phy10t4ucm&ulg
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s or x
UNIFORM CIRCULAR MOTION (UCM)
ANGULAR QUANTITIES
r
r
or
From 1 to 2 there is a change in
linear displacement (arc s), and alsoa change in angle from 0 to . As thearc length increases the angle in thecircle also increases
s s =r
x =r
1. ANGULAR DISPLACEMENT ( or)
= x / r
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=
v=
r
UNIFORM CIRCULAR MOTION (UCM)
ANGULAR QUANTITIES
2. ANGULAR VELOCITY ()
x r
This is the change indisplacement with respect to an
interal of time
tt = v / r
AVE=
t
INS=d
dt
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UNIFORM CIRCULAR MOTION (UCM)
ANGULAR QUANTITIES
=
a=r
. ANGULAR ACCELERATION( )
v r!
This is the change in elocit" with
respect to an interal of time
tt
= a / r
AVE=!
t
INS=d!
dt
* a = r (refers to !"#$%#!&"'acceleration)
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s C&r*+,%r%#% 2-r
UNIFORM CIRCULAR MOTION (UCM)
ANGULAR QUANTITIES
r
= 3
60 = 2 rad =1 rev
#eriod is Time for 1 reol$tion. %t is$s$all" in seconds orsecond&reol$tion
. PERIOD (T) /. FREQUENCY(,)A '$antit" that is the reciprocal of
#eriod .
%t is $s$all" in radians per secondsor reol$tion&second
f= 1 / T
T 0. ANGULAR FREQUENCY(,)This is the (aerage) ang$lar
elocit" for a reol$tion
f= !f f=
!/T
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UNIFORM CIRCULAR MOTION (UCM)
CENTRIPETAL FORCE(FC)T
% N%! ,or% !"! !%#s !o %3%! "# o45%! ,ro++o6$ " s!r"&$! 7"! "# "*s%s !o $o "&r*'"r 7"!. T&s &s &r%!% !o !% "x&s o, ro!"!&o#
CENTRIPETAL ACCELERATION ("C)
A'so 8#o9# "s !% r"&"' "%'%r"!&o#. T&s &s !%"%'%r"!&o# "sso&"!% 9&! %#!r&7%!"' ,or% "#,o''o9 !% NSLM: &! "'so $o%s !o !% "x&s o, ro!"!&o#.
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UNIFORM CIRCULAR MOTION (UCM)
CENTRIPETAL FORCE (FC)
6T
ac
FC
r
+
NS"# F#%! ;Fx +" and $%
o$servation t&ere is no vertical'ove'ent ;F< =F#%! +"
FC +"
" 6T2>r FC +(6T2> r)
(T)Tangential elocit" is also the inear
elocit". %f the ang$lar elocit" is *nownthen
6T r
FC +r2
"C r(2->T)2
(-2r)>T2
%f the ang$lar fre'$enc" is *nown (!f) +
FC +-2,2r
FC (+-2
r) >T2
" r2
%f the period or fre'$enc" is *nown +
"C r(2-,)2
-2,2r
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UNIFORM CIRCULAR MOTION (UCM)
%n -, the ang$lar elocit" /ang$lar acceleration at A
point or radial distance withinthat circle is constant.
6T
ac
+
12
?2
?1
??
6T
+616266
T, the linear or tangentialelocit" / ( acceleration )di3ers at A point or radialdistance within that circle.
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6F2 6O
2 @ 2"(x)
6F 6O@ "!
x xO @ 6O! @
"!2
L%"r Mo!&o# EB*"!&o#s Ro!"!&o#"' Mo!&o# EB*"!&o#s
x 6O! @
"!2
F2 O
2 @ 2?()
F O@ ?!
O @ O! @
?!2 O! @
?!2
is in radians
UNIFORM CIRCULAR MOTION (UCM)
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1 A oint on a +&eel rotatin, - rev/s and located . ' fro' t&e axis
exeriences +&at centrietal acceleration
r= .'
ac =
= - rev/s
ac = r
= (- rev/s)x(! rad/ 1 rev) = 1=-
r">sac = (. ')(1.!
