Eq Sheetv2
3
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Transcript of Eq Sheetv2
8/10/2019 Eq Sheetv2
http://slidepdf.com/reader/full/eq-sheetv2 1/3
!"#$%&$ ()* +,-- .*(/0%12345 6 789,2%:; <"332
=3&2:4$!
A i
!
B = AB cos ! = A x B x + A y B y
+ A z B z !
A !!
B = AB sin " , direction by right hand rule
!
A !!
B = ˆ
x A y B z " A z B y( )+ ˆ
y A z B x " A x B z( )+
ˆ
z A x B y " A z B y( )
>,423$%,; >::41%;,23$
!-,;3 ?:-,4 &::41%;,23$
@3A2:;B$ C,A$@3A2:;B$ D"433 C,A$E; ,; %;342%,- F4,53G
!
F NET
= 0 !!
v = const !
F NET
= M !
a !
F 1,2
= ! !
F 2,1
E;F:45,2%:; :; <:53 +:4&3$
H4,I%2#!
F Grav
= ! GM 1 M
2
r 2 r̂ g = 9.8 m / s
2! 10 m / s
2
+4%&2%:; f STATIC ! µ S N f KINETIC
= µ K N
J:?3 :; , ?9--3# N = Td !
K::L3B$ C,A F:4 <?4%;M$ F = ! kx
<%5?-3 K,45:;%& 0:2%:; d 2
xdt 2
+ ! 02 x = 0
H3;34,- $:-92%:; %$N x (t ) = A sin ! 0t + B cos !
0t :4 x (t ) = C sin( !
0t + " ) O
P"343 A = C cos ! B = C sin ! ,;1 C = A 2+ B 2 ! = tan
" 1 ( B / A )
+:4 , $#$235 :F ?,42%&-3$G!
PSYS =
!
p!
!
" = ; m!
v!
!
" d !
PSYS
dt =
!
F EXT O
!
v(t ) =dxdt
ˆ x +dydt
ˆ y +dzdt
ˆ z
!
a (t ) =d 2 xdt 2
ˆ x +d 2 ydt 2
ˆ y +d 2 zdt 2
ˆ z
!
v (t ) =dr
dt
r̂ + r d !
dt
ˆ!
!
a (t ) =d 2 r
dt 2 " r d !
dt # $ % &
' ( 2)
*+
,-. r̂ + r
d 2!
dt 2 + 2
dr
dt
d !
dt
)*+
,-.
ˆ!
8/10/2019 Eq Sheetv2
http://slidepdf.com/reader/full/eq-sheetv2 2/3
>3;234 :F 0,$$!
RCM
=
1
M T
m!
!
r!
!
" =
1
M T
!
r dm# =
1
M T
!
r $ dV #
!
V CM
=
1
M T
m!
!
v!
!
" =
1
M T
!
v dm# =
1
M T
!
v $ dV # !
F EXT
= M T
!
ACM
F dt = ! P Impulse"
J:&L32 789,2%:;m
dv
dt + v
EXH
dm
dt = F
EXT ! dm
dt v
EXH =
" thrust "
7;34M#
P:4L 1:;3 Q# , F:4&3 ,$ , ?,42%&-3 5:I3$ F4:5 ?:%;2 ( 2: ?:%;2 .W =
!
F i d !
r1
2
!
P:4LR7;34M# D"3:435N
!:23;2%,- 3;34M# F:4 , &:;$34I,2%I3 F:4&3N
D"3 ?:%;2 ! %$ ,; ,4Q%24,4# 43F343;&3 ?:%;2O S# 13F%;%2%:;G!"#$%*OE; :;3 1%53;$%:;
<?"34%&,--# $#55324%& F:4&3
E; 6 1%53;$%:;$
P"3; ,-- F:4&3$ ,43 &:;$34I,2%I3G
H3;34,-%T31 P:4LR7;34M# D"3:435N TE F ! TE
0 = W
NC
U2 -:&,- 5%;%595G O <:G -%L3 , $?4%;M A%2"
EFKE (q ) =
1
2 B !q 2 ,;1 PE (q ) =
1
2 Aq2 2"3; 2"3 $#$235 A%-- :$&%--,23 ,2 ! 0
= A / B
V%;32%& 7;34M# :F , <#$235NT = M
T ! R
CM
2+ T
rel to CM
V ( x ) ! V ( x 0 ) + 12
V ''( x 0 ) x " x0( )2 k = V ''( x0 )
KE 2 ! KE
1 =
!
F NET
i d !
r1
2
"
U (!
r ) = ! !
F i d !
rP
!
r
"
F ( x ) = ! dU
dx
F (r ) = ! dU
dr
KE 0
+ PE 0
= KE F
+ PE F
!
F = !" U = ! dU dx
ˆ
x +dU dy
ˆ
y +dU dz
ˆ
z# $ %
& ' (
8/10/2019 Eq Sheetv2
http://slidepdf.com/reader/full/eq-sheetv2 3/3
7-,$2%& &:--%$%:;$ %; (W X $?331 :F ,??4:,&" Y $?331 :F 5:I%;M ,?,42O
U;M9-,4 0:53;295!
LSYS =
!
ri ! !
pi ;d
!
LSYS
dt =
!
" EXT
i
#!
LSYS =
!
LORB +
!
LSPIN
!
LORB =
!
RCM ! M T
!
V CM
d !
LORB
dt =
!
RCM !!
F TOT
d !
LSPIN
dt =
! $rii
# !!
F i
%; 2"3 -,$2 389,2%:;G I3&2:4$ !" ,43 ?:$%2%:;$ :F ,-- ?,42%&-3$ 2,L3; A%2" 43$?3&2 2: 2"3
$#$235B$ &3;234 :F 5,$$O
!!
L SYS =
!
" EXT dt #
P"3; 2"3 ,Z%$ :F 4:2,2%:; %$ F%Z31G
! =
dL z
dt = I "
I = r 2 dm!
!,4,--3- ,Z%$ 2"3:435N I
A = I
CM + Md 2
!43&3$$%:; [M#4:$&:?3$G A"33-$ 4:--%;M %; , &%4&-3G 32&O\
d !
LS
dt = ! = " L
S
@:;R%;342%,- F4,53m
!""r =
!F ! " m
! A
