The seriesis valid only if dis in radians.Notice that if dis small (e <<l), eachterm in the series
is much lessin magnihrdethan the previousterm. The small angleapproximationarnountsto
rrsingonly &e fust term of the series.
If dis small,the error generatedby the small angleapproximationis nearlythe secondterm0'/6
andthe fractionalerror is roughly 02l6.For exarnple,the error is lessthan 1 pet cerLtif 02/6 <
0.01or d< 0.2 radians.In factthe errorfor 0= 0.2radiansis 0.7 percent.
Questionsand ExampleProblemsfrom Chapter L5
QuestionI
Which of the following describe/for the SHM of the figure below (a) -n < (< -d2,
S) n< Q<3td2,{c)-32d2<
d<-i?
z.l'1ti !'fit]Qe'\
rclrrylx r AA^{to.[t6fr\t
,c-rrvu-{^-. /,0qb
l! ]J" -L'N.ar&t-t
1 I
- -Tr<
l,
T/x +'rr
+ 7n/0.,
-,/, <_-.
-n <C 13n/^
Question 2
The figure below showsthe x(t) curvesfor threeexperimentsinvolving a particularspring-box
systemoscillatingin SHM. Rankthe curvesaccordingto (a) the system'sangularAequency;O)
the spring'spotentialenergyat time t : 0, (c) the box's kinetic energyat t : 0, (d) the box's
speedat t : 0, and(e) the box's maximumkinetic energy,greatestfirst.
(ur = ri,Kxt)
(b)
( v = J',/lt)
(.)
(r)l],}a
=\)r,-,^,)
( K"-^
(")
t")Jl)3,ei
/m (h"-*=vs,-")
r
Question3
-r-,c.".r*)
( u-, =2.-nf a f
(o)
':
i<'-1-'7-t
/
rt
' \
\
Theacceleration
a(t) of a particleundergoing
SHMiFSffiii-ii thefigurebelow.(a) Wlich of rhelabeled
-x.?
pointscorresponds
(b) At point4, is thevelocityofthe particlepositive,negative,or
to theparticleat
zero?(c) At point 5, is the particlo at -x., at +x., at 0, between-x,oand 0, or between0 and +x.?
(r)
F....jF^a
04-
Problem 1
An oscillatorconsistsof a block of mass0.500kg connectedto a spring.Whensetinto
oscillationwith amplitude35.0cm, the oscillatorrepeatsits motion every0.500s. Find (a) the
period,(b) the frequencn(c) the angularfrequency,(d) the springconstant,(e) the maximum
speed,and(f) the magnitudeof the maximumforceon the block from the spring.
("1lE "-.::(r) t= '/T =lT"."l+4
L^) =a..Tl
(.)
= An(a,o"/f") -
uo= le,b",qA/s
(r)
(
")
X = X- roa("ut*b)--+ V- d4 t= - r.a;lX,q,,ri.6*t.6;
--VV;el
V'.",=
wx* = ( tt.t"^"Q/s)(o"zl,i
(t)
F=-t<x---- lF;l
: KX" = (?1.oN/rD)(o.rs")
=fTf:tNl
( r.t
problem 2
-r.r-at Q a =
: r^ 9^
"rtx- )
"lA"
A body oscillateswith simplehannonicmotion accordingto the equation:
x = (6.0m) cos(3r radls)t +n /3radl. At t = 2.0 s, what are(a) the displacement,(b) the
velocity, (c) the acceleratiorLand (d) the phaseofthe motion?Also, what are(e) the frequency
and(f) the period of the motion?
\-/]
(") x(l=r,os)= (a.o.)c,o<l},r""t/,)(r.or)
*T/jn )"1 =ld
(b) v= dxAt= -(6'0l)(:'*%);"nf (:*-r4)* nr/t^^A7
v (f =:.o5) = - (5.o".)(3ffiV)/^r\
+ .yuA7 =
f(:n*17,1(l.oo3)
(.) q = d%t = ( -t rrr
"/s)(3rr"dl),t deI en *tl) t + T/za^,4f
a (t=r'o') = ( - s{.r'-'-1,)<na[( 3r'*%)( t.o)+T[^],1 = - A"7onf
tt
( J) 0)*^" -t ft"frd,h : vst+ fi = (3r *J^ ) ( 3.os)+ V3""L = lqTA^^4
(")
(+)
t =,/f
Problem3
In the figure below, two blocks (m : 1.8kg and M = 10kg) and a spring(k = 200 N/m) are
arrangedon a horizontal,frictionlessswface.The coefficientof staticfriction betweenthe two
.What
blocksis 0.40.
amplitudeof simpleharmonicmotion of the spring-blockssystemputs the
smallerblock on the vergeof slipping over the largerblock?
