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5.1.
Model:
We can assume that the ring is a single massless particle in static equilibrium.
Visualize:
Solve:
Written in component form, Newton’s first law is
FF
T
T
T
x
xxxx
net
N
()
==
+
+
=
Σ
123
0
T
T
T
y
yyyy
net
N
+
+
=
Σ
0
Evaluating the components of the force vectors from the freebody diagram:
TT
x
11
=−
T
2
x
=
0 N
x
33
30
=°
cos
T
1
y
=
0 N
y
22
=
y
30
°
sin
Using Newton’s first law:
−+
°
=
13
30
0
cos
N
23
30
0
−°
=
sin
N
Rearranging:
30
0 8666
=
(
)
=
cos
.
100 N
86.7 N
30
0 5
=
=
sin
.
100 N
50.0 N
Assess:
Since
r
T
3
acts closer to the
x
axis than to the
y
axis, it makes sense that
12
>
.
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Model:
We can assume that the ring is a particle.
Visualize:
This is a static equilibrium problem. We will ignore the weight of the ring, because it is “very light,” so the only
three forces are the tension forces shown in the freebody diagram. Note that the diagram
defines
the angle
θ
.
Solve:
Because the ring is in equilibrium it must obey
r
F
net
N
=
0
. This is a vector equation, so it has both
x
 and
y
components:
FT
T
x
net
N
()
=−
=
32
0
cos
⇒=
TT
cos
T
T
T
y
net
N
= ⇒
=
13
3
1
0
sin
sin
θθ
We have two equations in the two unknowns
T
3
and
. Divide the
y
equation by the
x
equation:
T
T
T
T
3
3
1
2
1
1 60
1 60
58 0
sin
cos
tan
.
tan
.
.
==
=
=
⇒
= ()
=°
−
80 N
50 N
Now we can use the
x
equation to find
T
T
3
2
°
=
cos
50 N
cos58.0
94.3 N
The tension in the third rope is 94.3 N directed 58.0
°
below the horizontal.
5.3.
Model:
We assume the speaker is a particle in static equilibrium under the influence of three forces:
gravity and the tensions in the two cables.
Visualize:
Solve:
From the lengths of the cables and the distance below the ceiling we can calculate
θ
as follows:
sin
.
sin
.
.
θθ
==
⇒
=
=
°
−
2 m
3 m
0 677
0 667
41 8
1
Newton’s first law for this situation is
FF
T
T
TT
x
xxx
net
N
N
()
+
=⇒
−
+
=
Σ
12
1
2
00
cos
cos
T
T
w
TTw
y
yyyy
net
N
+
+
+
−
=
Σ
1
2
sin
sin
The
x
component equation means
=
. From the
y
component equation:
2
1
Tw
sin
=
⇒
===
°
T
wm
g
1
22
2
4
1
8
sin
sin
sin
.
20 kg 9.8 m/s
196 N
1.333
147 N
2
Assess:
It’s to be expected that the two tensions are equal, since the speaker is suspended symmetrically from the
two cables. That the two tensions add to considerably more than the weight of the speaker reflects the relatively
large angle of suspension.
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Model:
We can assume that the coach and his sled are a particle being towed at a constant velocity by the
two ropes, with friction providing the force that resists the pullers.
Visualize:
Solve:
Since the sled is not accelerating, it is in dynamic equilibrium and Newton’s first law applies:
FF
T
T
f
x
xxx
x
net
k
N
()
==
+
+
=
Σ
12
0
T
T
f
y
yyy
y
net
k
N
+
+
=
Σ
0
From the freebody diagram:
TT
f
1
2
1
2
0
cos
cos
θθ
+
−=
k
N
1
2
1
2
00
sin
sin
−
+=
N
N
From the second of these equations
=
. Then from the first:
21
0
1
T
cos
°=
1000 N
⇒=
°
T
1
0
1000 N
1000 N
1.970
508 N
cos
Assess:
The two tensions are equal, as expected, since the two players are pulling at the same angle. The two add
up to only slightly more than 1000 N, which makes sense because the angle at which the two players are pulling is
small.
5.5.
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This note was uploaded on 03/06/2008 for the course PHYS 260 taught by Professor Chen during the Fall '08 term at Maryland.
 Fall '08
 CHEN
 Mass, Static Equilibrium

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