the other directionis 72 or the time it takesto go from ma,rimurndisplacementto zero
displaoemertis 74. Iflhese time intervalsor otltersaxegiven,you shouldbe ableto calculate
the period, flequency,and angulalftequency.You shouldalsolnow how to find the maximum
acceletationin tems of the angularfiequencyandatr,plrtudel.'r*= A.0 Dr'd
speedand_maximum
a.^ : Aal , Someptoblemsrequireyou to know the relationshipbetweenthe aagularfiequency
andthe approprialephysicalpropertiesofthe oscillating system:t =
t ^ for an undamped
"li
spring-objectsystem.
Someproblemscanbe solvedusingthe principle ofmechanicalenergyconse^rvation.
For a
=
+
+
system,
spring-object
the mechanioal
energyI is^given
by:I
% zt'
% ld
mgh.
Conservationofmechaaicalenergyyields '/zmvz' + % kxz' * mgh2= v, mvl + V2A,x/ + mghr.
Usually you will be given all quantitiesbut one.The equationcanthen be solvedthe unknown
quantlty.
Questionsand ExampleProbl€msfrom Chapter l0
Question1
The drawingshowsidentical springsthat are attachedto a box in two differenl ways,Inilially,
the springsareunshained.The box is thenpulled to the right andreleased.Irr eachcase,the
initial displacement
ofthe box is thesame.At themomentofrelease,whichbox, ofeither,
experiencesthe greaternet forcedueto the spring?Providea reaSonfor your answer.
<.---{
<------{
<-
<-
A
n.J Srrl."
2
Question
-M
R
l9s Aafto
Supposethat a grandfatherclock (a simplependulum)is runningslowly. That is, the time it takes
to completeeachcycle is longerthan it shouldbe. Shouldoneshortenor lengthenthe pendulum
to makethe clook keepthe oonecttime? Why?
T=>."ffi
jt" .!,^#.$
/3o F'otatl't*v'1
Problcm 1
The equilibrium lengthofa certainspdngwilh a force constantofk = 250 N/m is 0.20 m. (a)
What force is requiredto strelchthis springto twice its equilibrium length?(b) Is the force
requiredto compresstle springto half its length the sameasin part (a)?Explain.
(o)
X = A.1Af9^4t ^qn +,'d'r\ )r,v.&^tfu^A
,{ = o.3.o".r
K = A5ox/'.,
(b)
^vo* x--o"ron
I
a1 ))r,ttn\pv-!,rl,d
-!a.4A
F=-Kx= - ( rsor.r/^) (o.a-o,n)
r s.Oi lo' N
.Pp4'e
"r;k4
"6 l;Pa-"*"+)
Problem2
A personwho er'eighs670 N stepsonto a spring scalein the batbroom,andthe spring compresses
by 0.79 cm, (a) What is the spring coustad?(b) Wlat is the weight of anotherpersonwho
the springby 0.34 cm?
cornpresses
Kv
I
ft
=::1
f-v ,-o.tt..
1
G) Xf,=.oa,'o
Kx- mg'o -J Kx=n,?
L
,,
r\g
K, ^q/x = (61aN)
(o-,l1,to_t $ )
f----_---_-----..1
, (b)
B:
i K= z"s^lDeN/r.r
tI "
I
I,t n
Kx=m+
u.l=ng=Kx
x^"'rd
Ndov
--> t! -- (8.5'to{ x/.) (o.3,tr I o-am)
o = N83N
ProbleD3
ofup to 25 timesthe
A computerto beusedin a satellitemustbeableto withstandaccelerations
tlte computeris
acceleration
dueto gravity,In a testto seewhetherit meetsthis specification,
fialne
vibrated
harmonic
motion
at
boltedto a
thatis
backandforthin simple
a fiequencyof
9.5Hz. Whatis theminimumamplitudeofvibrationthatmustbeusedin tlis test?
,,5 g . E 5(e.ro *1r) = ),'t5 *fse
9.5 Hz
(
0^-r.= A*o
0^,*= A(a.{)"
A= 0*
4 'tr^$^
w= 2.it"f
= ?"'"{'A
4 'x t 71,silz)\
A= O'o(o? cq = 6.9 * lo-" r.j
Problem4
A block of massrz : 0.750kg is fastenedto an unstrainedhorizontalspdngwhosespring
coNtant is t = 82.0N/m. The block is given a displacementof+0.120 fl! wherethe + sign
indicatesthat tlle displacementis a.longthe +x axis, andthenreleasedfrom rest. (a) What is the
foree (rnagnitudeand direction)that the spring exertson the blockjust beforethe block is
released?(b) Find the angularftequencyor ofthe resultingoscillatorymotion. (c) WlB1 is t}le
maximumspeedofthe block? (d) Detemine the magnitudeofthe maximumaccelerationofthe
(o)
fo: O.?50KX
t./ -
F=-Q"zqN "n
3a.on7-
n
F= -Kx = -(ra,olrlm)(o"raa,)
x--O . l r . O r q
H
-}z}e"
W
F= ?,s,{N
( 8t,. N/o)
(o.tso91
to.5""lfs
o^.fr,tr,/4 ft =o.rao,.,-> V""-"= tJ)A : (rc.sA/s) (o.rao^)
V,,,""= l,).b"tj I
(J) o^"- = ur"A
Problem5
= ( ro.s"J/,)) (o.roo^)-__+O^*" = 13.4./s*
Theshockabsorbers
in thesuspension
systemofa cararein suchbadshapethattheyhaveno
effectonthebehaviorofthe springsattached
to ihe axles.Eachofthe identicalspringsattached
to theftont axlesupports320kg.A personpushesdownon themiddleof thefrontendofthe car
afldnotic€sthat.itvibratesthrcugh5 cyclesin 3.0s.Findthespdngconstafltofeithel spring.
