^) = - j, -y, = McG -M2y (^) , + (^) M3 Y (^) , + M i, (i-Y^ ,^ )^ = (32-^ Y^ , ) (^) (3- yi) (^) = Y (^) , (Ms -M2) + (^) y (M3 + (^) M2) =^ - Y2 Y (^) ,^ - By (^) , + (^) it,^2 = (^) Y - (^2424) , + (^) y,
/j,^ (Ms^ -^ M2)^ +^ Y^2 (M^ Me e) (-Y (^).^ - (2) = (- Y - (2) (-Y,^ - Yz) = (^) jj(Ms- M2) + (M3 + (^) M2) = i + y (^) , y +^ y (^) , y+ (^) y = Y +^ 2y (^) , y +^ y=
Ed
EG) M (^) - (^942) -Mye T = <M , y^ , + (^) EM2(32 - 3) (^) + (^5) M3( - Y - Y2) = =^ M29^ -^ M jm,^ y,^ + +M2 (i-2y, 32 +^ y, ) + (^) -Ms (y ,+^24 , y+^ Y2)
= g(M2- M3)
dibawah V^ =^ O V = -(m , (^) gy , + (^) M29(y2+ (^) (L- Y (^) , )) + m ,g((L-^ Y^ ,^ ) +^ (l- ya) = - M,9y ,^ - M2g(k+^ L^
M3g(l -^ y^ , +^ l-^ Yz)^ =JB d :^ T-V (^) jj(Ms- M2) + (M3 + (^) M2) = g(Mc - M3) = Em ,^ y,^ +TM2(i -2y,^32 +^ y,^ )^ +^ 5 Mz(y ,+^24 , y+^ Y2) i (^) =^9 /Mc^ -^ Mz)^ -^ y^ ,^ (M3^ -^ M2)
m (^) , (^) gy ,^ - M2g(y+^ L - y (^) , ) - M3g(2 - y (^) , +^ l- Yz)) (M3^ +^ M2) L=^ m^ ,^ y,^ + jm2 (i-2y, 32 +^ y, ) + (^) 5 Mz(y ,+^24 , y+^ Y2)
+ M,gy , + M2g(Y2+^ L - y , ) + M39(1-y, + 1 - Yz)
untuk (^) y (^) ,
(^8)! =^ Itm
, y^ +^ McFey, y +^ imay, + im, y
+ty Ms zyi/,the = M (^) , G (^) , -Mayc +^ M2Y (^) , + (^) M3Y (^) , + (^) M3Y · ) = /m , y - May + (^) Mey (^) , + (^) M3y + (^) M (^) iis = M^ , jj, -M2iz +^ Mij,^ +^ M3 (^) i, + (^) M3 (^) i = (^) Y (^) (M , + (^) M2 + (^) M3) + i (^) (M3 -M2)
y n
Y,^ =^0 D2 =^ l^ CUS # X (^) ,^ =^ X, (^) X2 = l sin o (^) + X (^) , = O is (^) ,^ =^0 is (^) al (^) cost (^) + y I I dest^ de
- =^ X (^) ,^ dt^ do dt i =^ -^1 sint 2 ·^ =^ 1 cost^ +^ &^ ,
- (^) = +^ (205) (, +^ 16 (05t) = + 2 1 x, #^ CO50 +^1 Cost is =^12 :^2 sin (i+^ y) =^ <+ 21 x, (058 + 12052 - + 2 : (^) sin + = + (^21) ,C85t + l(2052 - Fsin^
= * + (^) 21 x (^) , C050 + 1
T = jMX ↑ +
jMc(+^ ya) =E MX,^ + [M/ ,+^ 21 x^ ,(050^ +^ 2)^ / = & =D untu o =
"MX,^ +,M2X,^ +^ Mc^ Ix,(05t^ +^ Melt
V =^ - M291 Cost =^ Mlx,^ Co5 O^ +^ Mr: L =^ T-V indiamitmimimi -4) = 28 = Untuk^ e^ =-M2lx,^ sino-Meglan
2
E,^ IEMX,^ +^ -M2X,+^ M2lx,^ Ecos e^ ·^ &C ) = m1 <, Cost +^ m &^ / = MX (^) , +^ M2X (^) , + Mr ECO5t^ (tx,^1058 +^2 )^ = -Mano(,+ (^) ge)
l & ee
· (^) X
· Co58 + = -^ X f^ sint g Of
sin t
f (^) l Komponen X. Larangiannya Independen (x,^ (M^
- (^) M) + M2(5) =^0 Karena -^ x^ =^ Konstan^ ,^ X^ =^0
- =^ constant^ :. Cost + sin + = (^0) E + r E =^ - sin