rad/s)ac = 1020
'/s
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A sled +it& 'ass = -3, rest on a &ori4ontal s&eet of frictionless iceIt is attac&ed $% a - ' roe to a ost set in t&e ice 5nce 6s&ed7 t&esled revolves 6nifor'l% in a circle aro6nd t&e ost If t&e sled 'a3es8ve co'lete revol6tions er 'in6te 9ind t&e force exerted on it $% t&e
roeTo Vie+
= (- rev/'in) : (! rad/1rev) :
(1'in/;. sec) =./2
T F
+r2T (2/ 8$)(/+)(=./2
r">s)2T .22 N
'=- 3,
r=- '
T T
Isolating thesled
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UNIFORM CIRCULARMOTION U#&,or+ or&o#!"' C&r*'"r
Mo!&o#
9
6T
ac
6T
6T
ac
ac
< =
',
6T
ac
6T
ac
6Tac
6Tac
Tangential or inear elocit"(T) is constant and
perpendic$lar to the radial orcentripetal acceleration (a-)
!" #I$%
a&is
r
F'! #I$%(hal)
Fc or an*net force is not
drawn on the F4
;Fx F
+";F< =
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2 A . 3, $loc3 in t&e 8,6re is attac&ed to a vertical rod $% 'eans of t+ostrin,s o+ 'an% revol6tions er second '6st t&e s%ste' '6st 'a3e in order t&att&e tension in t&e 6er strin, s&all $e 1- N
($)
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2 A . 3, $loc3 in t&e 8,6re is attac&ed to a vertical rod $% 'eans of t+ostrin,s o+ 'an% revol6tions er second '6st t&e s%ste' '6st 'a3e in order t&att&e tension in t&e 6er strin, s&all $e 1- N
($)
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2 A . 3, $loc3 in t&e 8,6re is attac&ed to a vertical rod $% 'eans of t+ostrin,s o+ 'an% revol6tions er second '6st t&e s%ste' '6st 'a3e in order t&att&e tension in t&e 6er strin, s&all $e 1- N
($)
(=. 8$)
= 3+
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2 A . 3, $loc3 in t&e 8,6re is attac&ed to a vertical rod $% 'eans of t+ostrin,s o+ 'an% revol6tions er second'6st t&e s%ste' '6st 'a3e in order t&att&e tension in t&e 6er strin, s&all $e 1- N
($)
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UNIFORM VERTICAL CIRCULAR MOTION
Tangential or Linear
Velocity (vT) is constant
and perpendicular to the
radial or centripetal
acceleration (aC)
W = mg
vT
ac
vT
vT
ac
ac
vT
ac
vT
ac
vTa
c
FRONT VIEWW = mg
W = mg
W = mgW = mg
W = mg
F
F
F
F
F
F
Effect of Weight is
present and its
reaction force (F) is
considered in the
analysis.
;F< F
+";Fx =
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NON- UNIFORM VERTICAL CIRCULAR MOTION
Tangential or Linear
Velocity (vT) is not
constant and but still
perpendicular to the
radial or centripetal
acceleration (aC) which
also varies.
W = mg
vT
ac
vT
vT
ac
ac
vT
ac
vT
ac
vTa
c
FRONT VIEW
W = mg
W = mg
W = mgW = mg
W = mg
F
F
F
F
F
FEffect of Weight is
present and its
reaction force (F) is
considered in the
analysis.
Fc or anynet force is not
drawn on the F!
;F< F
+";Fx =
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Vertical Circ. Motion
1. Tarzan (m=85 kg) tries to cross a river by singing !rom a 1"m long vine. #is s$ee% at t&e bottom o!
t&e sing (as &e 'st clears t&e ater) is 8 ms. Tarzan %oesn*t kno t&at t&e vine &as a breaking
strengt& o! 1""" +. ,oes &e make it sa!ely across t&e river-
ac
vT= 8 m/smT= 85 kg
r = 10 m
Tmax= 1000 N
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UNIFORM CIRCULAR MOTION APPLICATIONS
ROAD CURVES DESIGN
FLAT CURVESTOP VIEW REAR VIEW
ris "radius of curvature#
axis
A FLAT !RVE" ROA" #A$ A %A&I%!% VELOIT' LI%IT IN
W#I# (ELOW T#I$ $PEE" T#E AR AN $AFEL' RO!N"
T#E !RVE WIT#O!T $)I""IN* FRO% T#E ROA"+ T,is can
-. caca.2 sing !% 3 N$L%
vTmax = 4
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UNIFORM CIRCULAR MOTION APPLICATIONS
ROAD CURVES DESIGN
FLAT CURVES F
ris 5radi$s of c$rat$re6
+
$
N
a&is
ac
fis t&e (net) side frictional force actin,on t&e car It is t&e onl% force alon, t&e
xFaxis rovidin, t&e net force Fcentrietal force
N
+$
ac
NS"# ;F +" and $% o$servation t&ere isno vertical 'ove'ent ;F< =
;Fx +"xH (@)
@ , @+" ;F< =K(@)@ N =
, (+6T+"x2) >
r
N
+$ = s
sN (+6T+"x2) >
rs+$
(+
6T+"x
2) > rs$ (6T+"x
2
) > r
6T+"x2 sr$
6T+"xr
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UNIFORM CIRCULAR MOTION APPLICATIONS
ROAD CURVES DESIGN
ANED CURVES!" #I$% '$' #I$%
C4'#$ '!5 '$ $ 78$ (9)"'I:'I8; F!' 5F$; '$5!5 4'I7 %$ !'F'!