? -^r"rr* Ms a^^^!\ 2tI4X\;^ /n lne ,"^rV "l
,t)t+p^narfr=fr,.", =r'f,\ = )J,nt
Frictionless
+dalr-Qr"q n
'
---,--5fr,^".*= mq*", +
Xn-feey
--/\
o
a A-lltvr.a
4fa1
vud
- tnat4
.-.rn |
Op/UJlJtdtl
(l m = Ll) . 1-X
r,r
^g
Ji, - J ' ,tl"" ot
l"drmg = tr)4,n.
--(
tlx''=o')3rn
x-=@
Problem 4
2oo N/ r'r
An oscillator
consists
of a blockattached
to i .pri"!
rg ftYfi
ft :400 N/m). At sometime t, the position
(measuredfrom the system'sequilibrium location),velocity, andaccelerationof the block are
x = 0.100m, v = -13.6m/s,anda = -123mlsz.Calculate(a)the frequencyof oscillation,(b) the
massof the block, and(c) the amplitudeof the motion.
(=
(o)
4ooX/r",
"*
t=osr
X=o"looo,
V =-|3.{onfs, Q =-1a3"'/5\
----?,^= =
q =:esAx
E m
=--F:;;l ,,^td
(u)
w=V, -'-J,=/*n=
[##)"-
c")
E = /ornvr-nrA
Kx\
y^rx"]=
E = l^ l(x-t = 2*^vj
/a.rnul 7*v'N X. ,l'
= '/"v
Xm
,\
rn = o.3e5 Kg
t*
x'
,
v - I<)
,,\
( t z . t "" 1 1 ) 4 +( o "t o o "; '
X" = 0,{oo.
Problem5
A simpleharmonicoscillatorconsistsof a block of mass2.00kg attachedto a springof spring
constant100N/m. Whent = 1.00s, thepositionandvelocityof the block arex = 0.129m and
v :3.415 m/s. (a) Whatis the ampiitudeof the oscillations?
Whatwerethe (b) positionand
(c) velocityofthe block at t = 0 s?
tt
(")
k=/t'\\+/)
? l Kxt_ TaKx] --a x.=
YKv'
X. = O.5oo r'.)
X: X. caa(wt+6)
V = -.oX',., a;"" (*++d)
( b)
Yx
-*fr4(*t*6)
r=ffi ="r.o'r^A/s
* .r-^r41r,!rAd
*z ZI
X +v af t'o
--- F,\(aL*d)'=-u/,nx --->6 t + Q =
'
-r /-v/
\
t r a . n\ , / , ' D r )
=^'fffffit".,r^>](t""1*t13111.f,.
t) - -S.lt"^[
e=r^"'(Yq-ot
X = (o "so-) r-arL(r"ol,Al)[o,) - r.]s^-I ] -- - O " ) 5 1 r . )
yv =- - (t.o?.t/r)(o.s""^l-.a,,"n
(^*+l) -(.)
rProblem
roDlem o
6
\r,/
\ ,'u,rqs/5/((-,,6.wn\
1 , , / . ) r \ 1t 1* c ' Y )
l13.66n/s
"'--"(iA'noscillatingblock-springsystemhasa mechanicalenergyof 1.00J, an amplitudeof 10.0cm,
anda maximumspeedof 1.20mis.Find (a)the springconstant,(b) themassof thebiock,and(c)
thefrequencyof oscillation.
7i= ).oo5
X-
=lD"Dcrr
Y* = 1-a'o'o/e
2..oox to'N/r1
(U) E= Y^rrvi----:,n = ATui = A(r'or)
( l'ao ) "
"7t
(.) u)--Arf =E -----)
f =r.?tr
-LC=
Jln
l.eo^ lo3NA
I.31Vg
f = t.rrtl*
Problcm 7
A physicalpendulumconsistsof a meterstick that is pivoted at a small hole tlrilled throughthe
stick a distanced from the 50 cm mark. The period of oscillationif 2.5 s. Find d.
