Download Physics 218 Honors Final Exam

Physics 218 Honors Final Exam; Secs. 201,202,203; Fall-07
_.,
Section. No:
NOTE: Points are noted on each problem
1.(10 pts) A 12.0 kg object hangs in equilibrium from a string with a total length of L = 5.00 m and a
linear mass density of .t= 1.00 x 1 0 kg/m. The string is wrapped around two light, frictionless pulleys
that are separated by a distance D 2.00 m.
a) Determine the tension in the string
b) At what frequency must the string between the pulleys vibrate to form the standing wave pattern
shown below?
D
I4-
m
i/f,
‘.
/
z
h)
2.
3
g
9
J
L
7
J.c,)
2.(l0 pts) A 420 kg wooden raft floats on a lake. When a 75 kg man stands on the raft, it sinks 3.5 cm deeper
into the water. When he steps off the raft vibrates for a long while.
Hint: Show that the motion is simple harmonic by showing that the restoring force is proportional to the
distance from equilibrium
a) Find the “effective spring constant k” for this oscillation
b) What is the frequency of vibration?
c) What is the total energy of the vibration?
//
3 .5tM
/
p
wIv-1
IPZcri
1.3
•1
‘Y,
C’
rJf
75
Ai
r A (.ss)
1
-.
9r
I’
1.
Id?
oGLc
1
&
“-i’
•0
-
•‘
)
P
:
fe!L5
>1
D >
1
D
r
2-17
A
=
—-
-—
_L
2-
Al
9
i4XiO
-/m
)JO
4
j
1
Z
A
-
3__
7cj d.
—
1
y b
2i-4fferf
PJe.v’r&n S
=
—
c(o3J
I ‘2j-(
‘
3-
£
12,d/J
3
3.(12 pts) A long uniform rod of length L and mass M is pivoted about a horizontal frictionless pin through one
end (‘pivot = 4L
/3) The rod is released from rest in a vertical position as shown below. At the instant the rod
2
is horizontal, find:
a) its angular speed,
b) the magnitude of its angular acceleration,
c) the x and y components of the acceleration of its center of mass, and
d) the components of the reaction force at the pivot
Caiic.
y
tz To(
oc
zJ-J-’’
j
713/_ 3
x
flu
?cce1
C-)
—co
—
Cbfl
i1p
+J
t’°
‘C,
d)
ce,, t
F?
t
F
F*
‘4,
PH
F
1
dafv/f.
/
1
4)bk’3YL
4.( 12 pts) A solid sphere of mass m and radius r (Icenter ofmass = 2m?15) rolls without slipping along the
track shown in the figure below. It starts from rest with the lowest point of the sphere at height h above
the bottom of the loop of radius R>>r.
a) What is the minimum value of h (in terms of R) such that the sphere completes the loop?
b) What are the horizontal and vertical force components on the sphere at the point P if h 3R?
1
C
’
1
+?1
2
m
to
7s
ri
7-
IZ -.LniL
2.-
a-
lL)
:
1
yL
l0
k .1?
(2,721 r7
,(
2
L
2
-f-.
)7)/3g
—
&tnypi-
j*iVsL)
5.( 12 pts) A thin rod of mass M and length L hangs from a pivot at its upper end (‘rod about end = MI}/3)
A ball of clay of mass m and horizontal velocity v strikes the lower end at right angles and remains stuck
(a totally inelastic collision).
a) Find the distance to the center of mass of the system of clay plus rod from the pivot point
b) How high will the center of mass rise after the collision?
NOTE: Express your answers in temis of m,M,v,g, and L.
b
c)
)L.
“4
i27L
‘1
L
eCg,i
fl7
4
7
1fr7
6)
‘awc.
of
4?143d4&fl
CIy
J_7;o
2L
j’-IVL
4-7J4) 2
IL’
I
w
I—
I—
I.
e
I.
e
I—
7. (12 pts) Two blocks are set in motion on a frictionless surface. A spring of force constant k is
attached to the near side of one of the blocks. The first block of mass m
1 has velocity Vi and the second
block of mass m
2 moves more slowly with velocity V2 as shown below. When m
1 collides with the
spring attached to m
2 and compresses the spring to its maximum value Of Xm, the velocity of the blocks
is v. In terms of m
,v
1
, and k, fmd
2
,m
1
,v
2
a) the velocity v at maximum compression
b) the maximum compression Xm
£)theie1eity ef each bluds. aftci in las 1ot wiilat with tTiespriig.
vi
v2
Itt
A1
,
H
‘
-
v—
V”2i’
*
-
‘‘1-2
,,1,Vf-M
L
1
-
-
v21 7LI27L
2
c—t3J
1
8.(10 pts) A pendulum comprising a light string of length L and a small sphere, swings in a vertical
plane. The string hits a peg located a distance d below the point of suspension. If the pendulum is
released from the horizontal position (9 = 90°) and is to swing in a in a complete circle centered on the
peg, find the minimum distance d (in terms of L) must be for this to occur.
rc..i_
Tc’
3
=Th7Z
e,,er
Cc91s
,-.
—
ft.::
‘I)—
J_)Ø1t-
-it)
1
21
9. (10 pts) A toaster of mass M is not plugged in. The coefficient of static friction between the toaster
and a horizontal countertop is
To make the toaster start moving you carelessly pull on its electric
cord.
a) Draw a free body diagram of the toaster showing all forces acting on it.
b) For the cord tension to be as small as possible , you should pull at what angle above the horizontal
(express your answer in terms of us)?
t Lhi 1
al
gle,15wir m
u4
—
7.
I
w
.c
rc--
‘S
T.i-& -,%-
T
T
I
4-’-i
7E
7
-
&
4
Se A
2I 1
t
7-
d
C---6
6
2 ‘1 2.)(