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UNIFORM CIRCULAR MOTION APPLICATIONS
ANED CURVES
F
ris 5radi$s of c$rat$re6
+$N
N< N os
+$
a&is
ac
ac
N
+$
ac
Nx N s
NS"# ;F +" and $% o$servation t&ere isno vertical 'ove'ent ;F< =
;Fx +"xH (@)
@ Nx @+"
;F< =K(@)@ N<
=N s +" N os
+$
N (+$) > os
$6t
ac = vT'ax /r
(+$ > os ) s +"+$ !"# +"
$ !"# "
!"# ">$
!"# 1
(">$)
!"# 1(6T+"x2>
(r$)J
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1 A Gat (6n$an3ed) c6rve on a &i,&+a% &as a radi6s of H. ' A carro6nds t&e c6rve at a seed of '/s s)2>(==,!)(2,!>s2)JW
12.2
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2 T&e radi6s of a 9erris +&eel is '7 and it 'a3es one revol6tion in1 s9ind t&e aarent +ei,&t (Nor'al 9orce) of an . 3, assen,er at t&e
&i,&est O lo+est oints
+$
N
NT
+$
ac
a
c
r
A ferris wheel is a ertical circle moing atconstant speed (%F78 98T%-A-%8-A8 7T%7). Apparent weight meansthe e3ect of feeling light or hea" at certain
portions of the ride as the ferris wheel isoperated. This is d$e to the normal force
e:erted ;" wheels ca; in reaction to "o$rweight and the motion of the wheel
T 12s%
"C (-2)(+)J>(12s)2
= 8$( . +>s2) N
"C (-2r)>T2
"C 2.1 +>s2
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NT
+$
ac +$
ac
NT
;F< +"s2X 2.1
+>s2
)NT 0=./0=
N
N
+$
ac
+$
N
;F< +"s2@ 2.1
+>s2
)
!% TOP o, !% F%rr&s%%'
!% OTTOM o, !% F%rr&s%%'
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H A cord is tied to a ail of +ater7 and t&e ail is s+6n, in a verticalcircle +it& radi6s 1H'
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T1W = mg
ac
vT
To get the $ini$u$ velocity% the tension
in the cord $ust also be the $ini$u$%which is &ero.
T1
W = mg
ac
Fy = may (+
!T1" W = "ma#
!T1" mg = "ma#
!(T1+ mg = "(mvT$ % &
a# = vT$% &
T1+ mg = (mvT$ % &
' + mg = (mvMIN$ % &
g = vMIN$% &
vMIN$
= g& = ()* m%$
(1), m = 1).$ m$
%$
vMIN= ).',m%
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T&is +as discovered $% Sir Isaac Ne+ton
States t&at P Ever% article of 'atter in t&e 6niverse
attracts ever% ot&er article +it& a force t&at is directl%roortional to t&e rod6ct of t&e 'asses of t&e articleand inversel% roortional to t&e sJ6are of t&e distance$et+een t&e'Q
+1 +
+2
F12
F21 F2
F2
F1F1
Ill6stration
r
r 1 r2
91= F 91
92= F 92
912= F 921
Us$ N%9!o#Zs T&r L"9 o,
Mo!&o#
T +To eer" actionthere is alwa"s opposedan e'$al reaction, same inmagnit$de ;$t opposite in
direction.
