This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: monster gomkws
Physics 214 PRELIM I (March 8, 2005 from 7:309:00 )
Name :
Signature:
Section # and TA: Please count your pages — they should be 13! Score: ° This examination is closed books and closed notes. NO calculators are permitted. A formulae sheet is provided. Use the provided space after each
question for your answers. The ﬁrst part consists of a few simple questions, and
no partial credit will be given. Part II, III, IV are problems for which you have to
show the work done arriving at the answer. Partial credit will be given for these answers. The perfect score of the exam is 100. The point value of the problem is
given in the parenthesis next to the problem.
° Please put your name on each sheet. ' By signing this exam you certify to adhere to the Cornell academic integrity code. Part 1: Short answer questions (35 points —— 5 points each) You do not need to show work. No partial credit. 32 1 a2
FYOCJFV—ZFYUJ) (A) YOU) = f(XVt)g(X+Vt) B; D
(B) y(x,t) = A [sin(xvt)]3, where A is a constant (C) Y(X,t) = ﬁx) g(XVT)
(D) y(x,t) = A elkom), where A is a complex number and k is a real number
(E) none of the above 1. Which wave is a solution of the wave equation 2. A string has the wavevelocity v and displacement y(x,t). What is the acceleration of a
segment on the string? 2 072
(A) V 33435,» E (E) none of the above. 3. An electric pulse with electric ﬁeld E = (0,0,f(xct)) is propagating with wavespeed c
in vacuum. In which direction is the pulse propagating? (A) +x direction (B) —x direction
(C) +y direction
(D) —y direction
(E) +z direction
(F) —2 direction 4. The general solution of the wave equation is y(x,t) = f(x+vt) + g(xvt) on a string.
What is the relationship between the function f and g given the initial condition that the string is ﬂat at t=0?
C
(A) f(X,t) = V g(X,t)
(B) f(X,t) = g(X,t)
(C) f(X,t) = ' g(X,t)
(D) f(x,t) = l/g(x,t)
(E) none of the above 5. The general solution of the wave equation is y(x,t) = f(x+vt) + g(xvt) on a string.
What is the relationship between the function f and g given the initial condition that the segment velocity is zero at t=0? E (A) f(Kat) = V g(X,t)
(B) ﬁxat) = g(X,t)
(C) f(XJ) =  g(xat)
(D) f(x,t) = 1/g(x,t)
(E) none of the above 6. A pulse of the form y(x,t) = f(xvt) is propagating towards a ﬁxed end. What is the
from of the pulse when it has been fully reﬂected from the ﬁxed end? (A) Inverted and mirrored. (B) Mirrored. (C) Inverted. (D) Same (E) none of the above 7. A pulse of the form y(x,t) = f(x+vt) is propagating towards a free end. What is the from
of the pulse when it has been fully reﬂected from the free end? (A) Inverted and mirrored. (B) Mirrored. (C) Inverted. (D) Same (E) none of the above Part II: Reﬂection of an Electromagnetic Pulse (20 points) You do need to show work. An electromagnetic pulse (plane wave) is traveling with speed c in the negative x
direction towards an inﬁnite conducting plate, which is placed in the yzplane at x=0. The
electric ﬁeld vector E(x,t) = (O,Ey(x,t),0) is polarized in the ydirection. At t=0 the
electric ﬁeld of the pulse has the shape shown below. conductor in E0 Since the conductor represents a boundary, our pulse will be reﬂected at x=0. It helps to imagine a right moving pulse coming from the left. We use the following convention
when sketching: ‘ left traveling pulse __________
' right travelingpulse ......................
’ superposition of the pulses (A) (4 points)
The boundary condition associated with a perfect conductor is that Ey(0,t) = 0 for all
times. Given this, sketch on the graph above the electric ﬁeld associated with the imagined reﬂected pulse for time t = 0. (Remember to use the conventions above for
left/right moving pulses. rid (B) (6 points)
Using the convention from the previous page, sketch the incoming and reﬂected electric ﬁeld pulses at time t= 4d/c on the upper graph, and sketch the net electric ﬁeld in the
physical region (x>0) at the same time t = 4d/c on the lower graph. % Ey (Wild/c)
§ single pulses
RUM PM +2
’ R: e .1
/\}\30 6h+ Ska/9 l"
/ \\ (cﬁf'n‘ch Mishke ‘l)
(K / x
/ \
i/ g 4 \ Kid
5 —4 —3 i2 —1 1 2 3 4 5 6
t , ">
‘ Ey (tz4d/c) superposition
_ E0
+3— qalaleol (ofrzcﬂj
1 2 ‘s ’4. 5 a (C) (4 points) ~
The sketch below shows the associated magnetic ﬁeld pulse B(x,t) = (0,0,Bz(x,t))
traveling to the left at t = 0. 4} 132 (1:0)
conductor » 1230‘ {a
w W M... MLMWW m w w WW. MW: Wééwvwwmwcgm N. x ""“i m»... m m M mm...
