Questionsand ExampleProblemsfrom Chapter 16
Question 1'
Figurea below gives a snapsholofa wavetraveling in the directionof positive x along a string
undertension.Four string elementsareindicatedby the letteredpoints.For eachofthose
elements,determinewhether,at the instantof the snapshot,the elementis moving upwardor
downwardor is momentarilyat rest.(Hint Lnaginethe wave as it movesthroughthe four string
elements.)
Figureb below givesthe displacementofa string elementlocatedat, say,x = 0 asa fimction of
time. At the letteredtimes, is the elementmoving upwardor downwardor is it momentarilyat
rest?
a) &g
b) ry
c ) dpu^\
r) )a^xr\
ptA-.
")
yr\t
4s
s\)g
\!-'
U) *
Question2
In the figure below, five points areindicatedon a snapshotof a sinusoidalwave.Wlat is the
phasedifferencebetweenpoint 1 and(a) point 2, (b) point 3, (c) point 4, and(d) point 5? Answer
in radiansand in termsof the wavelengthof the wave.The snapshotshowsa point of zero
displacementat x : 0. In termsof the period T of tle wave,whenwill (e) a peakand(f) the next
pointof zerodisplacement
reachx : 0?
I,/d
A)
6'a
,b
b) rr o,u
3rl* ar
")
r)
e'l
?4
6t
).T
-/u
M
I
c\
'rJ
e-
Question3
The following four wavesare sentalong stringswith the samelinear densities(x is in metersand
t is in seconds).Rankthe wavesaccordingto (a) their wave speedand (b) the tensionsin the
stringsalongwhich they travel, great€stfust:
(1) yr : (3 mm) sin(x- 3t)
(2) yz = (6 mm) sin(2x- t)
a)
ly ?1?7 3
b)
ll{,lr3
(3) y3: (1 mm) sin(4x-t)
(4) ya: (2 mm) sin(x - 2t)
V= \t =*/t<
Y = )-a"(Kx--t)
VqG
n=7
Question4
If you setup the seventhharmonicon a string, (a) how manynodesarepresent,and (b) is therea
node,antinode,or someinternediatestateat the midpoint?If you next setup the sixth harmonic,
(c) is its resonantwavelengthlonger or shorterthan that for the seventhhannonic,and (d) is the
resonantfrequencyhighgl 61l6v/61!
(')
(b)
^\
.)LOY\4pL
---==^/.
7 ,toa"a
<==:-'
!.r
/D,'IY\/W a?V
"==:=-
(r)
Xa^+t
---:=:
A
I
ProblemI
The equationofa transversewavetraveling along a very long string is
y : 6.0 sin(O.020rx + 4.07d),ryherex andy areexpressedin centimetersandt is in seconds.
Determine(a) the amplitude,(b) the wavelength,(c) the fiequencS(d) the speed,(e) the
directionofpropagationofthe wave,and(f) the maximumtansversespeedofa particle in the
string, (g) What is the transversedisplacementat x = 3.5 cm whent : 0.26 s?
\/ = (a"o."\ 3,, f(o.o>o,, "n,') x + (+"orr"[/)t]
4)
r)
(l)
ri = 6.0c.,
K=t./x -+ \-- xn/t<- *%r.oroo."-)
--t =-/^n = U9::t!)
?.lr
-
)=loo"-
d)
e) _X
r) q -- ey/at'=
(-";*
wl.
LL'cAa^*
.-a:(Kx*.ll)
Kx *ot)
u))vr =-.fui.o.,..j1rf
t".o",9
----zQ"r".x=
: "75 .o v/ 5
rtill
,e-p
2
2).
l$9P,1.1 =it
W
(a) Write an equationdescribinga sinusoidalFansveisewavetraveliirg6n a cord in the +x
directionwith a wavelengthof 10 crq a frequencyof 400 Hz" and an amplitudeof 2.0 cm.
(b) What is the maximumspeedof a point on the cord?(c) What is the speedof the wave?
(b)
q= Al/st = - w)., r.oz(Kx-*t)
(.) V=\S=Y7''/t<
=
*
= (x5:^A1s),*,1,
An^o = bY"l
=F;t"il
2,5t3 noAls
6A"9 "q-l
V:
4.O * 16rm/:
Problem3
A sinusoidalwave of frequency500 FIzhasa speedof350 rnls. (a) How far apartmetwo points
that differ it phasebynl3 rad?@) What is the phasedifferencebetweentwo displacementsat a
certainpoint at times 1.00ms apart?
v--\f -+ Y=Y?=
a -booftf
I
7 Soqls
500ftt
v-- Sson/s
to)
o,w c:w+,ef1 pf-ae ta t4t'ttnvl*'lt k
b
Po4ru+a\Ai
'T--
\ro)
l/| = /+=
a"Tb"-^;r"
/u aX q" 'cr/o[
I *,
/,\
u, ^^a^;"*
pd;Jro t"
t/
/5ool1Z = O.Dolos.
