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Unformatted text preview: Part 1: Short answer questions 1K (40 points — 4 points each) 1. The graph shows the potential energy U of a particle as a function
of its position along the xaxis. Near which point(s) are oscillations possible?
(A) a only 15
(B) b only 10
(C) c only
(D) (1 only 5
(E) e only
(F) b and e U(x) o
(G) a and c 5 i
10 a “La: Liislidllllimigmll 0 0‘5 1 1,5 2 2,5 3 X
2. The motion of an ideal massspring system is described by:
x(t) = A cos (cot + ¢0). Which of the constants A, a), ﬂy, are determined by the initial conditions (initial position and velocity)? (A) A and ¢0, but not a) (B) A and a), but not (/50 (C) A only
(D) a) only (E) ¢o only (F) all three
3. A transverse string wave is traveling in the ——x direction without
changing shape. The velocity of a string segment is proportional to
the of the segment.
(A) mass (B) displacement (C) slope
(D) acceleration (E) radius of curvature 4. The speed of sound in water is 4.3 times the speed of sound in air.
A whistle on land produces a sound wave with a frequency f0. When this sound wave enters the water, its frequency becomes (A) 4~3fo (B) fo (C) — 5. The length of a pipe closed at one end is L. In the standing wave
whose frequency is 7 times the fundamental frequency, what is the .D
closest distance between pressure nodes? (A) %L (B) Ila—L (C) %L (D) %L (E) %L (F) %L .. L
waif) Page 2 of 9 6. A string ﬁxed at both ends vibrates in a standing wave with no
nodes (other than at the ends) at frequency f when the tension is r. To what should the tension be changed so that the string has one t/M C! 1“
additional node (at its midpoint) when it vibrates at the same 3 ~51“ :1 L
frequency f? A m
3 2 1 1
A 4 B 2 C —— D — E — F 
()T ()T ()27 (>31 ()21()4T 7. A sound wave in a thin tube travels in a ﬂuid with bulk modulus Bo and equilibrium density pg toward an abrupt change in medium
at x = 0. The new medium has bulk modulus B1 and equilibrium density ,0]. In which case will there be no reﬂected wave? 20 3 (A) Bo/Po = Bl/pl (B) po = p1, even if Bo i 31
(C) Bo == Bl, even ifpo 36 p1 (D) Bopo = Blpl
@130po = 31% z!
8. An oscillator (mass m, spring constant k) has natural angular
frequency we E \/k / m and a drag force with damping coefﬁcient
1). It is driven by a driving force at angular frequency 60D > (00.
The steadystate motion of the oscillator has an angular frequency
(E) w > can cowhere
(A)a)<a)0 (B)a)=wo (C)a)o<a)<wp (D)a)=a)D 9. The equation of motion of a damped oscillator is solved (in the
complex representation) by x(t) = gem where _0__l = (—5 5—1) + i(12 s‘l). What is the time constant for the exponential decay of the amplitude? 1 1 1 5.12
(A) gs (B) Es (C) Es (D) 5+0 10. A transverse wave on an ideal string (no forces other than tension)
is described by y(x, t) = Alexi” + a”)
where A = (~1 mm) + i(3 mm), 03 = 4000 rad/s, and k = 80 rad/m.
What is the segment velocity at the point x = 0 at time t = 0? (A) +50 m/s (B) +12 m/s (C) +4 m/s (D) —4 m/s (E) —12 m/s S~3.SS Page 3 of 9 Part II: Waves on a piano string (20 points) You do need to show work. The propagation of transverse waves along a piano string is described by the modiﬁed wave
equation 02y _ 82_y _ 64y
'u 812 _ T 6x2 (3x4
where ,u and r are the linear mass density and tension, respectively, and F is an elastic parameter
that is independent of the tension. (a) Is the rinci le of su er osition valid for transverse waves on the piano strin ? Justif our
P P p P g _J’_X___ answer. 2 &2 37.3 33; ‘— rk r
.3: ) 38:}— ako linear, :‘> wag/4‘) Qh‘o 92694:")
x‘ X are soluti\ms) 44mm awful mud” he a solohcm. (b) Can a traveling wave of any arbitrag shape propagate on the string at constant speed Without changing shape? Justify your answer. If the answer is yes, ﬁnd the wave speed in terms of ,u, r,
and F (as needed). gee x? 39QCert> is a solo‘l't‘on. Ram 44% chain 1 30:3
31 37‘ I 7. 373 z A 33 F r:
\ q ’2.
