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s R
θ =
A
ANGULAR MOTION
WIND TURBINES such as these cangenerate signifcant energy in a
way thatis environ-mentally rien!ly an!renewa"le# The conce$ts o
rotationalacceleration% angular velocity% angular!is$lacement%
rotational inertia% an!other to$ics !iscusse! in this cha$ter
areuse ul in !escri"ing the o$eration o win!tur"ines#
Rotational Displacement,&onsi!er a !is' that rotates rom A
to B(
Angular displacement θ:
)easure! in revolutions% !egrees% orra!ians#
* rev + , . . + / π ra!
The "est measure or rotation o rigi!"o!ies is the radian #
Defnition o t!e Radian0ne ra!ian is the angle θ
su"ten!e! at the center o a circle "y anarc length s e1ual to
the ra!ius R o thecircle#
* Ra! + R R + 23#, 4
E5am$le * ( A ro$e is wra$$e! many
times aroun! a !rum o ra!ius 2. cm#6ow many revolutions o the
!rum arere1uire! to raise a "uc'et to a height o/. m7
8 + s/R + /.m9.#2m + :. ra!#Now% * rev + /; ra!
8 +
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8 +
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2 ( ) 2
f Since f
t
π α ω π
∆= =
E5am$le 2( The "loc' is li te! rom restuntil the angular
velocity o the !rum is* ra!9s a ter a time o : s# What is
theaverage angular acceleration7
ɑ +ω f − ω o
t or ɑ +ω f t
ɑ +16 rad / s
4 s + :#..rads
2
a + :#.. ra!9 s2
Angular and Linear $peedrom the !efnition o
angular!is$lacement(
s = 8R Linear vs. angular displacement
v + Δs Δt + <
Δθ · R Δt = + <
Δθ Δt = R
v = ωR
Linear speed = angular speed x radius
Angular and LinearAcceleration:
From the velocit relationship !ehave"v = !R Linear vs. angular
velocit
v + Δv Δt + <
Δv · R Δt = + <
Δv Δt =R
a + ɑ R
inear acceleration + Angularacceleration 5 Ra!ius
E5am$les (&onsi!er Cat rotating !is'(
R
h = /. m
R# $
BR
2
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0 f v va t
−=
R* + /. cm R / + :. cmω o = %& ω f = 2% ra!9s t +: s
What is fnal linear s$ee! at $oints A an!B7
v $f = ω $f R* =
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210 2 s v t at = + f ov v at = +
210 2t t θ ω α = + 21
2 f s v t at = −
2 2
02
f as v v= − 21
2 f t t θ ω α = −
f o t ω ω α = + 2 2
02
f αθ ω ω = −
inear E5am$le >( A car traveling initiallyat /. m9s comes to
a sto$ in a !istanceo *.. m# What was the acceleration7
Select E1uation(
/as + v f 2
- vo2
a =0 − vo
2
2 s +−( 20 m/s)2
2 (100 )
a = − 2 m / s2
Angular analogy @( A !is'
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2. rev + ,*: ra!
ɑ +0 − ω o
2
2 θ +−( 62.8 rad /s)2
2 (314 rad )
ɑ + - #/@ m9 s2
Answer
Gro"lem Solving Strategy(
Draw an! la"el s'etch o$ro"lem#
In!icate H !irection orotation#
ist givens an! state what isto "e oun!#
iven( JJJJ% JJJJJ% JJJJJ