'
t/1, (o.too*t)
tf
-- t^- t , t rr r a tr \
I-
a_pzal
J,-oD*s
+ Pt .rnl'b'."i l.oons'pA.a-q, PvPe";A ' 'J trEo-td.'r
-\
fr 'lx 4 ,v^Lawc(2,
,q:""
dltWe",rc
.= 7^(.r_*) =
T ^^)^;.,,)
Problem 4
A stretchedstring hasa massper unit length of 5.00 g/cm anda tensionof 10.0N. A sinusoidal
wave on this stringhasan arnplitudeof0.12 mm anda frequencyof 100Hz andis traveling in
the negativedirection of an x-axis. Write anequationfor this wavein the form:
y(x,t) : y' sin(kx t ot).
)
(xr|') --
/.
).0=
S
( K x +*t)
4-L u'. ;,1rq tgro x- 'du-;h-"t
O"J?,m\ - l.t*lo-e",
4r = A.rrt = 3.n-(toolf z) :
K = *n/>
G\Sn^Afs
--) rue
,ti gl* l(
"u""I \
\t=.,=[4--->=/*E
K= \-n/s=tW
-) K= (roouz)( D . s o l b / n )
lo.oN
yt(x,t) = ( t"a*,6-'.)A""
+ (6rr^^i/,)tf
L(ruo;,,)x
.= t,,n -l
I 1V {t\
Problem5
The equatioaof a tansv€rsewave on a string is
y = (2.0mm) sin [(20 m r)x - 1600s ')q
The tensionin the string is 15N. (a) What is the wave speed?(b) Find the linear densityof the
sfring in gams per meter.
:;.4"tI|*\
))rou<
t;=
.
I
r\
) txrt/
=
,rrX_-L)
)rr -a-'L\1 1
(o) v =*/K = 't:## -.._ V = 3 6 n / s
::::
tD = 6oo"^l1s
1
u.
(b)
f-v = ,lO^ --
v'. 2/l+_- A=
z/u^
---) A = l.to7t to-\yx/71
Jl = l3!-4aa'/s )*
r----------:--:--t
-l-{ :
lte.1 g/^
Problem 6
Two identicaltaveling waves,moving in the samedirection,areout of phaseby n/2 rad. What
is ths emplitudeof thJresultantwave in termsof fie-common amplitudea; oftletwo
combiningwaves?
Y
=
)* -a""( Kx--*) t yn .a/r\(fx- *t + d)
-= Ay" ..ro('/r
d).Ar" ( Kx-.,t *'/rd)
Q=n/'
y = Ay- or. ('/r)4/(Kx--*
*n/q)
r-,ac ( t'./u )
dY''\
/-.J
/-t
\
["4ty^
'
\ .,'
t---
Problem7
A string that is stretchedbetweenfixed supportsseparatedby 75.0 cm hasresonantfrequencies
of 420 and315Hz with no intemrediateresonantfrequencies.What me (a) the lowestresonant
frequencyand (b) the wave speed?
a.' '=nV
\Jn
"
-"a wl'e*41n2
--4 @
, t
di ffi!
UD
(^)
$ ,= 1osttt
X=
-75
l$ocr.r'l.5orn
(u) v= )l-ll*")
- r",
(r"s,+r)
-
Problem8
l/st.s nfs
What arethe threelowestfrequenciesfor standingwaveson a wirel0l0 m long havinga massof
I 00 g, which is stretchedundera tensionof 250N?
i'"
n
/-
P4.,ru-Itk lfrqytlr"4i"
\t= 19 - l-z = IZL
" {,,(t l{.t.1 J r')
(?t5oN)(lo.or",
-
a\\/,/
Y= " /Ut-
a^s Wnor\ ),ra
/'
y.
/,-^
v')=l)l,t)"-.
/ \
n (/5)F1/si =
t ( to.o.)
n (f.9Hz)
I > t 6l <
C o " l o o h)
n = ?, f= is.grfz
Problem 9
In an experimenton standingwaves,a sting 90 cm long is attachedto the prongof an
electricallydriven tuning fork that oscillatesperpendicularto the lengthof the shing at a
frequencyof 60 Hz. The massof the string is 0.044kg. What tensionmustthe string be under
(weightsareattachedto the other end)if it is to oscillatein four loops?
o,?Sn
\=q5.-,=
{= Go,}z
v - ) f --(o.,ts,)(a"pr) : \'tnfs
4o"u
r-1
I.>
l->
,,1 A
I
IZL
l_-
r/
-L __ n v/t
rl
v^ = 77n
(o.otqp{(}1nf s)^
( o"ro,.,)
? = 35,{, r\l