University of Rochester
Department of Physics and Astronomy
PHY 123, Spring 2012
1st Midterm Exam
February 22, 2012
Short questions
Answer the following questions in one sentence:
a) What limits the amplitude of motion of a real vibrating system that is driven at one of
its resonant frequencies?
Damping (and nonlinear effects) in the oscillation turns energy of vibration into internal
energy. (This is discussed in Giancoli, Chapter 14-8.)
b) Under which condition does a wave traveling in a medium invert upon reflection?
A wave inverts when it reflects off a medium in which the wave speed is smaller. (This is
discussed in Giancoli, Chapter 15-7 and Chapter 34-5.)
c) Is the following situation possible? An electromagnetic wave travels through empty
space with electric and magnetic fields described by
⇥
⇤
6
15
E = 9 ⇥ 103 cos (9
⇥
10
)
x
(3
⇥
10
)
t
⇥
⇤
B = 3 ⇥ 10 5 cos (9 ⇥ 106 ) x (3 ⇥ 1015 ) t
where all numerical values and variables are in SI units. explain why the equation
describe
!
The amplitudes of the electric and magnetic fields are in the correct ratio (!! = 𝑐), but
!
!
!
> 𝑐 , thus it is impossible.
Problem 1
A 200 g block connected to a light spring for which the force constant is 5.00 N/m is free
to oscillate on a frictionless, horizontal surface. The block is displaced 5.00 cm from the
equilibrium and released from rest. In SI units,
a) find the period of its motion,
b) express the position, velocity, and acceleration as functions of time,
c) determine the amplitude, maximum speed, and acceleration of the block.
d) Assuming that the block is released from the same initial position (5.00 cm) but with
an initial velocity of v (t = 0) = − 25 cm/s, answer questions in (a) and (b).
1
Solution
(See below.)
Problem 2
A harmonic wave is described by the wave function
D(x, t) = 0.10sin(x − 50t)
where x and D are in meters and t is in second. The mass per unit length of this string is
8.0 g/m. Determine
a) the speed of the wave,
b) the wavelength,
c) the frequency,
d) and the average power transmitted by the wave.
Solution
ω 50.0
=
= 50.0 m/s
k
1
2π 2π
b) λ =
=
= 2π m
k
1
ω 50.0
c) f =
=
Hz
2π
2π
1
1
1
d) P = µω 2 A 2 v = (8.0 × 10 −3 )(50)2 ( )2 50 = 5 W
2
2
10
a) v =
Problem 3
Two identical piano strings of length L are each tuned exactly to the same frequency. The
tension in one of the strings is then increased by 1%. If they are now struck together,
what is the beat frequency between the fundamentals of the two strings?
Solution
f2
(v2 /2L)
=
=
f1
(v1 /2L)
f2
f1 = f1
⇣p
r
T2
;
T1
1.01
f2
f1 =
⌘
1
2
r
T2
f1
T1
f1 = f1
r
T2
T1
!
1
u~
0,2
;m:::
k -::-S.O
N/fYY\
•
~(o ') :::;v.
S
0
A
x:
(WI
+
k
Xe::O
.(YYi
<-\J(o)
=
;<
0
+-
u}-;;<
=
0
W :=-
r
::n
'\=: .2 IT
W
r
51
.2-'\\
~ \T
=
S
r
~ A
;«tJ
'N
0V
L
x
09S
tI1
(o)
)0'(0)
)
K(O)
1
\J (OJ
=
lL-
(cut
d ~
T-
&--
-u
¢J
-r' ~
by
d..R-
+~ \~ \+t 'Q..-!
(I)---:£' flo-,,- s
XJ
~ 0'0
== o.'O~
=.
(YVI
<9
t< Lt') ::;
A
CDS (
w
t jt-
v (t) ~~ ~ - w/1
&"(t) ~
~~
=- - w(A-
9)
$'/V)
= (0 -os;
Cwt
T
fIV\)
w<;( (; t )
P) "'(-0 2.S t)
G.9s{w6 -1-1)
~C-1-,£5
~)
$,m(,t)
G0s(~t)
c)
'Ii -=
d)
Th-<
o.
S
0
;WI
do,,>,u..O+ d~p'-<Ad
(b->-:DJ
1+
T\......e.z...e-+D~
+It
lS
fL, ,,,,,-;1-,,,,' (D~5L'+,o.-~
o~
~~~.R..
2.LT
5
1x
(0)
) CU ( 0)
=
5
0.0
= -
<0_
-\>
C;
A.0I
f1
60S?
A
S \ {Vt ..p
=-i",>
~
2
IYY1
cP~
+Q.M
/s
::::.0.0
0,
-
<;
=1>
e.s
~
~
r ¢~
J-
U~
<-
'(((0)
= (-1
x
LC ') ~ ((9. 0 S {'(
rV
(t') ~ (- " 2.c;
~ (tJ ~ C -
i,L.s
--
CDS.p
MA)
S
C--9S
f~ Is )
W
fi "'If)
S
CDs
If
SIM
I
A --
(s; t -r- Z\)
I fY\
u> So
Cst
C s; t
f
4>
5i~1 =- CD s,
=-!>
-;}..oO
H
+
z)
if' :)
=
O.DS""
-
0.0
4
o. oS
C-
~
S
-L
{E