:
..N,
]*-= ZlNL^
-f = I.,-* Mh*= /vi{Ln* MJ'
-l-= r='
e" \fiU
P,,"1""^:
4+"F["AALA
\\)
l/,.,
t'*
,-,^.-.--
l-
h=J
J'
gJ
T'=4'T^f 44>'l --' ? Jr' = !E*'?rr'J'
4n'J'-?JT** :r'LA=o
.r J \ jt
-4 lt'
.%-.
=o
d +-
)
tfr1
"^H^e- J=o.osa,.,
d,=t"s{7
Problem8
valueofdistance x betweenthe stick's centerand its pivot point o givesthe leastperiod?
(b) What is that Period?
f --f."** m hr
\./
= Y6ML\ M xe
-b"-, pt^l^;.! Wd-b^,; --T=2-1t l-+1 mg\
"
I
(),'irMr\vv') =
l=Arr
I
r;*.
: a..1T, z - + t a Y
I
'l
-
d.
l0.gr
T' = 4 n* (?"" * Yr)
rt
j:
*-\.
)>.:x 2
"t-
h =x
:e"@
Iri-
'r ,t2.9x
- x-r I-Ll , ,u),,^
\ l)'91
;",a..TL
fig I g *e S*&
-i*ia
q'd vro,"n
*o.
.^,-*
a.",,1,,
q rr"r-'
rt 1
r-l/
. . ^ T -i;.
/'
L/t^
* /s
o'/a*:
L
J=o
-L\
o,/Z=o ----.X= /61. -=o.ts;2/G --lp-"s3,"]7
llax-
(f) 44 .;,fut4^- ^ ),'-rk r*lz__-i-J4
Problen 9
A masslessspringwith springconstant19N/m hangsvertically. A body of mass0.20 kg is
attachedto its free end andthenreleased.Find (a) how far below the initial position the body
descends,andthe @) fiequencyand(c) amplitudeof the resultingSHM.
x=o -r
rt
qt
I
oJ %urilbv"i^
t
Fy=f0QY=o
toz/v
/ N
K,
-)
ff
\l
Il
'.
f."tTi,.A
= @:3:J!s.ald
ltn7.
,D\, nr^"t uiAl 4tU,,J,tlfrlon'ns x tl9i,
,canh"ro )'o,nn Pr-'A
= D'163\
p""b-a +
t4.*;lt*
^, 'oayal )Jffirr* X L)}t fu'-g
f""A
--14.x
(^) AX= Ar = 11o"roan,)
-",\J
L
(u) f =1^"F^ = lrn fs-tb '--l f =
fq?
(.) a-4a,b,* = ,!r\.o.4;;W
Problem10
"*"il;
In the figure below,block 2 of mass2.0 kg oscillateson the endof a springin SHM
20 ms.Thepositionof theblock is givenby x = (1.0cm) cos(ot+ n/2).Block i of mass4.0 kg
slidestowardblock 2 with a velocity of magnitude6.0 rn/s,directedalongthe spring's length.
The two blocksundergoa completelyinelasticcollision at time t : 5.0 ms. (The durationof the
collision is much lessthanthe period of motion.) What is the amplitudeof the SHM after the
collision?
-f,^-!-P
/urtj
x\t''nL+,?il
).W-19,t
.^} 7{
nma.oa
\
coaQ,ut-a" \
,.,orurJ
w
w= E"tr/' = -$"-**t --) t,D- 3tq/''^Afs
V= -u,uXn,-^/,.,(*t*d)
X r 1 , =l , O . . q = O . o l ( , n
T/
/a
d=
v(t=5.o"r) = - ( :r I ^-l1r) ( o"oro ) -".i" lQt1"-tl') (s.or1oBs) - r/l t1J
"
Jatn:^4t*
lrlel\t,n
)
\^-+^]m
.t a\A-p^\rd.t 2\t "f, *ron/r,yYl
j
\/{
-'
(,t**.;_A-J
P,"$a\,rc.4g),uz^
(f'r,+rtr.)V; = f\,,V,;+ fl
,
---\
N),V,r
--...---..---._
( rn '+ tttl";
.
Q:tr\!::t:L
( z1.ot5+,1.o(z)
E= l^*v\'/r.Kx^''lrKx,l -'-. X"=
x = tx")
'1
= 4"oon/s
K= mwt
--
I' .9?*lotnrr
/ rr)