Gozick, Brandon – Homework 3 – Due: Sep 15 2006, 5:00 pm – Inst: D Weathers
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The due time is Central
time.
001
(part 1 of 2) 10 points
Consider four vectors
~
F
1
,
~
F
2
,
~
F
3
, and
~
F
4
,
where
their
magnitudes
are
F
1
=
56
N,
F
2
= 27 N,
F
3
= 12 N, and
F
4
= 60 N.
Let
θ
1
= 140
◦
,
θ
2
=

140
◦
,
θ
3
= 18
◦
, and
θ
4
=

54
◦
, measured from the positive
x
axis
with the counterclockwise angular direction
as positive.
What is the magnitude of the resultant vec
tor
~
F
, where
~
F
=
~
F
1
+
~
F
2
+
~
F
3
+
~
F
4
?
Correct answer: 31
.
1721 N.
Explanation:
Basic Concepts:
Vector components fig
ure1
Solution:
The
x
components of the forces
~
F
1
,
~
F
2
, and
~
F
3
are
F
1
x
=
F
1
cos(140
◦
) =

42
.
8985 N
F
2
x
=
F
2
cos(

140
◦
) =

20
.
6832 N
F
3
x
=
F
3
cos(18
◦
) = 11
.
4127 N
F
4
x
=
F
4
cos(

54
◦
) = 35
.
2671 N
.
and the
y
components are
F
1
y
=
F
1
sin(140
◦
) = 35
.
9961 N
F
2
y
=
F
2
sin(

140
◦
) =

17
.
3552 N
F
3
y
=
F
3
sin(18
◦
) = 3
.
70821 N
F
4
y
=
F
4
sin(

54
◦
) =

48
.
541 N
.
The
x
and
y
components of the resultant vec
tor
~
F
are
F
x
=
F
1
x
+
F
2
x
+
F
3
x
+
F
4
x
= (

42
.
8985 N) + (

20
.
6832 N)
+ (11
.
4127 N) + (35
.
2671 N)
=

16
.
902 N
F
y
=
F
1
y
+
F
2
y
+
F
3
y
+
F
4
y
= (35
.
9961 N) + (

17
.
3552 N)
+ (3
.
70821 N) + (

48
.
541 N)
=

26
.
192 N
Hence the magnitude of the resultant vector
k
~
F
k
is
k
~
F
k
=
q
F
2
x
+
F
2
y
=
q
(

16
.
902 N)
2
+ (

26
.
192 N)
2
= 31
.
1721 N
002
(part 2 of 2) 10 points
What is the direction of this resultant vector
~
F
?
Note:
Give the angle in degrees, use coun
terclockwise as the positive angular direction,
between the limits of

180
◦
and +180
◦
from
the positive
x
axis.
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 Fall '08
 Weathers
 mechanics, Pythagorean Theorem, Work, Correct Answer, resultant vector, Gozick

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