hernandez (ejh742) – statics and elasticity problems – Turner – (58120)
1
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001
10.0 points
A cylindrical stone column of diameter 2
R
=
1
.
35 m and height
H
= 2
.
69 m is transported
in standing position by a dolly.
a
side view
When the dolly accelerates or decelerates
slowly enough, the column stands upright,
but when the dolly’s acceleration magnitude
exceed a critical value
a
c
, the column top
ples over.
(For
a >
+
a
c
the column topples
backward; for
a <

a
c
the column toppes
forward.)
Calculate the magnitude of the critical ac
celeration
a
c
of the dolly. The acceleration of
gravity is 9
.
8 m
/
s
2
.
Correct answer: 4
.
91822 m
/
s
2
.
Explanation:
Let :
g
= 9
.
8 m
/
s
2
,
R
= 0
.
675 m
,
and
H
= 2
.
69 m
.
In the noninertial frame of the accelerating
dolly, the column is subject to the horizontal
inertial force
vector
F
in
=

mvectorg .
Together, the gravity and the inertial force
combine into the
apparent weight
force
vector
W
app
=
m
(
vectorg

vectora
)
in the direction
θ
= arctan
parenleftbigg
a
g
parenrightbigg
from the vertical.
From the torque point of view, this appar
ent weight force applies at the center of mass
of the column.
The column is stable in the
vertical position when the line of this force
goes through the column’s base
CM
W
app
but when this line misses the base, the column
topples over
CM
W
app
For the critical acceleration
a
c
, the line goes
through the edge of the base, hence the direc
tion of the apparent weight force must deviate
from the vertical by the angle
θ
c
where
tan
θ
c
=
R
h
cm
=
2
R
H
.
Consequently, the critical acceleration of the
dolly is
a
c
=
g
tan
θ
c
=
g
2
R
H
= (9
.
8 m
/
s
2
)
2 (0
.
675 m)
2
.
69 m
=
4
.
91822 m
/
s
2
.
002
10.0 points
The system shown in the figure is in equilib
rium. A 13 kg mass is on the table. A string
attached to the knot and the ceiling makes an
angle of 60
◦
with the horizontal. The coeffi
cient of the static friction between the 13 kg
mass and the surface on which it rests is 0
.
31.
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hernandez (ejh742) – statics and elasticity problems – Turner – (58120)
2
13 kg
m
60
◦
What is the largest mass
m
can have and
still preserve the equilibrium? The accelera
tion of gravity is 9
.
8 m
/
s
2
.
Correct answer: 6
.
98017 kg.
Explanation:
Let :
M
= 13 kg
,
m
= 6
.
98017 kg
,
and
θ
= 60
◦
.
For the system to remain in equilibrium,
the net forces on both
M
and
m
should be
zero, so the tension in the rope has an upper
bound value
T
max
,
where
T
max
cos
θ
=
μ M g
(1)
T
max
=
μ M g
cos
θ
=
(0
.
31) (13 kg) (9
.
8 m
/
s
2
)
cos 60
◦
= 78
.
9881 N
.
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 Spring '08
 Turner
 Physics, Force, Correct Answer

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