01quiz_11

01quiz_11 - Physics 9A-C 4/1/2011 Practice Quiz Week #1...

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Unformatted text preview: Physics 9A-C 4/1/2011 Practice Quiz Week #1 Name: ID (last 4): When
two
vectors
a√
and
b√
are
drawn
from
a
common
point
as
shown,
the
angle
between
the
 them
is

ƒ.

Using
vector
techniques: a)
(7
pts)

Show
that
the
magnitude
of
their
vector
sum
is

 a + b = a 2 + b 2 + 2 ab cos φ To
add
vectors
place
them
head-to-tail
as
shown
(
3
pts) Solution
1(law
of
cosines):

The
angle
opposite 
a√
+
b√

is

π
-
ƒ,
so
the
law
of
cosines
applied
to
this
triangle
is: 
|
a√
+
b√
| 

=

a 
+
b 
-

2
a
b
cos(π
-
ƒ).

(3
pts) 2 
2 a√ + b√ 
2 But,

cos(π
-
ƒ)

=

-
cos
ƒ.

õ
(1
pt) π-ƒ ƒ a√ a√ ƒ b√ 
|
a√
+
b√
| 

=

a
 

+

b 

+

2
a
b
cos
ƒ
.

 2 2 
2 Solution
2
(components):


 Choose
the
x
axis
along
vector
b√.

Resolve
into
components
and
add: 2 ˆ a = a cos φ i + a sin φ ˆ j a + b = ( a cos φ + b) + a 2 sin 2 φ ˆ b= bi +0 ˆ j 



(4
pts)
.

 = ( a 2 cos 2 φ + 2 ab cos φ + b 2 ) + a 2 sin 2 φ 




(3
pts). ˆ a + b = ( a cos φ + b)i + a sin φ ˆ j = a 2 + b 2 + 2 ab cos φ b)
(3
pts)

When
the
angle
between
them
is
90°,
show
that

 a + b 
is
given
by
the
Pythagorean
 theorem.

 When
the
angle
between
them
is
90°
use
the
above
result

note
that
cos
90°
=
0. a + b = a 2 + b 2 + 2 ab cos 90° = a 2 + b 2 

which
is
the
Pythagorean
theorem. ...
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