'Tl'us \S no‘i 6k Solo‘l'lovi uni—€55 ﬁg). : MESA. ax“ $XL where WW}: 1"" (7°C W
OR From (c)J Phase veiOu‘b YeS/NO:
ATMS on K, Nave uni/k “dank/4Y3 ha a Can ams EUHCY‘ Iers,z)=
:omgoncw‘i'S er" dUG’péro/xf k‘s
~Hrwtl' in»le oi“ olhcfewozil‘ SPQEAS :37? giaan changes Page 4 of 9 (c) Can a traveling harmonic (sinusoidal) wave y(x, t) = gem“ i w) propagate on the string at constant speed without changing shape? Justify your answer. If the answer is yes, ﬁnd the wave speed in
terms of k, ,u, r, and F (as needed). 2 t a1 i 2.
Take. denuxhuas; 3425; : (inc...) 23 21%; ck) 3 Page 5 of 9 Part III: Longitudinal wave on a spring (20 points) You do need to show work. Consider a spring that obeys Hooke's law with spring constant K, total mass M, and relaxed (unstretched) length L0. The spring is stretched to a length L > L0 and its ends attached to keep it
under tension. We want to derive the wave equation for longitudinal waves on the spring and to ﬁnd the wave speed. Consider 1c, M, L0, and L to be known quantities.
L (a) What is the tension to in the spring when its length is L and there is no wave traveling on it? To: ﬁance exeva In?) Spring m eds!
37— K ‘ 003W inﬂame) Karl») To: (b) The function s(x, I) gives the displacement of any point on the spring from its equilibrium . . . . . . . as x,t
p051tion x due to a longitudinal wave. The tensron at pornt x is then t(x,t) = To + KL fax ) .
What is the net force on a short section of spring whose ends have equilibrium positions x and
x + Ax? AX é—"") Page 6 of 9 (c) What is the mass of the section of spring? a came“ .43: mam +2411 mass AXM A mass = L. ((1) Apply Newton's second law (ZF=ma) to the short section of spring and take an appropriate limit
to derive the wave equation. State the wave speed in terms of K, M, and Lo, and L (as needed). 7.
ammo, asoo _, 91 M aim
KL< ax ax ”" L 6*
W?” M
Tana [BX—>0
K 615 M 315
L = 2’?
3X L. a KLZ 7—,...
“This 1S ~er wave eq. bod/in \l ——— M 31,2; Page 7 of 9 Part IV: Massive ring boundary condition (20 points) You do need to show work. A string is ﬁxed at x = O. The other end (x = L) is connected to a
slip ring that can slide without friction on a pole along the y axis.
If the slip ring were massless, this would represent an ideal free
end, but in this case the ring has mass M i 0. The string has
tension T and mass per unit length n. (a) Derive an equation for the boundary condition at x = L. [Hint
Apply Newton’s second law to the ring] Your equation may
contain y(x,t) and its derivatives, evaluated at x = L, and the
constants L, M, T, and u, as needed. Boundary condition equation: Page 8 of 9 (b) Consider a standing wave y(x,t) = A sin 2326» cos cot. Apply the boundary condition at x = L to A
ﬁnd an equation for the allowed values of A. (Do not t to solve for xi.) Only if you did not get part (a), use this (incorrect) boundary condition equation to do part (b): ( y — x =0. x=L Us ms 44m. meoﬂ‘cc‘l’ B.C1 Equation for allowed xl’s: (c) In the limit M —> 0, what are the allowed values of 1 according to your equation from (b)? Does
it reduce to somethin ou alread knew? Ex lain. which \5 wka’l' we ‘Q’P
4&6 Vlad gee é'xA ‘ Page 9 of 9 ...
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 Spring '07
 GIAMBATTISTA,A

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