Someproblemsmakeuseof the relationshipsamongangular frequency,frequency,andperiod
for simpleharmonic motion: at:Znf, and,f= 1/2. Occasionallythe period is given indirectly by
describinga time intewal. You must then know, for example,that the time the oscillator takesto
in the other
go from maximumdisplacement
in one directionto maximumdisplacement
go
zero
from maximum displacementto
displacementis
direction is T/2 or rhe tirrle it takesto
74. If thesetime intervals or othersare given, you should be able to calculatethe period,
frequency,and angularfrequency.You shouldalso know how to find the maximum speedand
in teimsof the angularfrequencyand amplifide: v^*: Ao anda.*=
maximumacceleration
lar2. Someproblems requireyou to know the relationshipbetweenthe angularfrequencyand the
appropriatephysical propertiesof the oscillating system; 6 = "l k I m for an undampedspringobjectsystem.
For a
.Someproblemscanbe solvedusing the principle ofmechanical energyconse^rvation.
spring-objectsystem,the mechanicalenergyE is givenby: E : Yzmv" + t/zkx" + mgh.
Questionsand Example Problemsfrom Chapter 14
QuestionI
The drawing showsidentical springsthat are attachedto a box in two different ways. Initially,
the springsare unstrained.The box is ihen pul1edto the right and released.In eachcase,the
initial displacernentof the box is the same.At the momentofrelease,which box, of either,
experiencesthe greaternet force due to the spring?Provide a reasonfor yorlr answer.
F{
€-
A
+-"
+-
B
Question2
Supposethat a grandfatherclock (a simple pendulum)is n:nning slowly. That is, the time it takes
to completeeachcycle is longer than it shouidbe. Should one shortenor lengthenthe pendulum
to makethe clock keepthe correcttime? Why?
-
).tl
r7
t4
)+@)ila-afi",lA
4 )9,! Whfur"\
Problem I
The equilibrium length of a csrtain spring with a force constantof k = 250 N/m is 0.20 m' (a)
Whal force is required to stretchthis spring to twice its equilibrium lengfh?(b) Is the force
requiredto compressthe springto half its lengththe sameasin part (a)?Explain.
(")
(b)
X=o.1orr
( = 2.5oNfr,.,
-(x
F -
= -(AEoN/") (o.la^)
,/r"c.- X =-b'ior')
(-o.on): Futq#*T
F =-Ky=-(asoN/*)
ffffi*
Problem2
An air-trackglider attachedto a springoscillatesbetweenthe 10 cm mark andthe 60 cm mark on
the tmck. The glider completes10 oscillationsin 33 s. What arethe (a) period, (b) frequency,(c)
amplitude,and(d) maximumspeedof the glider?
(o) ,ar;rv"r)$"
c-,"+*X.
4i&
p-pa,qd.r.r l-' =- 33i i'
lo oan-plpb"rv r/rr 33s,
3"3s/
..
/6".uoaktn
lo Ytal!)at
lgro
f = 3"3s
(b) f =t/r - ,=+a, -----,)+-o"e.4=]
f -. o "hLlz
(c)
&M
!)eD-n
,"ret^h"
/O."'t * 6ocrr N,aaq\-)
}wNeI a,oUa.bw.<to$ 5o.nr
frb a/rrupdlJo
)-
Problem3
A computerto be usedin a sateilitemustbe ableto withstandaccelerationsof up to 25 timesthe
accelerationdueto gravity,ln a testto seewhetherit meetsthis specification,the computeris
bolted to a framethat is vibratedbackand forth in simpleharmonicmotion at a frequencyof
9.5 Hz. What is the minimum amnlitudeof vibration that must be usedin this test?
0.^n
= x5z =
t5{q.80">le>)
t= q.5fi?
/1
9
A'?
,"t q,r
-
A =
O"eav
e---
/
|
4 r'."'
A
/,,-
L d
= AqS n/s'
f\r
n
,, T ,/
r,_)
111" '
f,
t
/i
A=
(e,asr/s')
4 n'(e-5il")e
- 0.oGi5\ = 6"?n{o-"".\
Problem4
=
A bl:c\
9f Tas^slr_ 0.750kg is fastenedro an instrainedhorizonlalspringwhosesprins
constantis *=
82.0N/m. The btbckis given a displacementof+0.Dd;;h;r;;#.6.
indicatesthat the displacement
is_alonpfthe
+x *i., -Jtrr", *reased from rest.(a) what
is the
fr,*" (T:*tg"-and
direction) that,thespring exertson tir"-ulo"r..;,rrr uefore
the
block
is
released?(b) Find the frequencvofthe reiultig osc'laio;
;;;r"". (c) whar is the maximum
speedof the block?(d) DeterminethemagnituJeof the
maximumacceleration
of the block?
Nr= o.15ol%
(")
K: 8r;o N/rn
g'?9N
F=
x= O . l 0 O r n ,
-/r
= -(?a.oN/.r)(o"rao^)
P=-Kx
ffi
t-
\
tb/ .
\-=-J-
t
1*
<
arr{6
I
An
F = t.66Fl'z
(i) v.^.- arr{n
A = O. tlon,, avlre ),aa.0\L ^rb-"J
- A r,-( r.ooH*) (0"ltor,r)---)
Vrn..r
(d)
probfem5
Qmn*= (arr{;'
IW- 1 rJ 5 "/J
= 4n,(l.wt{z),(a.iloe)
\
r*^t
8*
The shock absorbersin the suspensionsystemofa car
are il suchbad shapethat they have no
13"I n/,
unu"r,"o,o
rn ux"r.sl"lort identical
springs
anached
:fl;"",*ji::,""t:::::lrl:"-p-Tg.
320
kg.