4 )J.a)"y,^g
?
jtu IfW-.*l
-,w
/rt,D,nh 5 @r.!.e4 3, os , )9*t
u
$ rv<'U^.alIA
t'
f= l.G'1tlt
m-
^t-p/-k/
tr-r-- A" {
3).o (q
r;-'
'lo.
t<=?
rrtT
V" -)
K = 4 n^ ( :aor, ) ( t.,,?tt=)*--'
t -#.a"4
LI-
x"f=F)1= 4-'.f"
K = 3.s " to?N/,.,
Protrlem6
In preparationfor shootinga ball in a pinball machine,a spring ft = 675N/m) is compressedby
0.0650m relativeto its unstrainedlength.The ball (m - 0.0585kg) is at rcst againstthe spdng at
point A. wlen the spring is released,tlte ball slides(without rolling) to point B, which is 0.300m
higherthanpoint A. How fast is the ball moving at point B.
A
'/,^vio mg\ = /" px.a-
vi. Xz\+= K/rx)
(or s r/.) ( o.ooso.)1I (+,rc"1s')
go.zo+.)
(o,osrsft)
V9= 6'ss./s
| r
Problen7
Ar archerpullsthebowstringbackfor a disknceof 0,470m beforereleasingtheanow.The
bowandsaingactlike a spdngwhosespringconstanlis 425N/m. (a) Whatis theelastic
potentialenergyofthe draranbow?(b) Theanowhasa massof0.0300kg. How fastis it
traveling whenit leavesthe bow?
X = O,q?o r-r
(o)
6 = ta.s N/m
j/rKx*
'/"(txsN/.) (o.rro.)a
4(,.1f
(b)
s\\bnA i
4"-. ^-,at'r"-.b:a 44r"",r,"4,^);'!
,.4
(Er+lE1= l<E"+lp"
4 o
= l/errV"*o ro2h" + YrK/-:
.rr'E *"
( o.+ro.) ---. ve= 55-9 ^/s
O , o 3 o al g
'
Problem 8
kg block is restingon a horizontalfiictionless surfaceandis attachedto a
A 1.00x 10-'?
horizonlal springwhosesprirg constantis 124N/m. The block is shovedparallel to the spring
axis and is given an initial speedof 8.00rrls, while the springis initially unshained.What is the
arnplitudeof the resultingsimpleharmonicmotion?
rc)= l.oo' l0-1K$
> 4-.
K = tt.eN/m
,r,./,^1 X
o -!,a4^X/^r4!/\P^;^,2 Y,iAatv"Y
= z' KA'
/!\ SHM)
= Orn V = g.oo-fs
tt ., ^1
A=3
/eKH
/r KA =
^a
A=
\t v ..) * rr
.i
= /J
AX ,/}nv
\\
o
r/
a.
/I c.l v
m,i
/K v
l
---J
A = [t"..r*r,t
1 ta, NA
Problem9
ftom resta
A 0.40kg massis attached
to a splingwith a forceconstantof26 N/m andrel€ased
position
of thespring.Whatis thespeedof themass
distanceof 3.2 cmfrom theequilibdum
whenit is halfuayto theequilibriumposition?
en = o.qDK?
( = O.te
Nlrr.
t " = 3 - ! r "1 t ' l n ,
V" = o -/s
l.br lo-2ro
X1=
tt
-
?
-\"
'-nn
", ,t-,vtt
o$ O",-o-".a|'.
o
o
lI
/l
\
m V ". , - f n q ' l J
r/,,.1
+ /a l\x.
J r o 9 D o + i d .A X D
/a.rnvi*TeKxi = 7.Kxo'
mvrr= K(*,'-xi)
r.. ------ ---:-
vr=/tr (xj-ra/
v+ = (39f@
O.qDK?
.-
| [a.r'to".)'- ( t.r"*to'-)^ l.
-\J
Problem 10
Ashonautson a distanlplanet setup a simplependulumof length I .2 m. The pendulm e)<ecutes
simplehaxmonicmotion afldmakes 100completevibmtionsin 280 s. What is the accelention
dueto gravity?
/=l.l--- o,?s"tHt
c) = lDo iu"r^"1
,
/ eXos
11
>,"$17
t=* lt
4"'S'= e/,
t'
2= 4n't
= 4 n'(t.a^) (o'as74*;*
g -- 6.o9^/s'
v/
Problem11
lfthe periodofa simplependulum
is to be2.0s,whatshouldbeits length?
| = a.os
uo = 2'r + =
) t^ , t-t*,'6fu9t"Ar!" ,
/'l
)_.-_
/
t/
I = /t
-'Ltl
| -3'lL
-- :;..-tll L /
|/q
^ -.1-l
,
/=Ltr
(?
"""2
_-
?n^
/=
/
--)
,-r }.
|
/,
= 7T
lt+l
= (1.so '/s")(a.o')'
? n'
O.99 r"r
), / L1 \
E