—5 »‘‘«l —3 I —1 —l A l 2 \ x 3 4 I I
' \ I
I ’ ‘ K i
" » 2 s. ’
s ' , f \ I
,, I I
r . > \ \ x ’ f
. I L. r \ ’
v ‘ "Ev C +2 nedqith FK’SQ A A
3‘) X (3) r X “H "63‘” flitc
$0 §: __1\ Tl ru‘ﬂl’lt shq‘oc On the axes above, sketch the (imagined) magnetic reﬂected pulse that is moving from
left to right, at t=0. Hint: Pay attention to the relation between the direction of the electric ﬁeld vector, the
magnetic ﬁeld vector, and the direction of propagation. (D) (6 points)
Using the same conventions as above for the electric ﬁeld, sketch the incoming and reﬂected ﬁeld pulses at time t = 4d/c on the upper graph, and sketch the net magnetic
ﬁeld in the physical region (x>0) at the same time t = 4d/c on the lower graph. ‘2 Bl (tz4d/C)
single pulses Eo/C xid
. . l 1‘ t . F
—n —5 4 —3 —.\ 1 i /o 2 1 4 5 6
9 \ /‘
o x g z“
o o
‘l
V —E(}/C +9 nb’ﬁ 63’7“;
+9~ r901» shape 3.2: ([1:4(l/c) superposition
C l (and Umj Hro’ ibnm £)) +01 «Mug Cometl. ’J Kid Part III: Lab Experiment (20 points) You do need to show work. ‘31
I
frictionless rod
1 / :ﬂf/
/ QTfTTfW"W"‘”—'MTQTZTTTV{ ”WWW ‘‘‘‘‘ 7"me shaker
(freq l) (length L) ,/ adjustable _
massless ring In your ﬁrst laboratory experiment you studied a Vibrating string with two ﬁxed ends.
Consider now the modiﬁcation of the experiment in which the right hand end 'of the string is attached to a small massless ring which slides frictionless up and down a
horizontal post. The post is attached through additional strings to a pulley on the other
side of which a spring scale provides constant tension (see Figure above). L is the length of the string between the rod and the shaker. The frequency of the shaker
is f, the tension is F and the u is the constant mass per unit length of the string. The shaker vibrates in and out of the page. Assume that the left end is ﬁxed at the shaker,
while the horizontal motion on the other end is free. (A) Fundamental (7 points) Sketch the shape of the standing wave corresponding to the lowest (fundamental) mode.
Find an expression for the lowest (fundamental) frequency f] in terms of L, F, and u. 1;; V (B) Higher Modes at Fixed Length (7 points)
Sketch the shape of the standing wave corresponding to the second mode. What should the frequency f2 of the shaker be for this mode to be formed? (Express f2 in terms of L, F,
and u.) (B) Higher Modes at Fixed Frequency (6 points)
Now suppose you vary the length of the string. Find the new string length l at which the 2"d standing wave mode will be formed if the frequency of the shaker is ﬁxed to f] and
the tension F remains constant. Express 1 in terms of the original length L. ’5 _E___ \fE
'QZHLM— Lu.” Part VI: Wave Equation(25 points) You do need to Show work. A thin string of linear mass density {J and length L is ﬁxed to the ceiling of a room. Due to its own weight it is under a tension F(x). Choose the origin of the coordinate system at
the ceiling . (A) What will the wavespeed be along the string? (4 points) (a) Larger towards the bottom?
(b) Smaller to towards the bottom? (c) Same?
Fiﬁ Af:\j AA FOX) frat/€57! ed” 7(0/6 10 (B) Calculate the tension F(x) (5 points). (El/x) : Al;(L”/X) ¢=O of (cl/I314? +¢ cllré'c‘llrou ‘lwqrcl {XMP (C) Show graphically the force balance for a segment of length Ax ( 4 points). ll (D) Derive the total transverse force of a curved segment of length Ax for a tension F(x). Hint: remember the derivation in lecture for a constant tension (4
points). : ’ M4212
2F? {’(4 .)\ 0‘44 l4+b¢ , «9‘?
F“) 3: . 4. (E) calculate the acceleration of the string segment ( 4 points). 35:”: W14
f «2
g. a
\f— :xMA’K
4.1:? at?
EF/‘y, 8277 AAA/X 0152 12 (F) Use the results from above to calculate the wave equation ( 4 points). ’2 22,2 \ Z
.AIAV gt THE END 13 ...
View
Full Document
 Spring '07
 GIAMBATTISTA,A

Click to edit the document details