"
person
A
pushes
o"*"
.,
,rr"
*ijai#ilffi;-.i;
ll Pj 1"', Tt: :"nports
#H:::;
''
ard norlcestharit vibr"testhrough5 cycresir 3-0s.
Findthespringconstant
rp;"g.
"r"itrr".
)
i4 Jlrr -a,1o-*,g r.,"r$.uotld ir\"D,€A
-tlr, $-r.rrg."u,*cbo-{ zw{rta}r+q
| = l.t"1ltt
I
_ 5,c*+er:o = ).("'7Hz
17
+/ _- -?aTi I EI..,
r\ = 3aD[?
V-?
^r
S <-A"r-t:. )rq 3.os, *g*4
3,a*
----)
{
?,r.fl' - {E
'/ n'{' = V,
K = 4 rrtf
tr.n
K = 4 rrl(t.r,?Hz)*(3roEs)
K
= 3.5 ^
iDeN/m
Yrn
Problem 6.
A 200 g massattachedto a horizontal spring oscillatesat a ftequencyof 2.0 Hz. At one lnstant,
the massis at x: 5.0 cm and hasspeedv = -30 crn/s.Determine:(a) The period. (b) The
amplitude. (c) The maximum speed.(d) The total energy'
g"aDoFg.
p=
f = 2.ol{*
x= _5.O.1
.
V
(U)
= -b6,/9
I-
o.5bs
Tr
E=
I
,
TaKna- t/smv'+ )/aJ(xr
\
'/s,K fta Yl rn,v\* 7aKy
A*: T/'Kv\+Y\*
h6-
A=
? r"*
- 3 t,6N/"\
l( = (rrrf )*rq -- ?n\Ca.oHr1a(o.rooFq)
rt
.( c)
.V.,-.= A.r,FA
({)
probremz
= a'"-n
( a-olte)[o 'os5nr)=l o'bq '/g f
F
tl/xKAo'=
r/ar,,v,il, = 7e( o"loarg;1o"61"'l)}
An archerpul1sthe bowstring back for a distanceof0.470 m before releasingthe arrow. The
bow and string act like a springwhose spring constantis 425 N/m. (a) What is the-ela:ti9 r=
;ffi i-";;ffid;60;;;;*;;i,6:!-@_
;ilJr;:x#'1tri':HT"l'1"6H;ffi
tle
bow?
it
leaves
traveline when
k^-,^l:-^
--'L--
,;+ l6'.ao
+ha
X=O.l?o"n
K.
h^ull
L-
(q)
qls N/bo
gr,
t/a[.x^
= ,1a-(rttslv/.)(o-'rfo"')q
fr@
( u) 4'*.
nz,wr.C,r,',rrh-ari +d tnre.grot' ,t.c ll'l
t/.
/7.
a r. ' - l +
pt]*tft'd
i
,zr'ic"o;-,
zjs=,2.*i;*'lr(xl* *z;,
/a.vi = t/,vxf ----- uo=ffi n,'
( o.rta ",)
er:N/c
.D3oo
O
fg
V+= ss"z r/s
\--"
Problem 8. A 1.00x 10-'kg block is resting on a horizontal frictionless surfaceand is attachedto a
horizontal spring whosespring constantis 124N/m. The block is shovedparallel to the spring
axisandis given an initial speedof 8.00m/s,while the springis initially unsfained.Whatis the
amplitude of the resulting simple harmonicmotion?
eP?
lA = I"m* lo-
-)
)( = lall N/no
u"rrl"f f :Drfl1
+""
q J du,\/-"bL A'V*-A ,pilllAr'A
]/n sHM) E w:
VEE.aDfrt/5
YaKAe
I/tll/-^ -- /a l(x'* 7a'qv*
A=?
"o
,/aKAa= /a.mvu
f:frV*----> ft: @"
A -ffi
Probrem
n
(t'oo"'/e)
A= ?.t8* ID
arr\
A 0.40 kg massis attachedto a spring with a force constantof26 N/m and releasedfrom rest a
distanceof 3.2 cm from the equilibrium position of the spring.What is the speedof the mass
when it is halh,vayto the equilibriumposition?
rq = O.lOl{g
-1,",o..r,t-"rar^t,raltrct)
D
K = abN/r",
-i)
r/
X; = 3-?'xlu *'''t
V.' = O.t/5
X5 = 1.6*lo "t
V.T
.?
!t,t
"{
d\
) / 3r . r V a * T aK X t t m g y {
r"-.r.}*"/"t-l
t
'/r
tto
'/^V
*
x.
%.vi" hKxf
b Vt' = K (r;'- xi )
?
V+=,'JlEro tx.?-xi)
\ /v f
:
a'
g'Nt!
o. qoYz
Vt " o"ea'*fg
N V"
\\
MV,I
+ [ KY,.+mfly;
\\
6
Problem lll
Astronautson a distantplanet set up a sinple pendulum of length 1.2m. The pendulumexecutes
simple harmonicmotion andmakes 100 cornpletevibrations in 280 s. What is the acceleration
due to gravity?
= o-35"7t+z
f = 1o;o^"lzr. /mOt
^"f=E
S=-*E,
4n'l^=Z
o.sS"Fl-z)t(t'a.)
*= 4 TI +AL
Problem ll
If tle period ofa sinple pendulumis to be 2.0 s, what shouldbe its length?
r'-a
f
= !_Ds
+ &-raal,,^ryt'$"!"'"t^'I/^^"
f= #l+
, ,,IL
L--?
-r-
X
t-
,.
/)
a -T)
L--
4'n^
/ ^ * O- Q
wo
1-
/ \
1 r"" 1t21
|
| Y\1
-J
, --
L
(1.abu/s")(a.-o1)*
'----------/ ) *rt
\
'7