UNIVERSAL LA OF GRAVITATION
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UNIVERSAL LA OF GRAVITATION
Ronsider T+o 5$ects
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MR
9#E= BM ME
RE2
G 0.0 x 1= 11N+2>8$2
FORCE OFGRAVITY
ME /. x 1=2
8$
(+"ss o, !%
%"r!)RE 0. x 1=0
+(r"&*s o, !%%"r!)
(0.0 x 1= 11 N+2>8$2)M (/. x 1= 2
8$)(0. x 1=0 +)2
9#E= (0 '/s) #
M [ (+"ss o, "#
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1 T&e 'ass of t&e 'oon is a$o6t 127 and its radi6s is -7 t&at oft&e eart& Ro'6te for t&e acceleration d6e to ,ravit% on t&e 'oonUss6rface fro' t&is data
ME /. x 1=2
8$ (+"ss o, !% %"r!)RE 0. x 1=
0 + (r"&*s o, !%
%"r!)G 0.0 x 1= 11N+2>8$2 (U#&6%rs"') Gr"6&!"!&o#"'Co#s!"#!
M+ (=.=12)(/. x 1=2 8$) . x 1=228$
R+ (=.2/)(0. x 1=0 +) 1.// x 1=0+
,'= ?#'/ '
,'= (;;x1.F11N'/3,)(22x1.3,)/
(1-0-x1.;')$+ 1.0
+>s2
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,'= ?#'/ '
$+ 1.2
+>s2
,E= ?#E/
E ? = ?
,E
E
/ #E
=,'
'
/ #'
,' = ,E(#'/#E)(E/')
1 T&e 'ass of t&e 'oon is a$o6t 127 and its radi6s is -7 t&at oft&e eart& Ro'6te for t&e acceleration d6e to ,ravit% on t&e 'oonUss6rface fro' t&is data
9ro' ,iven #'= (..12)#EO '=
(.-)E,E= 0 '/s
,' = (0 '/s)(..12#E/#E)(E/.-E)
,' = (0 '/s)(..12)(H)
ALTERNATIVE SOLUTION
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At +&at oint $et+een t&e Eart& and t&e #oon is t&e ,ravitational6ll of t&e Eart& eJ6al in 'a,nit6de to t&at of t&e 'oon Ass6'e ano$ect +it& 'ass # in $et+een t&e eart& and t&e 'oon (Avera,e
distance $et+een Eart& O #oon 2Hx1.
')D = 2Hx1.'
SE S#
ME /. x
1=2 8$
MM .0x
1=22 8$
M
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At +&at oint $et+een t&e Eart& and t&e #oon is t&e ,ravitational6ll of t&e Eart& eJ6al in 'a,nit6de to t&at of t&e 'oon Ass6'e ano$ect +it& 'ass # in $et+een t&e eart& and t&e 'oon (Avera,e
distance $et+een Eart& O #oon 2Hx1.
')D = 2Hx1.'
SE S#
ME /. x
1=2 8$
MM .0x
1=22 8$
MFES FSE
FES
FSE
FSM FMS
FSM FMS
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At +&at oint $et+een t&e Eart& and t&e #oon is t&e ,ravitational6ll of t&e Eart& eJ6al in 'a,nit6de to t&at of t&e 'oon Ass6'e ano$ect +it& 'ass # in $et+een t&e eart& and t&e 'oon (Avera,e
distance $et+een Eart& O #oon 2Hx1.
')
ME /. x
1=2 8$
SE S#
MFSE FSM
FSE GMEM > RSE2FSM GMMM >
RSM2
FSE FSM
GMEM> RSE2
GMMM>RSM2
ME>RSE2 MM>RSM
2
GMEM>RSE2
GMMM>RSM2
MERSM2 MMRSE
2
RSM2 (MM>ME)RSE
2
RSM2 (=.=12)RSE
2
RSM
=.11=RSE
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At +&at oint $et+een t&e Eart& and t&e #oon is t&e ,ravitational6ll of t&e Eart& eJ6al in 'a,nit6de to t&at of t&e 'oon Ass6'e ano$ect +it& 'ass # in $et+een t&e eart& and t&e 'oon (Avera,e
distance $et+een Eart& O #oon 2Hx1.
')
SE S#
MFSE FSM
RSM =.11=RSE
D = 2H x 1.'
D RSM @ RSE
.x1=+ =.11=RSE
@ RSE.x1=+ 1.11=RSE
(.x1=+)>1.11= RSE
RSE
.0x1=+
RSM
.x1=0+
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Satellite#otion
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